Traffic System Design Handbook

Traffic System Design Handbook

Telecommunications Handbook Series Whitham D. Reeve, Series Editor The Telecommunications Handbook Series is designed to provide the engineer and technical practitioner with working information in the three basic fields of telecommunications: inside plant, outside plant, and administration and regulatory. This integratedseriesof handbooksprovidespracticalinformationon the 1ink between field experience and formal telecommunication industry standards and practices. These books are essential tools for engineers and technical practitioners who require day-to-day engineering and technical information on telecommunication systems. Other proposed and forthcoming books include coverage on: • Introduction to telecommunications • Grounding methods and measurements • Telecommunication protection • Power system design • Fiber-in-the-Ioop design • Telecommunicationconstruction methods • Satellite earth station design • Terrestrial radio system design • Telecommunications regulatory • Personal communications • Switching and networking If you are interestedin becoming an author, contributor, or reviewerof a book in this series, or if you wouldlike additional informationabout forthcomingtitles, please contact: Whitham D. Reeve Series Editor, Telecommunications Handbook Series PO Box 190225 Anchorage, Alaska 99519-0225

Traffic System Design Handbook Timesaving Telecommunication Traffic Tables and Programs

James R. Boucher, P.E.

A volume in the Telecommunications Handbook Series, Whitham D. Reeve, Series Editor

~

T

.IEEE PRESS

~

Reviewed and endorsed by the

~ IEEE Communications Society

The Institute of Electrical and Electronics Engineers, Inc., New York

IEEE PRESS 445 Hoes Lane, PO Box 1331 Piscataway, NJ 08855·1331

1992 Editorial Board William Perkins, Editor in Chie! K. K. Agarwal R. S. Blicq R. C. Dorf D. M. Etter 1. J. Farrell III

K. Hess J. D. Irwin A. Michel E. K. Miller J. M. F. Moura 1. G. Nagle

A. C. Schell L. Shaw M. Simaan Y. Sunahara J. W. Woods

Dudley R. Kay, Executive Editor Carrie Briggs, Administrative Assistant Denise Gannon, Production Supervisor IEEE Communications Society Jack Holtzman, Liaison to the IEEE PRESS

© 1993 by the Institute of Electrical and Electronics Engineers, Inc.

345 East 47th Street, New York, NY 10017-2394 All rights reserved. No part of this book may be reproduced in any form, nor may it be stored in a retrieval system or transmitted in any form, without written permission from the publisher.

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ISBN 0-7803-0428-4 IEEE Order Number: PP0325-1

Library of Congress Cataloging-in-Publication Data Boucher, James R. (date) Traffic system design handbook: timesaving telecommunication traffic tables and programs / James R. Boucher. p. cm.-(Telecommunication handbook) Includes bibliographicalreference and index. ISBN 0-7803-0428-4 I. Telecommunication-Traffic 2. Telecommunication-Switching systems. I. Title. II. Series. TK5102.5.B683 1993 621.382-dc20 92-30770 CIP

To my wife, Barbara

Contents

Preface

XI

List of Tables

1

XIII

Traffic System Design Overview

1.1 Traffic Units 1.2 Traffic Calculations

1 2

1.2.1 Trunk-Group Traffic, 2 1.2.2 Server-Pool Traffic, 8

1.3 Traffic Assumptions 1.3.1 1.3.2 1.3.3 1.3.4

10

General Assumptions, 10 Number of Sources, 11 Disposition of Blocked Calls, 11 Holding-Time Distributions, 11

1.4 Grade of Service 1.5 Traffic Formulas and Tables 1.6 Computer Programs

12 14 16 vii

Contents

viii

2

3

4

5

6

Poisson Distribution

19

2.1 Poisson Formula 2.2 Poisson Computer Program 2.3 Poisson Traffic Capacity Tables

19

20 20

Erlang B Distribution

33

3.1 3.2 3.3 3.4 3.5

33 34 34 48 73

Erlang B Formula Erlang B Computer Program Erlang B Traffic Capacity Tables High-Usage Traffic Capacity Tables Peaked Traffic Capacity Tables

Erlang C Distribution

79

4.1 Erlang C Formula 4.2 Erlang C Computer Program 4.3 Erlang C Delay Loss Probability Tables

79 80 81

Binomial Distribution

131

5.1 Binomial Formula 5.2 Binomial Computer Program 5.3 Binomial Loss Probability Tables

131 132 132

Engset Distribution

147

6.1 Engset Formula 6.2 Engset Computer Program 6.3 Engset Loss Probability Tables

147 148 149

ix

Contents

Defi nitions

165

Glossary

171

Selected Bibliography

173

Index

175

Preface

This handbook is intended for engineers, technicians, and others who specify, design, test, operate, or maintain telephone systems and networks. It is the handbook I would have referred to most often if it had been available earlier in my career of engineering telecommunication switching and control systems for domestic, international, and military applications. Instead, I amassed a considerable traffic-theory library, including many formulas, tables, curves, and charts. From these I developed a family of timesaving computer programs that greatly simplified my traffic system design tasks. The most commonly used of these diverse design resources, with examples of their use in practical applications, are now available in a single volume. Where typical values of parameters are given, they are indicative of industry practice-individual companies may use different values based on engineering requirements or policy. For those new to traffic system design, introductory information, definitions of common terms and abbreviations, and an extensive bibliography are included. I would be remiss if I did not acknowledge and thank the staff who developed and conducted theGTE Traffic EngineeringSchoolheldat Norwalk, CT in 1974. The notes for that course introduced me to traffic system design and became the basis for the course I teach at Northeastern University. I must xi

xii

Preface

also thank my colleagues, Armand L. Damienand John C. Rothwell, for their critical review of the manuscriptfor this handbook. To Louis K. Pollen,mymentor,colleague,andfriend,I extendmy sincere appreciation for his guidance through the years, and my best wishes for his retirement. In addition,I amindebtedto DudleyR. Kay,IEEEPressExecutiveEditor, for encouraging me to take on this project; Whitham D. Reeve, Telecommunications Handbook Series Editor, for his sound advice and comments along theway;andtoDeniseGannon,ProductionSupervisor,andall thestaffatIEEE Press who helped bring this handbook to fruition. J. R. Boucher

List of Tables

1-1 1-2 1-3 1-4 1-5 2-1 2-2 3-1 3-2 3-3 3-4 3-5 3-6 3-7 3-8 3-9 3-10

Traffic - Unit Conversion Chart Typical Signaling Register Holding Times in Seconds Typical Call-Attempt Dispositions Typical Grade of Service Specifications Traffic Formula Selection Guide Poisson Traffic Capacity in Erlangs Poisson Traffic Capacity in CCS Erlang B Traffic Capacity in Erlangs Erlang B Traffic Capacity in CCS High-Usage Traffic Capacity in CCS (Trunks 1 to 4) High-Usage Traffic Capacity in CCS (Trunks 5 to 8) High-Usage Traffic Capacity in CCS (Trunks 9 to 12) High-Usage Traffic Capacity in CCS (Trunks 13 to 16) High-Usage Traffic Capacity in CCS (Trunks 17 to 20) High-Usage Traffic Capacity in CCS (Trunks 21 to 24) B.01 Peaked Traffic Capacity in CCS (Z = 1.0 to 2.4) B.01 Peaked Traffic Capacity in CCS (Z = 2.6 to 4.0)

3 8 10 13 14 22 27 36 42 50 56 61 65 68 71 75 76 xiii

xiv

3-11 3-12 4-1 4-2 4-3 4-4 4-5 4-6 4-7 4-8

4-9 4-10 4-11 4-12

4-13 4-14 4-15

4-16 4-17 4-18

4-19 4-20

4-21 4-22 4-23

4-24 5-1 5-2 5-3 5-4

5-5 5-6 6-1

6-2 6-3

List of Tables

B.02 Peaked Traffic Capacity in CCS (2 = 1.0 to 2.4) B.02 Peaked Traffic Capacity in CCS (2 = 2.6 to 4.0) Erlang C Delay Loss Probability (N = 2 Servers) Erlang C Delay Loss Probability (N = 3 Servers) Erlang C Delay Loss Probability (N =4 Servers) Erlang C Delay Loss Probability (N = 5 Servers) Erlang C Delay Loss Probability (N = 6 Servers) Erlang C Delay Loss Probability (N = 7 Servers) Erlang C Delay Loss Probability (N = 8 Servers) Erlang C Delay Loss Probability (N = 9 Servers) Erlang C Delay Loss Probability (N = 10 Servers) Erlang C Delay Loss Probability (N = 11 Servers) Erlang C Delay Loss Probability (N = 12 Servers) Erlang C Delay Loss Probability (N = 13 Servers) Erlang C Delay Loss Probability (N = 14 Servers) Erlang C Delay Loss Probability (N = 15 Servers) Erlang C Delay Loss Probability (N = 16 Servers) Erlang C Delay Loss Probability (N = 17 Servers) Erlang C Delay Loss Probability (N = 18 Servers) Erlang C Delay Loss Probability (N = 19 Servers) Erlang C Delay Loss Probability (N = 20 Servers) Erlang C Delay Loss Probability (N = 21 Servers) Erlang C Delay Loss Probability (N = 22 Servers) Erlang C Delay Loss Probability (N = 23 Servers) Erlang C Delay Loss Probability (N = 24 Servers) Erlang C Delay Loss Probability (N = 25 Servers) Binomial Loss Probability (S = 10 Sources) Binomial Loss Probability (S = 20 Sources) Binomial Loss Probability (5 = 30 Sources) Binomial Loss Probability (S = 40 Sources) Binomial Loss Probability (S = 50 Sources) Binomial Loss Probability (S = 60 Sources) Engset Loss Probability (5 = 10 Sources) Engset Loss Probability (5 = 20 Sources) Engset Loss Probability (5 = 30 Sources)

77 78 83 85 87 89

91 93 95 97 99

101 103 105

107 109 III

113

115 117

119 121 123 125

127 129 134

135 137 139 141 144 150 151 153

List of Tables

6-4 6-5 6-6

Engset Loss Probability (5 = 40 Sources) Engset Loss Probability (S = 50 Sources) Engset Loss Probability (S = 60 Sources)

xv

155 158

161

1 Traffic System Design Overview

Traffic system design is a process that considers the entire telecommunication system and the interrelationship of its components. Total system and subsystem performance (i.e., service) objectives are specified, and conflicts are resolved to achieve an optimum configuration. Therefore, traffic system design ensures the cost-effective dimensioning of switching and transmission equipment (traffic-handling resources or servers) to provide the required service objectives (grade ofservice) economically. Telephone traffic (teletraffic) theory--drawing on many disciplines including electronics, mathematics, statistics, probability, queuing theory, reliability, and economics-is at the heart of traffic system design. 1.1 TRAFFIC UNITS

Traffic units are a measure of traffic intensity, the average traffic density during a one-hour period. The international unit of traffic intensity is the Erlang, * where one Erlang represents a circuit occupied for one hour.

*

Named for A.K. Erlang, the father of telephone traffic theory [Brockmeyer, 1948].

Chapter 1

2


The Erlang defines the efficiency (percent occupancy) of a traffic resource and represents the total time in hours to carryall calls. It is the traffic unit used exclusively in classic traffic theory. In the North American public switched telephone network (PSTN), the standard traffic unit is the unit call (UC), which is expressed in seconds. The UC is defined in centum-call-seconds (CCS) or more commonly, hundred-callseconds. Equation 1.1 gives the relationship between Erlangs and CCS. Table 1-1 is an Erlang-to-CCS conversion chart for selected traffic levels up to 200 Erlangs (7200 CCS). 1 Erlang

= 1 call-hour = 3600 call-seconds = 36 CCS

(1.1)

1.2 TRAFFIC CALCULAliONS Before common-equipment pools such as trunk groups, signaling registers, and operator positions can be dimensioned, their busy-hour traffic intensities must be determined. Trunks are assigned to serve calls on an immediate basis and are held for the duration of the call. Signaling registers, operator positions, and similar servers normally serve calls on a delayed basis and are held only long enough to serve their specific functions.

1.2.1 Trunk-Group Traffic Routing plans specify a mix of direct-route and alternate-route trunk groups to provide least-cost routing of interswitch traffic through the network. The selected routing technique determines, to some extent, the level of traffic offered to each trunk group. Offered trunk-group traffic is the total of all traffic offered to the group. If the trunk group were large enough, it would carry all offered traffic but such a trunk group probably would not be economical. Instead, trunk groups are engineered to block a fraction of the offered busyhour traffic, typically one to ten percent.

Chapter 1

Traffic System Design Ovcrview

3

Table 1-1. Traffic-Unit Conversion Chart CCS

Erlangs

CCS

Erlangs

CCS

Erlangs

CCS

Erlangs

CCS

0.05 0.10 0.15 0.20 0.25

1.8 3.6 5.4 7.2 9.0

2.05 2.10 2.15 2.20 2.25

73.8 75.6 77.4 79.2 81.0

4.05 4.10 4.15 4.20 4.25

145.8 147.6 149.4 151.2 153.0

6.05 6.10 6.15 6.20 6.25

217.8 219.6 221.4 223.2 225.0

8.05 8.10 8.15 8.20 8.25

289.8 291.6 293.4 295.2 297.0

0.30 0.35 0.40 0.45 0.50

10.8 12.6 14.4 16.2 18.0

2.30 2.35 2.40 2.45 2.50

82.8 84.6 86.4 88.2 90.0

4.30 4.35 4.40 4.45 4.50

154.8 156.6 158.4 160.2 162.0

6.30 6.35 6.40 6.45 6.50

226.8 228.6 230.4 232.2 234.0

8.30 8.35 8.40 8.45 8.50

298.8 300.6 302.4 304.2 306.0

0.55 0.60 0.65 0.70 0.75

19.8 21.6 23.4 25.2 27.0

2.55 2.60 2.65 2.70 2.75

91.8 93.6 95.4 97.2 99.0

4.55 4.60 4.65 4.70 4.75

163.8 165.6 167.4 169.2 171.0

6.55 6.60 6.65 6.70 6.75

235.8 237.6 239.4 241.2 243.0

8.55 8.60 8.65 8.70 8.75

307.8 309.6 311.4 313.2 315.0

0.80 0.85 0.90 0.95 1.00

29.8 30.6 32.4 34.2 36.0

2.80 2.85 2.90 2.95 3.00

100.8 102.6 104.4 106.2 108.0

4.80 4.85 4.90 4.95 5.00

172.8 174.6 176.4 178.2 180.0

6.80 6.85 6.90 6.95 7.00

244.8 246.6 248.4 250.2 252.0

8.80 8.85 8.90 8.95 9.00

316.8 318.6 320.4 322.2 324.0

1.05 1.10 1.15 1.20 1.25

37.8 39.6 41.4 43.2 45.0

3.05 3.10 3.15 3.20 3.25

109.8 111.6 113.4 115.2 117.0

5.05 5.10 5.15 5.20 5.25

181.8 183.6 185.4 187.2 189.0

7.05 7.10 7.15 7.20 7.25

253.9 255.6 257.4 259.2 261.0

9.05 9.10 9.15 9.20 9.25

325.8 327.6 329.4 331.2 333.0

1.30 1.35 1.40 1.45 1.50

46.8 48.6 50.4 52.2 54.0

3.30 3.35 3.40 3.45 3.50

118.8 120.6 122.4 124.2 126.0

5.30 5.35 5.40 5.45 5.50

190.8 192.6 194.4 196.2 198.0

7.30 7.35 7.40 7.45 7.50

262.8 264.6 266.4 268.2 270.0

9.30 9.35 9.40 9.45 9.50

334.8 336.6 338.4 340.2 342.0

1.55 1.60 1.65 1.70 1.75

55.8 57.6 59.4 61.2 63.0

3.55 3.60 3.65 3.70 3.75

127.8 129.6 131.4 133.2 135.0

5.55 5.60 5.65 5.70 5.75

199.8 201.6 203.4 205.2 207.0

7.55 7.60 7.65 7.70 7.75

271.8 273.6 275.4 277.2 279.0

9.55 9.60 9.65 9.70 9.75

343.8 345.6 347.4 349.2 351.0

1.80 1.85 1.90 1.95 2.00

64.8 66.6 68.4 70.2 72.0

3.80 3.85 3.90 3.95 4.00

136.8 138.6 140.4 142.2 144.0

5.80 5.85 5.90 5.95 6.00

208.8 210.6 212.4 214.2 216.0

7.80 7.85 7.90 7.95 8.00

280.8 282.6 284.4 285.2 288.0

9.80 9.85 9.90 9.95 10.00

352.8 354.6 356.4 358.2 360.0

Erlangs

(table continues)

4

Chapter I


Table 1-1. Traffic-Unit Conversion Chart (Continued) CCS

Erlangs

CCS

Erlangs

CCS

Erlangs

CCS

Erlangs

CCS

10.1 10.2 10.3 10.4 10.5

363.6 367.2 370.8 374.4 378.0

14.1 14.2 14.3 14.4 14.5

507.6 511.2 514.8 518.4 522.0

18.1 18.2 18.3 18.4 18.5

651.6 654.2 658.8 662.4 666.0

22.1 22.2 22.3 22.4 22.5

795.6 799.2 802.8 806.4 810.0

26.1 26.2 26.3 26.4 26.5

939.6 943.2 946.8 950.4 954.0

10.6 10.7 10.8 10.9 11.0

381.6 385.2 388.8 392.4 396.0

14.6 14.7 14.8 14.9 15.0

525.6 529.2 532.8 536.4 540.0

18.6 18.7 18.8 18.9 19.0

669.6 673.2 676.8 680.4 684.0

22.6 22.7 22.8 22.9 23.0

813.6 817.2 820.8 824.4 828.0

26.6 26.7 26.8 26.9 27.0

957.6 961.2 964.8 968.4 972.0

11.1 11.2 11.3 11.4 11.5

399.6 403.2 406.8 410.4 414.0

15.1 15.2 15.3 15.4 15.5

543.6 547.2 550.8 554.4 558.0

19.1 19.2 19.3 19.4 19.5

687.6 691.2 694.8 698.4 702.0

23.1 23.2 23.3 23.4 23.5

831.6 835.2 838.8 842.4 846.0

27.1 27.2 27.3 27.4 27.5

975.6 979.2 982.8 986.4 990.0

11.6 11.7 11.8 11.9 12.0

417.6 421.2 424.8 428.4 432.0

15.6 15.7 15.8 15.9 16.0

561.6 565.2 568.8 572.4 576.0

19.6 19.7 19.8 19.9 20.0

705.6 709.2 712.8 716.2 720.0

23.6 23.7 23.8 23.9 24.0

849.6 853.2 856.8 860.2 864.0

27.6 27.7 27.8 27.9 28.0

993.6 997.2 1000.8 1004.2 1008.0

12.1 12.2 12.3 12.4 12.5

431.6 439.2 442.8 446.4 450.0

16.1 16.2 16.3 16.4 16.5

579.6 583.2 586.8 590.4 594.0

20.1 20.2 20.3 20.4 20.5

723.6 727.2 730.8 734.4 738.0

24.1 24.2 24.3 24.4 24.5

867.6 871.2 874.8 878.4 882.0

28.1 28.2 28.3 28.4 28.5

1011.6 1015.2 1018.8 1022.4 1026.0

12.6 12.7 12.8 12.9 13.0

453.6 457.2 460.8 464.4 468.0

16.6 16.7 16.8 16.9 17.0

597.6 601.2 604.8 608.4 612.0

20.6 20.7 20.8 20.9 21.0

741.6 745.2 748.8 752.4 756.0

24.6 24.7 24.8 24.9 25.0

885.6 889.2 892.8 896.4 900.0

28.6 28.7 28.8 28.9 29.0

1029.6 1033.2 1036.8 1040.4 1044.0

13.1 13.2 13.3 13.4 13.5

471.6 475.2 478.8 482.4 486.0

17.1 17.2 17.3 17.4 17.5

615.6 619.2 622.8 626.4 630.0

21.1 21.2 21.3 21.4 21.5

759.6 763.2 766.8 770.4 774.0

25.1 25.2 25.3 25.4 25.5

903.6 907.2 910.8 914.4 918.0

29.1 29.2 29.3 29.4 29.5

1047.6 1051.2 1054.8 1058.4 1062.0

13.6 13.7 13.8 13.9 14.0

489.6 493.2 496.8 500.4 504.0

17.6 17.7 17.8 17.9 18.0

633.6 637.2 640.8 644.4 648.0

21.6 21.7 21.8 21.9 22.0

777.6 781.2 784.8 788.4 792.0

25.6 25.7 25.8 25.9 26.0

921.6 925.2 928.8 932.4 936.0

29.6 29.7 29.8 29.9 30.0

1065.6 1069.2 1072.8 1076.2 1080.0

Erlangs

(table continues)

Chapter 1

5


Table 1-1. Traffic-Unit Conversion Chart (Continued) Erlangs

CCS

Erlangs

CCS

Erlangs

CCS

Erlangs

CCS

Erlangs

CCS

30.1 30.2 30.3 30.4 30.5

1083.6 1087.2 1090.8 1094.2 1098.0

34.1 34.2 34.3 34.4 34.5

1227.6 1231.2 1234.8 1238.4 1242.0

38.1 38.2 38.3 38.4 38.5

1371.6 1375.2 1378.8 1382.4 1386.0

42.1 42.2 42.3 42.4 42.5

1515.6 1519.2 1522.8 1526.4 1530.0

46.1 46.2 46.3 46.7 46.5

1659.6 1663.2 1666.8 1670.4 1674.0

30.6 30.7 30.8 30.9 31.0

1101.6 1105.2 1108.8 1112.4 1116.0

34.6 34.7 34.8 34.9 35.0

1245.6 1249.2 1252.8 1256.4 1260.0

38.6 38.7 38.8 38.9 39.0

1389.6 1393.2 1396.8 1400.4 1404.0

42.6 42.7 42.8 42.9 43.0

1533.6 1537.2 1540.8 1544.4 1548.0

46.6 46.7 46.8 46.9 47.0

1677.6 1681.2 1684.8 1688.4 1692.0

31.1 31.2 31.3 31.4 31.5

1119.6 1123.2 1126.8 1130.4 1134.0

35.1 35.2 35.3 35.4 35.5

1263.6 1267.2 1270.8 1274.4 1278.0

39.1 39.2 39.3 39.4 39.5

1407.6 1411.2 1414.8 1418.4 1422.0

43.1 43.2 43.3 43.4 43.5

1551.6 1555.2 1558.8 1562.4 1566.0

47.1 47.2 47.3 47.4 47.5

1695.6 1699.2 1702.8 1706.4 1710.0

31.6 31.7 31.8 31.9 32.0

1137.6 1141.2 1144.8 1148.4 1152.0

35.6 35.7 35.8 35.9 36.0

1281.6 1285.2 1288.8 1292.4 1296.0

39.6 39.7 39.8 39.9 40.0

1425.6 1429.2 1432.8 1436.4 1440.0

43.6 43.7 43.8 43.9 44.0

1569.6 1573.2 1576.8 1580.4 1584.0

47.6 47.7 47.8 47.9 48.0

1713.6 1717.4 1720.8 1724.4 1728.0

32.1 32.2 32.3 32.4 32.5

1155.6 1159.2 1162.8 1166.4 1170.0

36.1 36.2 36.3 36.4 36.5

1299.6 1303.2 1306.8 1310.4 1314.0

40.1 40.2 40.3 40.4 40.5

1443.6 1447.2 1450.8 1454.4 1458.0

44.1 44.2 44.3 44.4 44.5

1587.6 1591.2 1594.8 1598.4 1602.0

48.1 48.2 48.3 48.4 48.5

1731.6 1735.2 1738.8 1742.4 1746.0

32.6 32.7 32.8 32.9 33.0

1173.6 1177.2 1180.8 1184.4 1188.0

36.6 36.7 36.8 36.9 37.0

1317.6 1321.2 1324.8 1328.4 1332.0

40.6 40.7 40.8 40.9 41.0

1461.6 1465.2 1468.8 1472.4 1476.0

44.6 44.7 44.8 44.9 45.0

1605.6 1609.2 1612.8 1616.4 1620.0

48.6 48.7 48.8 48.9 49.0

1749.6 1753.2 1756.8 1760.4 1764.0

33.1 33.2 33.3 33.4 33.5

1191.6 1195.2 1198.8 1202.4 1206.0

37.1 37.2 37.3 37.4 37.5

1335.6 1339.2 1342.8 1346.4 1350.0

41.1 41.2 41.3 41.4 41.5

1479.6 1483.2 1486.8 1490.4 1494.0

45.1 45.2 45.3 45.4 45.5

1623.6 1627.2 1630.8 1634.4 1638.0

49.1 49.2 49.3 49.4 49.5

1767.6 1771.2 1774.8 1778.4 1782.0

33.6 33.7 33.8 33.9 34.0

1209.6 1213.2 1216.8 1220.4 1224.0

37.6 37.7 37.8 37.9 38.0

1353.6 1357.2 1360.8 1364.4 1368.0

41.6 41.7 41.8 41.9 42.0

1497.6 1501.2 1504.8 1508.4 1512.0

45.6 45.7 45.8 45.9 46.0

1641.6 1645.2 1648.8 1652.4 1656.0

49.6 49.7 49.8 49.9 50.0

1785.6 1789.2 1792.8 1796.4 1800.0

(table continues)

Chapter 1

6


Table 1-1. Traffic-Unit Conversion Chart (Continued) CCS

Erlangs

CCS

Erlangs

CCS

Erlangs

CCS

Erlangs

CCS

50.5 51.0 51.5 52.0 52.5

1818 1836 1854 1872 1890

70.5 71.0 71.5 72.0 72.5

2538 2556 2574 2592 2610

90.5 91.0 91.5 92.0 92.5

3258 3276 3294 3312 3330

121.0 122.0 123.0 124.0 125.0

4356 4392 4428 4464 4500

161.0 162.0 163.0 164.0 165.0

5796 5832 5868 5904 5940

53.0 53.5 54.0 54.5 55.0

1908 1926 1944 1962 1980

73.0 73.5 74.0 74.5 75.0

2628 2646 2664 2682 2700

93.0 93.5 94.0 94.5 95.0

3348 3366 3384 3402 3420

126.0 127.0 128.0 129.0 130.0

4536 4572 4608 4644 4680

166.0 167.0 168.0 169.0 170.0

5976 6012 6048 6084 6120

55.5 56.0 56.5 57.0 57.5

1998 2016 2034 2052 2070

75.5 76.0 76.5 77.0 77.5

2718 2736 2754 2772 2790

95.5 96.0 96.5 97.0 97.5

3438 3456 3474 3492 3510

131.0 132.0 133.0 134.0 135.0

4716 4752 4788 4824 4860

171.0 172.0 173.0 174.0 175.0

6156 6192 6228 6264 6300

58.0 58.5 59.0 59.5 60.0

2088 2106 2124 2142 2160

78.0 78.5 79.0 79.5 80.0

2808 2826 2844 2862 2880

98.0 98.5 99.0 99.5 100.0

3528 3546 3564 3582 3600

136.0 137.0 138.0 139.0 140.0

4896 4932 4968 5004 5040

176.0 177.0 178.0 179.0 180.0

6336 6372 6408 6444 6480

60.5 61.0 61.5 62.0 62.5

2178 2196 2214 2232 2250

80.5 81.0 81.5 82.0 82.5

2898 2916 2934 2952 2970

101.0 102.0 103.0 104.0 105.0

3636 3672 3708 3744 3780

141.0 142.0 143.0 144.0 145.0

5076 5112 5148 5184 5220

181.0 182.0 183.0 184.0 185.0

6516 6552 6588 6624 6660

63.0 63.5 64.0 64.5 65.0

2268 2286 2304 2322 2340

83.0 83.5 84.0 84.5 85.0

2988 3006 3024 3042 3060

106.0 107.0 108.0 109.0 110.0

3816 3852 3888 3924 3960

146.0 147.0 148.0 149.0 150.0

5256 5292 5328 5364 5400

186.0 187.0 188.0 189.0 190.0

6696 6732 6768 6804 6840

65.5 66.0 66.5 67.0 67.5

2358 2376 2394 2412 2430

85.5 86.0 86.5 87.0 87.5

3078 3096 3114 3132 3150

111.0 112.0 113.0 114.0 115.0

3996 4032 4068 4104 4140

151.0 152.0 153.0 154.0 155.0

5436 5472 5508 5544 5580

191.0 192.0 193.0 194.0 195.0

6876 6912 6948 6984 7020

68.0 68.5 69.0 69.5 70.0

2448 2466 2484 2502 2520

88.0 88.5 89.0 89.5 90.0

3168 3186 3204 3222 3240

116.0 117.0 118.0 119.0 120.0

4176 4212 4248 4284 4320

156.0 157.0 158.0 159.0 160.0

5616 5652 5688 5724 5760

196.0 197.0 198.0 199.0 200.0

7056 7092 7128 7164 7200

Erlangs

Chapter 1

7


Figure 1-1 can be used to facilitate an understanding of traffic routing terms. Interswitch traffic is routed over the primary route trunk group provided there are idle trunks available in the group. In an alternate-routing system, blocked trunk-group traffic overflows to other alternate-route trunk groups or to final-route trunk groups as indicated by the curved arrows. Trunk groups provided with alternate routes are often referred to as high-usage trunk groups. Final-route trunk groups do not have alternate routes; therefore, blocked traffic in a final-route trunk group is lost. THIRD & FINAL

ROUTE

THIRD & FINAL

ROUTE

FINAL ROUTE

Figure 1-1. Interswitch Trunk Traffic Routing Diagram

Trunk-group traffic is the product of the number and duration of calls handled by the group. Equation 1.2can be used to calculate trunk-group traffic, expressed in Erlangs. (1.2) A =N· Tc where A = Offered traffic in Erlangs N = Number of calls during the busy hour Tc = Mean call-holding time in hours Number of calls refers to the total number of calls offered to the trunk group. Call-holding time is the total elapsed time between seizure of a trunk to serve the call and its subsequent release. The mean call-holding time is the arithmetic average of all call-holding times, expressed in hours.

Chapter 1

8


Example 1-1 Deterrnine the traffic in Erlangs and CCS for a trunk group carrying 1000 calls during the busy hour with an average call-holding time of 3 minutes. A = (1000 calls/hour)(3 min/call)( 1 hour/60 min) = 50 Erlangs (50 Erl)(36 CCS/Erl) = 1800 CCS

1.2.2 Server-Pool Traffic Server pools are groups of traffic resources, such as signaling registers and operator positions, that are used on a shared basis. Service requests that cannot be met immediately are placed in a queue and served on a first-in, firstout (FIFO) basis. Server-pool traffic is directly related to offered traffic, server-holding time, and call-attempt factor, and inversely related to callholding time as expressed in Equation 1.3.

A = S

where

AT· Ts • C

(1.3)

Tc

As = Server-pool traffic in Erlangs AT =Total traffic served in Erlangs T, = Mean server-holding time in hours T; = Mean call-holding time in hours C = Call-attempt factor (dimensionless)

Total traffic served refers to the total offered traffic that requires the services of the specific server pool for some portion of the call. For example, a dual-tone multifrequency (DTMF) receiver pool is dimensioned to serve only the DTMF tone-dialing portion of total switch traffic generated by DTMF signaling sources. Table 1-2 presents representative server-holding times for typical signaling registers as a function of the number ofdigits received or sent. Table 1-2. Typical Signaling Register Holding Times in Seconds Number of Digits Received or Sent Signaling Register

Local Dial-Pulse (DP) Receiver Local DTMF Receiver Incoming MF Receiver Outgoing MF Sender

3.7 2.3 1.0 1.5

4

7

10

11

8.3 5.2 1.4

12.8 8.1 1.8 2.3

17.6 11.0 2.2 2.8

19.1 12.0 2.3 3.0

1.9

Chapter 1


9

The mean server-holding time is the arithmetic average of all serverholding times for the specific server pool. Equation 1.4 is a general equation to calculate mean server-holding time for calls with different holding-time characteristics.

TS = a-T + b-T 2 + ••• + k-T n

( 1.4)

J

where

Ts = Mean server-holding time in hours T), T2, ••• , Tll = Individual server-holding times in hours a, b, ••• ,k = Fractions of total traffic served (a + b + ••• + k = 1)

Example 1-2 Determine the mean DTMF receiver-holding time for a central office (CO) where subscribers dial local calls using a 7-digit number and toll calls using an II-digit number. Assume that 70 percent of the calls are local calls, the remainder are toll calls, and that the typical signal ing register holding times of Table 1-2 are applicable.

Ts = (0.7)(8.1 sec) + (0.3)(12.0 sec)

= 9.27 sec

Call-attempt factors are dimensionless numbers that adjust offered traffic intensity to compensate for call attempts that do not result in completed calls. Therefore, call-attempt factors are inversely proportional to the fraction of completed calls as defined in Equation 1.5. I

C=T where C k

( 1.5)

= Call-attempt factor (dimensionless) = Fraction of calls completed (decimal fraction)

Example 1-3 Table 1-3 presents representative subscriber call-attempt dispositions based on empirical data amassed in the North American PSTN. Determinethe call-attempt factor for these data, where 70.7 percent of the calls were completed (k =0.707).

1

C =k-

0.707

= 1.414

Chapter I

10


Table 1-3. Typical Call-Attempt Dispositions Call-Attempt Disposition

Percentage

70.7 12.7 10.1 2.6 1.9

Call was completed Called subscriber did not answer Called subscriber line was busy Call abandoned without system response Equipment blockage or failure

1.6 0.4

Customer dialing error Called directory number changed or disconnected

Example 1-4 Using Equation 1.3, determine the server-pool traffic in CCS and Erlangs for the DTMF receivers of Example 1-2, assuming total offered busy-hour subscriber traffic of 2000 CCS, a call-attempt factor of 1.5, and a mean call-holding time of 3 minutes (180 seconds). (9.27 sec) As = (2000 CCS) (1.5) (180 sec) (154.5 CCS)

( 1 Erl) (36 CCS)

= 154.5 CCS

= 4.29 Erlangs

1.3 TRAFFIC ASSUMPTIONS Traffic formulas are based on a set of assumptions regarding the behavior of traffic and its sources. These assumptions are not always precisely true. If variations from these assumptions are small or known to have little effect, however, they can be used with confidence. 1.3.1 General Assumptions The following assumptions are applicable to traffic formulas in general: • The system is in statistical equilibrium. • Connection and disconnection of sources to servers occur instantaneous}y. • The anticipated traffic density is the same for all sources. • Busy sources initiate'no calls. • Every source has equal access to every server (full availability).

Chapter 1


II

• The number of busy servers in a group is equal to the number of busy sources in its group of sources. 1.3.2 Number of Sources

The number of sources that can originate calls affects the service these sources can expect to obtain. As the number of sources increases, the effect of adding more sources diminishes. Eventually, a point is reached where there is negligible difference in the probability of congestion regardless of how many new sources are added. It is this point that distinguishes between finite and infinite sources. Traffic formulas for applications where the number of sources in relation to the number of servers is very large assume infinite sources (worst case for blocking). This simplifies the mathematics and minimizes the number of required tables. 1.3.3 Disposition of Blocked Calls

Many assumptions for the disposition of blocked calls (which are also referred to as lost calls) have been proposed, of which the three common cases are: • If an idle server is not immediately available, the call is cleared from the system and the source becomes idle. This is commonly called the blocked calls cleared assumption.

• If an idle server is not immediately available, the call is held for an interval equal to its holding time, and then the source becomes idle. If an idle server becomes available during the waiting period, it will be seized and held for an interval equal to the remaining portion of its mean holding time. This is commonly called the blocked calls held assumption.

• If an idle server is not immediately available, the call is queued until an idle server is available. When an idle server becomes available, it will be seized to serve the next call in queue and held for the full callholding time. This is commonly called the blocked calls delayed assumption.

1.3.4 Holding-Time Distributions

A negative-exponential curve usually provides a reasonable fit for the variation in holding times encountered with nondelayed traffic-handling resources. Substituting a constant holding time equal to the average of varying holding times has a negligible effect for these applications. The effects of

12

Chapter 1


holding-time variations may be significant, however, when predicting the duration of delays. For example, the Crommelin-Pollaczek formulas are often used to determine service delays for resources with essentially constant holding times, such as dial-tone markers and intertoll trunks. Molnar's Delay Probability Charts for Telephone Traffic Where the Holding Times Are Constant graphically present data for these and similar applications.

1.4 GRADE OF SERVICE

Grade of service (GaS) is defined as the probability that offered traffic will be blocked or delayed. An absolutely nonblocking system has a GaS of zero, whereas a GaS of one indicates an absolutely blocking system. That is, the closer the grade of service is to zero, the better the system. Every traffic problem involves three interrelated parameters: offered traffic, traffic-handling resources (servers), and service objective (grade of service). This interrelationship can be pictured as a triangle, as shown in Fig. 1-2. For a given service objective (base of triangle held constant), increasing offered traffic requires a commensurate increase in the number of servers. Similarly, decreasing the number of servers requires a corresponding decrease in the level of offered traffic. It is important to understand that a server's GOS is a prediction of the probability of congestion (i.e., a call is blocked or delayed) at a given level of

SERVICE OBJECTIVE

Figure 1-2.

Grade of Service Concept Diagram

Chapter 1

13


offered traffic, not an absolute value. That is, a trunk-group grade of service of 0.0 1 does not mean that exactly one call in a hundred will be blocked during the busy hour. Rather, it means that, given a large volume of traffic, the probability of congestion will tend toward one in a hundred. Table 1-4 lists typical grade of service specifications for traffic system design. Matching loss as used in this table refers to congestion (blocking) in a switching matrix such that input and output terminations cannot be interconnected via the interstage links. Switching matrix matching loss is not covered in this handbook but the author's Voice Teletraffic Systems Engineering contains an entire chapter on the subject. Table 1-4. Typical Grade of Service Specifications Parameter

Specification

Trunk group loss probability

0.010

Intraoffice line-to-line loss probability

0.020

Line-to-trunk outgoing matching loss probability

0.010

Trunk-to-line incoming matching loss probability

0.020

Trunk-to-trunk tandem matching loss probability

0.005

Probability dial tone delay exceeds 3 seconds

0.015

Probability operator answer delay exceeds 10 seconds

0.050

The traffic formulas found in this handbook, used to predict grades of service, are all based on probability distributions. Probability distributions are bounded by the values zero and one; therefore, a grade of service (probability of congestion) cannot be negative nor can it exceed unity. Because of this property, the probability of a call not ex periencing congestion is one minus the probability of congestion, and vice versa. These relationships are expressed in Equations 1.6 and 1.7. p= l-Q

( 1.6)

Q= I-P

( 1.7)

where P = Probability of congestion Q = Probability of no congestion

Chapter 1

14


1.5 TRAFFIC FORMULAS AND TABLES

Table 1-5 is a selection guide for the traffic formulas contained in this handbook as a function of their typical applications. The Poisson, Erlang B, and Erlang C formulas, basedon the assumption of infinitesources, are referred to as the major traffic formulas. The Binomial and Engset formulas, based on the assumption of finite sources, are used in lieu of the major traffic formulas when the number of sources is small. Figure 1-3 is a decision tree to facilitate traffic formula selection on the basis of the standard traffic assumptions. Table 1-5. Traffic Formula Selection Guide Number of Sources

Blocked-Call Disposition

Holding-Time Distribution

Traffic Formula

Final trunk groups in North American PSTN

Infinite

Held

Constant or exponential

Poisson

Trunk groups and other nondelayed server pools

Infinite

Cleared


Erlang B

Delayed server pools

Infinite

Delayed

Exponential

Erlang C

Small PBX or remote switch trunk groups

Finite

Held


Binomial

Small line concentrators

Finite

Cleared


Engset

Typical Applications

Representative full-availability traffic tables, selected on the basis of common telephone industry practice, are provided for the Poisson, Erlang B, Erlang C, Binomial, and Engset distributions. Full availability refers to the assumption that every source hasequal access to every server. This assumption is normally true for modem traffic systems. Some older systems, however, many of which are still in use, may be limited-availability systems. Limitedavailability tables, such as those found in Siemens' Telephone Traffic Theory Tables and Charts and ITTStandard Electrik's Teletraffic Engineering Manual, can be used for those systems.

til

~

ERLANG C FORMULA (CHAPTER 4)

EXPONENTIAL

POISSON FORMULA (CHAPTER 2)

HELD

DELAY FORMULA

*

DELAYED

* NOT INCLUDED IN THIS HANDBOOK

BINOMIAL FORMULA (CHAPTER 5)

HELD

FINITE

Traffic Formula Selection Decision Tree

CROMMELINPOLLACZEK FORMULA*

CONSTANT

Figure 1-3

DELAYED

ERLANG B FORMULA (CHAPTER 3)

CLEARED

INFINITE

ENGSET FORMULA (CHAPTER 6)

CLEARED

16

Chapter 1


Tabulated traffic data values in this handbook are rounded off to the leastsignificant digit as applicable to the specific table. For example, loss probability values have been rounded off to five decimal places. This level of accuracy should be more than adequate for practical applications-very low loss probabilities may indicate overdesign, which is not economically sound. Where the parameters of a specific application do not coincide with table values, interpolation can be used. However, linear interpolation techniques are not generally satisfactory for these highly nonlinear formulas. Adequate results may be obtained with a graphic technique using semilogarithmic (semilog) graph paper, where loss probability is plotted logarithmically along the ordinate (vertical axis), and offered traffic is plotted linearly along the abscissa (horizontal axis). Figure 1-4 (page 17), a comparison of typical loss probabilities for the Poisson and Erlang B distributions, is an example of the graphic technique. Among other things it shows that, for a given loss probability, less traffic can be offered to a trunk group dimensioned using the Poisson distribution than to one containing the same number of trunks but dimensioned using the Erlang B distribution. That is, the Poisson distribution results in a more conservative design.

1.6 COMPUTER PROGRAMS Computer programs, useful for interpolating between table values or to determine more precise values for specific applications, are provided in subsequent chapters for the Poisson, Erlang B, Erlang C, Binomial, and Engset formulas. These programs are written in BASIC because it is an easy-to-Iearn language and is highly standardized. It is the universal programming language for the personal computers found in homes as well as engineering offices. The programs are formatted in an interactive (i.e., dialogue) style to facilitate the user's entry of traffic parameters and include separate lines ofcode for each step in an attempt to make them more easily understood by those with little or no programming experience. Readers adept at computer programming may prefer to rewrite these traffic programs, combining a number of steps into a single line of code. Alternatively, the programs can be converted to a language such as FORTRAN, which was specifically designed for computational problem solving. In any case, newly entered programs should be validated by running them against benchmarks, such as the examples in this book, before relying on their output data.

Chapter 1


17

0.1000

0.0500 POISSON

ERLANG B

0.0200

0.0100

>-

I-

:J

co

0.0050

CD

a:

Q.. (j) (j)

0

0.0020

...J

0.0010

0.0005

0.0002

0.0001 OFFERED LOAD - ERLANGS

Figure I-4.

Graphic Comparison of Poisson and Erlang B Distributions

A word of caution-s-computers are subject to overflow when dealing with very large numbers. This limitation is a function of the computer hardware and software, which can only process numbers within a finite range. Overflow often occurs when calculating traffic formulas, which typically involve calculation of factorials, numbers raised to the nth power, or infinite sums. The traffic programs provided herein have been written to avoid overflow conditions where possible. Overflow may still occur, however, when calculating the loss probability for a high traffic volume offered to a large number of servers, or some combination of these or other traffic parameters is encountered.

2 Poisson Distribution

The Poisson distribution is used in the North American PSTN for dimensioning final trunk groups, even though blocked calls are cleared from the system and not held. This practice is justified by extensive studies that indicate that the Poisson distribution more nearly matches actual PSTN experience than the Erlang B distribution used for foreign and military networks. It is also a more conservative distribution; that is, traffic capacity is less for a given number of

trunks and loss probability (see Fig. 1-4). The Poisson distribution is based on the following assumptions: • • • •

Calls are served in random order. There are an infinite number of sources. Blocked calls are held. Holding time is exponential or constant.

2.1 POISSON FORMULA The Poisson Formula, also known as Poisson's Exponential Binomial Limit, is given in Equation 2.1. Equation 2.2, based on the property of probability 19

Chapter 2

20


distributions that P = 1 - Q, is an equivalent form of Equation 2.1, which obviates the need to sum terms to infinity. 00

;

P= £-A ~ ~ £.J

i =N

(2.1 )

0'

l.

N-I

;

P = l_£-A ~ ~ £.J

i =0

0'

(2.2)

l.

where P = Poisson loss probability N = Number of trunks in full-availability group A = Traffic offered to group in Erlangs e = Natural logarithm base (2.71828 ...)

2.2 POISSON COMPUTER PROGRAM The following computer program can be used to calculate Equation 2.2 to determine Poisson loss probabilities. Required inputs are the number of trunks in the trunk group and the traffic offered to the group expressed in Erlangs. 100 110 120 130 140 150 160 170 180 190 200 210

REM POISSON LOSS PROBABILITY CALCULATION INPUT "ENTER NUMBER OF SERVERS {N)";N INPUT "ENTER OFFERED TRAFFIC IN ERLANGS (A)"iA LET X=l LET Y=l FOR 1=1 TO (N-1) LET X=X*A/I LET Y=X+Y NEXT I LET Q=Y*EXP(-A) PRINT USING "P = #.#####";l-Q END

2.3 POISSON TRAFFIC CAPACITY TABLES Poisson traffic capacity tables are used to determine the maximum amount of traffic that can be offered to a group of N trunks such that the specified loss probability (grade of service) will not be exceeded. Tables 2.1 (pages 22-26)

Chapter 2


21

and 2.2 (pages 27-31) present Poisson traffic capacities for 1 to 200 trunks with typical loss probabilities ranging from 0.001 to 0.1. These Poisson loss probabilities are commonly written as P.OO 1 to P.I.

Example 2-1 Determine the traffic capacity in Erlangs and CCS for a 24-channel final trunk group such that the loss probability will not exceed 2 percent (P.02). InTable 2-1, select the N row for 24 trunks and the P column for .02 and read 15.1 Erlangs at the intersection. In Table 2-2, select the N row for 24 trunks and the P column for .02 and read 543.6 CCS at the intersection.

Example 2-2 Determine the number of trunks required in a final trunk group to handle 720 CCS of offered traffic at a grade of service of P.Ol. In Table 2-2, select the P.OI column and read down until 730.8 CCS is found. Read across that N row to determine that 32 trunks are required.

Example 2-3 Determine the grade of service for a 48-channel final trunk group with offered traffic of 0.75 Erlangs per channel. Total offered traffic =(48 chan)(O.75 Erl/chan) =36 Erlangs In Table 2-1, select the N row for 48 trunks and read across until 37.2 Erlangs is found. Read up that P column to determine that the grade of service is P.05.

22

Chapter 2


Table 2-1. Poisson Traffic Capacity in Erlangs Traffic (A) in Erlangs for P =

No. of Trunks (N)

.001

.002

.005

.010

.020

1 2 3 4 5

.001 .044 .192 .428 .739

.002 .065 .244 .519 .868

.005 .104 .338 .673 1.08

.011 .150 .436 .822 1.28

6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

1.11 1.52 1.97 2.45 2.97 3.50 4.03 4.61 5.19 5.78 6.42 7.03 7.67 8.31 8.97 9.61 10.3 11.0 11.6 12.3 13.0 13.8 14.4 15.1 15.9

1.27 1.72 2.21 2.72 3.26 3.82 4.40 5.00 5.61 6.23 6.87 7.52 8.17 8.84 9.52 10.2 10.9 11.6 12.3 13.0 13.7 14.4 15.2 15.9 16.6

1.54 2.04 2.57 3.13 3.72 4.32 4.94 5.58 6.23 6.89 7.57 8.25 8.94 9.65 10.4 11.1 11.8 12.5 13.3 14.0 14.7 15.5 16.3 17.0 17.8

1.79 2.33 2.91 3.50 4.14 4.78 5.43 6.11 6.78 7.47 8.18 8.89 9.61 10.4 11.1

.021 .214 .567 1.02 1.55 2.11 2.69 3.31 3.94 4.61

31 32 33 34 35 36 37 38 39 40

16.6 17.3 18.1 18.8 19.5 20.3 21.0 21.8 22.5 23.3

17.4 18.1 18.9 19.6 20.4 21.1 21.9 22.7 23.4 24.2

18.5 19.3 20.1 20.9 21.6 22.4 23.2 24.0 24.8 25.6

.050

.100

.053 .358 .817 1.37 1.97

.106 .531 1.10 1.75 2.44

11.8 12.6 13.3 14.1 14.9 15.6 16.4 17.2 18.0 18.8

5.31 6.00 6.69 7.42 8.14 8.89 9.64 10.4 11.1 11.9 12.7 13.5 14.3 15.1 15.9 16.6 17.4 18.2 19.0 19.9

2.61 3.28 3.97 4.69 5.42 6.17 6.92 7.69 8.47 9.25 10.1 10.8 11.6 12.4 13.3 14.1 14.9 15.7 16.6 17.4 18.2 19.1 19.9 20.8 21.6

3.14 3.89 4.67 5.42 6.22 7.03 7.83 8.64 9.47 10.3 11.1 12.0 12.8 13.7 14.5 15.4 16.3 17.1 18.0 18.8 19.7 20.6 21.5 22.4 23.2

19.5 20.3 21.1 21.9 22.7 23.5 24.3 25.1 26.0 26.8

20.7 21.5 22.3 23.1 23.9 24.8 25.6 26.4 27.3 28.1

22.5 23.3 24.2 25.1 25.9 26.8 27.6 28.5 29.4 30.2

24.1 25.0 25.9 26.8 27.7 28.6 29.4 30.3 31.3 32.1

(table continues)

Chapter 2

23


Table 2-1. Poisson Traffic Capacity in Erlangs (Continued) No. of Trunks (N)

Traffic (A) in Erlangs for P = .001

.002

.005

.010

.020

.050

.100

41 42 43 44 45

24.0 24.8 25.5 26.3 27.1

25.0 25.8 26.5 27.3 28.1

26.4 27.2 28.0 28.8 29.6

27.6 28.4 29.2 30.1 30.9

28.9 29.8 30.6 31.5 32.3

31.1 32.0 32.9 33.7 34.6

33.1 33.9 34.9 35.8 36.7

46 47 48 49 50

27.9 28.6 29.4 30.2 31.0

28.9 29.7 30.5 31.3 32.1

30.4 31.2 32.0 32.9 33.7

31.7 32.5 33.4 34.2 35.0

33.2 34.0 34.9 35.7 36.6

35.5 36.4 37.2 38.1 39.0

37.6 38.5 39.4 40.3 41.2

51 52 53 54 55

31.8 32.5 33.3 34.1 34.9

32.9 33.7 34.5 35.3 36.1

34.5 35.3 36.1 37.0 37.8

35.9 36.7 37.6 38.4 39.2

37.4 38.3 39.2 40.0 40.9

39.9 40.8 41.6 42.5 43.4

42.1 43.0 43.9 44.8 45.7

56 57 58 59 60

35.7 36.5 37.3 38.1 38.9

36.9 37.7 38.5 39.3 40.1

38.6 39.3 40.3 41.1 41.9

40.1 40.9 41.8 42.6 43.5

41.8 42.6 43.5 44.3 45.2

44.3 45.2 46.1 47.0 47.9

46.6 47.6 48.5 49.4 50.3

61 62 63 64 65

39.7 40.5 41.3 42.1 42.9

40.9 41.8 42.6 43.4 44.2

42.8 43.6 44.4 45.3 46.1

44.3 45.2 46.0 46.9 47.7

46.1 46.9 47.8 48.7 49.6

48.8 49.6 50.5 51.4 52.3

51.2 52.1 53.1 54.0 54.9

66 67 68 69 70

43.7 44.5 45.3 46.1 47.0

45.0 45.9 46.7 47.5 48.4

46.9 47.8 48.7 49.5 50.3

48.6 49.4 50.3 51.2 52.0

50.4 51.3 52.2 53.1 53.9

53.2 54.1 55.0 55.9 56.8

55.8 56.7 57.7 58.6 59.5

71 72 73 74 75

47.8 48.6 49.4 50.3 51.1

49.2 50.0 50.8 51.7 52.5

51.2 52.0 52.9 53.7 54.6

52.9 53.8 54.6 55.5 56.3

54.8 55.7 56.6 57.4 58.3

57.7 58.6 59.5 60.4 61.3

60.4 61.4 62.3 63.2 64.1

76 77 78 79 !80

51.9 52.7 53.5 54.4 55.2

53.4 54.2 55.0 55.9 56.7

55.4 56.3 57.1 58.0 58.9

57.2 58.1 58.9 59.8 60.7

59.2 60.1 60.9 61.8 62.7

62.3 63.2 64.1 65.0 65.9

65.1 66.0 66.9 67.9 68.9

(table continues)

24

Chapter 2



Traffic (A) in Erlangs for P=

.001

.002

.005

.010

.020

.050

.100

81 82 83 84 85

56.0 56.8 57.7 58.5 59.3

57.5 58.4 59.2 60.1 60.9

59.7 60.6 61.4 62.3 63.1

61.5 62.4 63.3 64.2 65.0

63.6 64.5 65.4 66.3 67.2

66.8 67.7 68.6 69.5 70.4

69.7 70.6 71.6 72.5 73.4

86 87 88 89 90

60.2 61.0 61.8 62.7 63.5

61.8 62.6 63.5 64.3 65.2

64.0 64.9 65.7 66.6 67.5

65.9 66.8 67.7 68.5 69.4

68.1 68.9 69.8 70.7 71.6

71.4 72.3 73.2 74.1 75.0

74.4 75.3 76.3 77.2 78.1

91 92 93 94 95

64.4 65.2 66.0 66.9 67.7

66.0 66.9 67.7 68.6 69.4

68.3 69.2 70.2 70.9 71.8

70.3 71.2 72.1 72.9 73.8

72.5 73.4 74.3 75.2 76.1

75.9 76.8 77.7 78.6 79.6

79.1 80.0 80.9 81.9 82.8

96 97 98 99 100

68.6 69.4 70.2 71.1 71.9

70.3 71.1 72.0 72.8 73.7

72.7 73.5 74.4 75.3 76.2

74.7 75.6 76.4 77.3 78.2

77.0 77.9 78.8 79.7 80.6

80.5 81.4 82.3 83.2 84.1

83.7 84.7 85.6 86.6 87.5

101 102 103 104 105

72.8 73.6 74.5 75.3 76.2

74.6 75.4 76.3 77.1 78.0

77.0 77.9 78.8 79.6 80.5

79.1 80.0 80.8 81.7 82.6

81.4 82.4 83.2 84.1 85.0

85.1 86.0 86.9 87.8 88.7

88.3 89.3 90.2 91.2 92.1

106 107 108 109 110 111 112 113 114 115 116 117 118 119 120

77.0 77.9 78.7 79.6 80.4

78.9 79.7 80.6 81.4 82.3 83.2 84.0 84.9 85.8 86.6 87.5 88.4 89.3 90.1 91.0

81.4 82.3 83.2 84.0 84.9

83.5 84.4 85.3 86.2 87.1

85.8 86.7 87.5 88.4 89.3 90.2 91.1 92.0 92.8 93.7

87.9 88.8 89.7 90.6 91.5 92.4 93.3 94.2 95.1 96.0

85.9 86.8 87.7 88.6 89.5 90.4 91.3 92.3 93.1 94.1 94.9 95.9 96.8 97.7 98.6

89.6 90.6 91.5 92.4 93.3 94.3 95.2 96.1 97.0 97.9

93.0 94.0 94.9 95.8 96.8 97.7 98.7 99.6 100.5 101.5 102.4 103.4 104.3 105.3 106.2

81.3 82.1 83.0 83.8 84.7 85.6 86.4 87.3 88.1 89.0

98.9 99.8 100.7 101.6 102.6

(table continues)

Chapter 2

2S




.002

.005

.010

.020

.050

.100

121 122 123 124 125

89.8 90.7 91.6 92.4 93.3

91.9 92.7 93.6 94.5 95.4

94.6 95.5 96.4 97.3 98.1

96.9 97.8 98.7 99.6 100.4

99.5 100.4 101.3 102.2 103.1

103.5 104.4 105.3 106.3 107.2

107.1 108.1 109.0 109.9 110.9

126 127 128 129 130

94.1 95.0 95.9 96.7 97.6

96.2 97.1 98.0 98.9 99.7

99.0 99.9 100.8 101.7 102.6

101.3 102.3 103.1 104.1 104.9

104.0 104.9 105.8 106.8 107.6

108.1 109.0 110.0 110.9 111.8

111.8 112.8 113.7 114.7 115.6

131 132 133 134 135

98.4 99.3 100.2 101.1 101.9

100.6 101.5 102.4 103.2 104.1

103.5 104.3 105.2 106.1 107.0

105.8 106.8 107.6 108.5 109.4

108.6 109.5 110.4 111.3 112.2

112.8 113.7 114.6 115.5 116.4

116.6 117.5 118.4 119.4 120.3

136 137 138 139 140

102.8 103.7 104.5 105.4 106.3

105.0 105.9 106.8 107.6 108.5

107.9 108.8 109.7 110.6 111.5

110.3 111.2 112.1 113.1 113.9

113.1 114.0 114.9 115.9 116.8

117.4 118.3 119.3 120.2 121.1

121.3 122.2 123.2 124.1 125.1

141 142 143 144 145

107.1 108.0 108.9 109.8 110.6

109.4 110.3 111.2 112.0 112.9

112.4 113.3 114.2 115.0 115.9

114.8 115.8 116.6 117.5 118.4

117.7 118.6 119.5 120.4 121.3

122.1 123.0 123.9 124.8 125.8

126.0 126.9 127.9 128.8 129.9

146 147 148 149 150

111.5 112.4 113.2 114.1 114.9

113.8 114.7 115.6 116.5 117.4

116.8 117.7 118.6 119.5 120.4

119.4 120.3 121.2 122.1 123.0

122.3 123.2 124.1 125.0 125.9

126.7 127.6 128.6 129.5 130.4

130.7 131.7 132.6 133.6 134.5

151 152 153 154 155

115.9 116.7 117.6 118.5 119.4

118.2 119.1 120.0 120.9 121.8

121.3 122.2 123.1 124.0 124.9

123.9 124.8 125.7 126.6 127.5

126.8 127.8 128.7 129.6 130.5

131.4 132.3 133.2 134.2 135.1

135.5 136.4 137.4 138.3 139.3

156 157 158 159 160

120.2 121.1 122.0 122.9 123.8

122.7 123.6 124.5 125.3 126.2

125.8 126.7 127.6 128.5 129.4

128.4 129.3 130.2 131.1 132.1

131.4 132.3 133.3 134.2 135.1

136.0 137.0 137.9 138.8 139.8

140.2 141.2 142.1 143.1 144.0

(table continues)

Chapter 2

26




.002

.005

.010

.020

.050

.100

161 162 163 164 165

124.6 125.5 126.4 127.3 128.1

127.1 128.0 128.9 129.8 130.7

130.3 131.2 132.1 133.0 133.9

132.9 133.9 134.8 135.7 136.6

136.0 136.9 137.9 138.8 139.7

140.1 141.6 142.6 143.5 144.4

145.0 145.9 146.9 147.8 148.8

166 167 168 169 170

129.0 129.9 130.8 131.7 132.3

131.6 132.5 133.4 134.3 135.2

134.8 135.7 136.6 137.5 138.4

137.5 138.4 139.3 140.2 141.1

140.6 141.5 142.4 143.4 144.3

145.4 146.3 147.3 148.2 149.1

149.7 150.7 151.6 152.6 153.5

171 172 173 174 175

133.4 134.3 135.2 136.1 136.9

136.1 137.0 137.9 138.8 139.7

139.3 140.2 141.1 142.1 143.0

142.1 142.9 143.9 144.8 145.7

145.2 146.1 147.1 148.0 148.9

150.1 151.0 151.9 152.9 153.8

154.5 155.4 156.4 157.3 158.3

176 177 178 179 180

137.8 138.7 139.6 140.5 141.4

140.6 141.5 142.3 143.2 144.1

144.0 144.9 145.8 146.7 147.6

146.6 147.5 148.4 149.3 150.3

149.8 150.8 151.7 152.6 153.5

154.8 155.7 156.6 157.6 158.5

159.2 160.2 161.1 162.1 163.0

181 182 183 184 185

142.3 143.1 144.0 144.9 145.8

145.0 145.8 146.7 147.6 148.5

148.5 149.4 150.3 151.3 152.2

151.2 152.1 153.0 153.9 154.8

154.4 155.4 156.3 157.2 158.1

159.4 160.4 161.3 162.3 163.2

164.0 164.9 165.9 166.8 167.8

186 187 188 189 190

146.7 147.6 148.5 149.4 150.3

149.4 150.3 151.2 152.1 153.0

153.1 154.0 154.9 155.8 156.7

155.7 156.6 157.6 158.5 159.4

159.1 160.0 160.9 161.8 162.8

164.1 165.1 166.0 167.0 167.9

168.8 169.7 170.6 171.6 172.6

191 192 193 194 195

151.1 152.0 152.9 153.8 154.7

153.9 154.8 155.7 156.6 157.5

157.6 158.5 159.4 160.3 161.2

160.3 161.2 162.1 163.1 164.0

163.7 164.6 165.5 166.5 167.4

168.8 169.8 170.7 171.7 172.6

173.5 174.5 175.4 176.4 177.3

196 197 198 199 200

155.6 156.5 157.4 158.3 159.1

158.4 159.3 160.2 161.1 162.0

162.1 163.0 163.9 164.8 165.7

164.9 165.8 166.7 167.6 168.6

168.3 169.3 170.2 171.1 172.0

173.5 174.5 175.4 176.4 177.3

178.3 179.2 180.2 181.1 182.1

(table continues)

Chapter 2

27


Table 2-2. Poisson Traffic Capacity in CCS No. of Trunks (N)

Traffic (A) in CCS for P = .020

.050

.100

.396 5.40

.756 7.70

1.91 12.9

3.82 19.1

12.2 24.2 38.9

15.7 29.6 46.1

20.4 36.7 55.8

29.4 49.3 70.9

39.6 63.0 87.8

45.7 61.9 79.6 97.9 117.4

55.4 73.4 92.5 112.7 133.9

64.4 83.9 104.8 126.0 149.0

76.0 96.8 119.2 141.8 166.0

94.0 118.1 142.9 168.8 195.1

113.0 140.0 168.1 195.1 223.9

126.0 145.1 166.0 186.8 208.1

137.5 158.4 180.0 202.0 224.3

155.5 177.8 200.9 224.3 248.0

172.1 195.5 220.0 244.1 268.9

191.2 216.0 240.8 267.1 293.0

222.1 249.1 276.8 304.9 333.0

253.1 281.9 311.0 340.9 370.8

16 17 18 19 20

231.1 253.1 276.1 299.2 322.9

247.3 270.7 294.1 318.2 342.7

272.5 297.0 321.8 347.4 374.4

294.5 320.0 346.0 374.4 399.6

320.0 347.0 374.4 399.6 428.4

363.6 388.9 417.6 446.4 478.8

399.6 432.0 460.8 493.2 522.0

21 22 23 24 25 26 27 28 29 30

346.0 370.8 396.0 417.6 442.8 468.0 496.8 518.4 543.6 572.4

367.2 392.4 417.6 442.8 468.0 493.2 518.4 547.2 572.4 597.6

399.6 424.8 450.0 478.8 504.0 529.2 558.0 586.8 612.0 640.8

424.8 453.6 478.8 507.6 536.4 561.6 590.4 619.2 648.0 676.8

457.2 486.0 514.8 543.6 572.4 597.6 626.4 655.2 684.0 716.4

507.6 536.4 565.2 597.6 626.4 655.2 687.6 716.4 748.8 777.6

554.4 586.8 615.6 648.0 676.8 709.2 741.6 774.0 806.4 835.2

31 32 33 34 35

597.6 622.8 651.6 676.8 702.0

626.4 651.6 680.4 705.6 734.4

666.0 695.8 723.6 752.4 777.6

702.0 730.8 759.6 788.4 817.2

745.2 774.0 802.8 831.6 860.4

810.0 838.8 871.2 903.6 932.4

867.6 900.0 932.4 964.8 997.2

36 37 38 39 40

730.8 756.0 784.8 810.0 838.8

759.6 788.4 817.2 842.4 871.2

806.4 835.2 864.0 892.8 921.6

846.0 874.8 903.6 936.0 964.8

892.8 921.6 950.4 982.8 1011.6

964.8 993.6 1026.0 1058.4 1087.2

1029.6 1058.4 1090.8 1126.8 1156.6

.001

.002

.005

1 2

.036 1.58

.072 2.34

.180 3.74

3 4 5

6.91 15.4 26.6

8.78 18.7 31.2

6 7 8 9 10

40.0 54.7 70.9 88.2 106.9

11 12 13 14 15

.010

(table continues)

Chapter 2

28


Table 2-2. Poisson Traffic Capacity in CCS (Continued) No. of Trunks (N)

Traffic (A) in CCS for P =

.001

.002

.005

.010

.020

.050

.100

41 42 43 44 45

864 893 918 947 976

900 929 954 983 1012

950 979 1008 1037 1066

994 1022 1051 1084 1112

1040 1073 1102 1134 1163

1120 1152 1184 1213 1246

1192 1220 1256 1289 1321

46 47 48 49 50

1004 1030 1058 1087 1116

1040 1069 1098 1127 1156

1094 1123 1152 1184 1213

1141 1170 1202 1231 1260

1195 1224 1256 1285 1318

1278 1310 1339 1372 1404

1354 1386 1418 1451 1483

51 52 53 54 55 56 57 58 59 60

1145 1170 1199 1228 1256 1285 1314 1343 1372 1400

1184 1213 1242 1271 1300 1328 1357 1386 1415 1444

1242 1271 1300 1332 1361 1390 1415 1451 1480 1508

1292 1321 1354 1382 1411 1444 1472 1505 1534 1566

1346 1379 1411 1440 1472

1436 1469 1498 1530 1562

1516 1548 1580 1613 1645

1505 1534 1566 1595 1627

1595 1627 1660 1692 1724

1678 1714 1746 1778 1811

61 62 63 64 65 66 67 68 69 70

1429 1458 1487 1516 1544 1573 1602 1631 1660 1692

1472 1505 1534 1562 1591 1620 1652 1681 1710 1742

1541 1570 1598 1631 1660 1688 1721 1753 1782 1811

1595 1627 1656 1688 1717 1750 1778 1811 1843 1872

1660 1688 1721 1753 1786 1814 1847 1879 1912 1940

1757 1786 1818 1850 1823 1915 1948 1980 2012 2045

1843 1876 1912 1944 1976 2009 2041 2077 2110 2142

71 72 73 74 75

1721 1750 1778 1810 1840

1771 1800 1829 1861 1890

1843 1872 1904 1933 1966

1904 1937 1966 1998 2027

1973 2005 2038 2066 2099

2077 2110 2142 2174 2207

2174 2210 2243 2275 2308

76 77 78 79 80

1868 1897 1926 1958 1987

1922 1951 1980 2012 2041

1994 2027 2056 2088 2120

2059 2092 2120 2153 2185

2131 2164 2192 2225 2257

2243 2275 2308 2340 2372

2344 2376 2408 2444 2480

(table continues)

Chapter 2

29




.001

.002

.005

.010

.020

.050

.100

81 82 83 84 85 86 87 88 89 90

2016 2045 2077 2106 2135 2167 2196 2225 2257 2286

2070 2102 2131 2164 2192 2225 2254 2286 2315 2347

2149 2182 2210 2243 2272 2304 2336 2365 2398 2430

2214 2246 2279 2311 2340 2372 2405 2437 2466 2498

2290 2322 2354 2387 2419 2452 2480 2513 2545 2578

2405 2437 2470 2502 2534 2570 2603 2635 2668 2700

2509 2542 2578 2610 2642 2678 2711 2747 2779 2812

91 92 93 94 95

2318 2347 2376 2408 2437

2376 2408 2437 2470 2498

2459 2491 2527 2552 2585

2531 2563 2596 2624 2657

2610 2642 2675 2707 2740

2732 2765 2797 2830 2866

2848 2880 2912 2948 2981

96 97 98 99 100

2470 2498 2527 2560 2588

2531 2560 2592 2621 2653

2617 2646 2678 2711 2743

2689 2722 2750 2783 2815

2772 2804 2837 2869 2902

2898 2930 2963 2995 3028

3013 3049 3082 3118 3150

101 102 103 104 105

2621 2650 2682 2711 2743

2686 2714 2747 2776 2808

2772 2804 2837 2866 2898

2848 2880 2909 2941 2974

2930 2966 2995 3028 3060

3064 3096 3128 3161 3193

3179 3215 3247 3283 3316

106 107 108 109 110

2772 2804 2833 2866 2894

2840 2869 2902 2930 2963

2930 2963 2995 3024 3056

3006 3038 3071 3103 3136

3092 3125 3157 3190 3222

3226 3262 3294 3326 3359

3348 3384 3416 3449 3485

111 112 113 114 115

2927 2956 2988 3017 3049

2995 3024 3056 3089 3118

3089 3121 3150 3182 3215

3164 3197 3229 3262 3294

3254 3287 3323 3352 3388

3395 3427 3460 3492 3524

3517 3553 3586 3618 3654

116 117 118 119 120

3082 3110 3143 3172 3204

3150 3182 3215 3244 3276

3247 3280 3312 3341 3373

3326 3359 3391 3424 3456

3416 3452 3485 3517 3550

3560 3593 3625 3658 3694

3686 3722 3755 3791 3823

(table continues)

30

Chapter 2


Table 2-2. Poisson Traffic Capacity in CCS (Continued) Traffic (A) in CCS for P =

No. of Trunks (N)

.001

.002

.005

.010

.020

.050

.100

121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 ·153 154 155 156 157 158 159 160

3233 3265 3298 3326 3359 3388 3420 3452 3481 3514 3542 3575 3607 3640 3668 3701 3733 3762 3794 3827 3856 3888 3920 3953 3982 4014 4046 4075 4108 4136 4172 4201 4234 4266 4298 4327 4360 4392 4424 4457

3308 3337 3370 3402 3434 3463 3496 3528 3560 3589 3622 3654 3686 3715 3748 3780 3812 3845 3874 3906 3938 3971 4003 4032 4064 4097 4129 4162 4194 4226 4255 4288 4320 4352 4385 4417 4450 4482 4511 4543

3406 3438 3470 3503 3532 3564 3596 3629 3661 3694 3726 3755 3787 3820 3852 3884 3917 3949 3982 4014 4046 4079 4111 4140 4172 4205 4237 4270 4302 4334 4367 4399 4432 4464 4496 4529 4561 4594 4626 4658

3488 3521 3553 3586 3614 3647 3683 3712 3748 3776 3809 3845 3874 3906 3938 3971 4003 4036 4072 4100 4133 4169 4198 4230 4262 4298 4331 4363 4396 4428 4460 4493 4525 4558 4590 4622 4655 4687 4720 4756

3582 3614 3647 3679 3712 3744 3776 3809 3845 3874 3910 3942 3974 4007 4039 4072 4104 4136 4172 4205 4237 4270 4302 4334 4367 4403 4435 4468 4500 4532 4565 4601 4633 4666 4698 4730 4763 4799 4831 4864

3726 3758 3791 3827 3859 3892 3924 3960 3992 4025 4061 4093 4126 4158 4190 4226 4259 4295 4327 4360 4396 4428 4460 4493 4529 4561 4594 4630 4662 4694 4730 4763 4795 4831 4864 4896 4932 4964 4997 5033

3856 3892 3924 3956 3992 4025 4061 4093 4129 4162 4198 4230 4262 4298 4331 4367 4399 4435 4468 4504 4536 4568 4604 4637 4676 4705 4741 4774 4810 4842 4878 4910 4946 4979 5015 5047 5083 5116 5152 5184

(table continues)

Chapter 2

31



161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200


.001

.002

.005

.010

.020

.050

.100

4486 4518 4550 4583 4612 4644 4676 4709 4741 4763 4802 4835 4867 4900 4928 4961 4993 5026 5058 5090 5123 5152 5184 5216 5249 5281 5314 5346 5378 5411 5440 5472 5504 5537 5569 5602 5634 5666 5699 5728

4576 4608 4640 4673 4705 4738 4770 4802 4835 4867 4900 4932 4964 4997 5029 5062 5094 5123 5155 5188 5220 5249 5281 5314 5346 5378 5411 5443 5476 5508 5540 5573 5605 5638 5670 5702 5735 5767 5800 5832

4691 4723 4756 4788 4820 4853 4885 4918 4950 4982 5015 5047 5080 5116 5148 5184 5216 5249 5281 5314 5346 5378 5411 5447 5479 5512 5544 5576 5609 5641 5674 5706 5738 5771 5803 5836 5868 5900 5933 5965

4784 4820 4853 4885 4918 4950 4982 5015 5047 5080 5116 5144 5180 5213 5245 5278 5310 5342 5375 5411 5443 5476 5508 5540 5573 5605 5638 5674 5706 5738 5771 5803 5836 5872 5904 5936 5969 6001 6034 6070

4896 4928 4964 4997 5029 5062 5094 5126 5162 5195 5227 5260 5296 5328 5360 5393 5429 5461 5494 5526 5558 5594 5627 5659 5692 5728 5760 5792 5825 5861 5893 5926 5958 5994 6026 6059 6095 6127 6160 6192

5044 5098 5134 5166 5198 5234 5267 5303 5335 5368 5404 5436 5468 5504 5537 5573 5605 5638 5674 5706 5738 5774 5807 5843 5875 5908 5944 5976 6012 6044 6077 6113 6145 6181 6214 6246 6282 6314 6350 6383

5220 5252 5288 5321 5357 5389 5425 5458 5494 5526 5562 5594 5630 5663 5699 5731 5767 5800 5836 5868 5904 5936 5972 6005 6041 6077 6109 6142 6178 6214 6246 6282 6314 6350 6383 6419 6451 6487 6520 6556

3 Erlang B Distribution

The Erlang B distribution is used in the North American PSTN for dimensioning high-usage trunk groups in alternate routing systems (blocked calls offered to another high-usage trunk group or to a final trunk group), and for all trunk group dimensioning in foreign and military networks. In addition, it is applicable to common equipment server pools which are not provided with a waiting queue (immediate service). The Erlang B distribution is based on the

following assumptions: • • • •

3.1

Calls are served in random order. There are an infinite number of sources. Blocked calls are cleared. Holding times are exponential or constant. ERLANG B FORMULA

The Erlang B Formula, also known as the Erlang formula of the first kind [E,(N, A)], is given in Equation 3.1.

33

Chapter 3

34

N


Ai

(3. I)

L-., l. ;=0

where B =Erlang B loss probability N = Number of trunks in full-availability group A = Traffic offered to group in Erlangs 3.2

ERLANG B COMPUTER PROGRAM

The following computer program can be used to calculate Equation 3.1 to determine Erlang B loss probabilities. Required inputs are the number of trunks in the trunk group and the traffic offered to the group expressed in Erlangs. 100 REM ERLANG B LOSS PROBABILITY CALCULATION 110 INPUT "ENTER NUMBER OF SERVERS (N) ";N 120 INPUT "ENTER OFFERED TRAFFIC IN ERLANGS (A) ";A 130 LET X=l 140 LET Y=l 150 FOR 1=1 TO N 160 LET X=X*A/I 170 LET Y=X+Y 180 NEXT I 190 PRINT USING "B

#.#####";X/Y

200 END

3.3

ERLANG B TRAFFIC CAPACITY TABLES

Erlang B traffic capacity tables are used to determine the maximum amount of traffic that can be offered to a group of N trunks such that the specified loss probability (grade of service) will not be exceeded. Tables 3.1 (pages 36-41) and 3.2 (pages 42-47) present Erlang B traffic capacities for 1 to 200 trunks with typical loss probabilities ranging from 0.001 to 0.1. These Erlang Bloss probabilities are commonly written as B.OOI to B.l. The Erlang B formula yields relatively linear traffic capacity values for trunk groups with more than

Chapter 3


35

200 trunks. Therefore, linear interpolation can be used between the given table values for those applications. The following examples illustrate typical table usage:

Example 3-1 Determine the traffic capacity in Erlangs and CCS for a 24-channel high-usage trunk group such that the loss probability will not exceed 2 percent (B.02). In Table 3-1, select the N row for 24 trunks and the B column for .02 and read 16.6 Erlangs at the intersection. In Table 3-2, select the N row for 24 trunks and the P column for .02 and read 597.6 CCS at the intersection.

Example 3-2 Determine the number of trunks required in a high-usage trunk group to handle 720 CCS of offered traffic at a grade of service of B.O I. In Table 3-2, select the B.01 column and read down until 730.8 CCS is found. Read across that N row to determine that 30 trunks are required.

Example 3-3 Determine the grade of service for a 48-channel high-usage trunk group with offered traffic of 0.75 Erlangs per trunk (36 Erlangs). A = (48 chan) (0.75 Erl/chan) = 36 Erlangs

In Table 3-1, select the N row for 48 trunks and read across until 36.1 Erlangs is found. Read up that B column to determine that the grade of service is B.Ol.

Chapter 3

36

ErIang B Distribution

Table 3-1. Erlang B Traffic Capacity in Erlangs Traffic (A) in Erlangs for B =

No. of Trunks

(N)

.001

.002

.005

.010

.020

.050

.100

1

2 3 4 5

.001 .046 .194 .440 .763

.002 .066 .249 .536 .900

.005 .106 .349 .702 1.13

.011 .153 .456 .870 1.36

.021 .224 .603 1.09 1.66

.053 .382 .900 1.52 2.22

.111 .595 1.27 2.05 2.88

6 7 8 9 10

1.15 1.58 2.05 2.56 3.09

1.33 1.80 2.31 2.86 3.43

1.62 2.16 2.73 3.33 3.96

1.91 2.50 3.13 3.78 4.46

2.28 2.94 3.63 4.34 5.08

2.96 3.74 4.54 5.37 6.22

3.76 4.67 5.60 6.55 7.51

11 12 13 14 15

3.65 4.23 4.83 5.45 6.08

4.02 4.64 5.27 5.92 6.58

4.61 5.28 5.96 6.66 7.38

5.16 5.88 6.61 7.35 8.11

5.84 6.62 7.41 8.20 9.01

7.08 7.95 8.83 9.73 10.6

8.49 9.47 10.5 11.5 12.5

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

6.72 7.38 8.05 8.72 9.41

7.26 7.95 8.64 9.35 10.1

8.10 8.83 9.58 10.3 11.1

8.87 9.65 10.4 11.2 12.0

9.83 10.7 11.5 12.3 13.2

13.5 14.5 15.6 16.6 17.6

10.1 10.8 11.5 12.2 13.0 13.7 14.4 15.2 15.9 16.7

10.8

11.9 12.6 13.4 14.2 15.0

14.0 14.9 15.8 16.6 17.5 18.4 19.3 20.2 21.0 21.9

18.7 19.7 20.7 21.8 22.8

15.8 16.6 17.4 18.2 19.0

12.8 13.7 14.5 15.3 16.1 17.0 17.8 18.6 19.5 20.3

11.5 12.5 13.4 14.3 15.3 16.2 17.1 18.1 19.0 20.0 20.9 21.9 22.9 23.8 24.8

31 32 33 34 35 36 37 38 39 40

17.4 18.2 19.0 19.7 20.5 21.3 22.1 22.9 23.7 24.4

18.4 19.2 20.0 20.8 21.6 22.4 23.2 24.0 24.8 25.6

19.9 20.7 21.5 22.3 23.2 24.0 24.9 25.7 26.5 27.4

21.2 22.1 22.9 23.8 24.6 25.5 26.4 27.3 28.1 29.0

22.8 23.7 24.6 25.5 26.4 27.3 28.3 29.2 30.1 31.0

25.8 26.8 27.7 28.7 29.7 30.7 31.6 32.6 33.6 34.6

29.2 30.2 31.3 32.4 33.4 34.5 35.6 36.6 37.7 38.8

11.5

12.3 13.0 13.8 14.5 15.3 16.1 16.8 17.6

23.9 24.9 26.0 27.1 28.1

(table continues)

Chapter 3

37


Table 3-1. Erlang B Traffic Capacity in Erlangs (Continued) No. of Trunks

Traffic (A) in Erlangs for B =

.001

.002

.005

.010

.020

.050

.100

41 42 43 44 45

25.2 26.0 26.8 27.6 28.5

26.4 27.2 28.1 28.9 29.7

28.2 29.1 29.9 30.8 31.7

29.9 30.8 31.7 32.5 33.4'

31.9 32.8 33.8 34.7 35.6

35.6 36.6 37.6 38.6 39.6

39.9 40.9 42.0 43.1 44.2

46 47 48 49 50

29.3 30.1 30.9 31.7 32.5

32.5 33.4 34.2 35.1 36.0

34.3 35.2 36.1 37.0 37.9

45.2 46.3 47.4 48.5 49.6

33.3 34.2 35.0 35.8 36.6

36.9 37.7 38.6 39.5 40.4

38.8 39.7 40.6 41.5 42.4

36.5 37.5 38.4 39.3 40.3 41.2 42.1 43.1 44.0 44.9

40.5 41.5 42.5 43.5 44.5

51 52 53 54 55

30.5 31.4 32.2 33.0 33.9 34.7 35.6 36.4 37.3 38.1

45.5 46.5 47.5 48.5 49.5

50.6 51.7 52.8 53.9 55.0

56 57 58 59 60

37.5 38.3 39.1 40.0 40.8

38.9 39.8 40.6 41.5 42.4

41.2 42.1 43.0 43.9 44.8

45.9 46.8 47.8 48.7 49.6

50.5 51.5 52.6 53.6 54.6

56.1 57.1 58.2 59.3 60.4

61 62 63 64 65

41.6 42.5 43.3 44.2 45.0

43.2 44.1 44.9 45.8 46.7

45.6 46.5 47.7 48.3 49.2

43.3 44.2 45.1 46.0 46.9 47.9 48.8 49.7 50.6 51.5

50.6 51.5 52.5 53.4 54.4

55.6 56.6 57.6 58.6 59.6

61.5 62.6 63.7 64.8 65.8

66 67 68 69 70 71 72 73 74 75 76 77 78 79 80

45.8 46.7 47.5 48.4 49.2

47.5 48.4 49.2 50.1 51.0 51.9 52.7 53.6 54.5 55.3 56.2 57.1 58.0 58.8 59.7

50.1 51.0 51.9 52.8 53.7

52.4 53.4 54.3 55.2 56.1 57.0 58.0 58.9 59.8 60.7 61.7 62.6 63.5 64.4 65.4

55.3 56.3 57.2 58.2 59.1

60.6 61.6 62.6 63.7 64.7

66.9 68.0 69.1 70.2 71.3

60.1 61.0 62.0 62.9 63.9 64.9 65.8 66.8 67.7 68.7

65.7 66.7 67.7 68.7 69.7 70.8 71.8 72.8 73.8 74.8

72.4 73.5 74.6 75.6 76.7 77.8 78.9 80.0 81.1 82.2

(N)

50.1 50.9 51.8 52.7 53.5 54.4 55.2 56.1 57.0 57.8

54.6 55.5 56.4 57.3 58.2 59.1 60.0 60.9 61.8 62.7

(table continues)

38

Chapter 3


Table 3-1. Erlang B Traffic Capacity in Erlangs (Continued) No. of Trunks

Traffic (A) in Erlangs for B

=

(N)

.001

.002

.005

.010

.020

.050

.100

81 82 83 84 85

58.7 59.5 60.4 61.3 62.1

60.6 61.5 62.4 63.2 64.1

63.6 64.5 65.4 66.3 67.2

66.3 67.2 68.2 69.1 70.0

69.6 70.6 71.6 72.5 73.5

75.8 76.9 77.9 78.9 79.9

83.3 84.4 85.5 86.6 87.7

86 87 88 89 90

63.0 63.9 64.7 65.6 66.5

65.0 65.9 66.8 67.7 68.6

68.1 69.0 69.9 70.8 71.8

70.9 71.9 72.8 73.7 74.7

74.5 75.4 76.4 77.3 78.3

80.9 82.0 83.0 84.0 85.0

88.8 89.9 91.0 92.1 93.1

91 92 93 94 95

67.4 68.2 69.1 70.0 70.9

69.4 70.3 71.2 72.1 73.0

72.7 73.6 74.5 75.4 76.3

75.6 76.6 77.5 78.4 79.4

79.3 80.2 81.2 82.2 83.1

86.0 87.1 88.1 89.1 90.1

94.2 95.3 96.4 97.5 98.6

96 97 98 99 100 101 102 103 104 105 106 107 108 109 110

71.7 72.6 73.5 74.4 75.2 76.1 77.0 77.9 78.8 79.7 80.5 81.4 82.3 83.2 84.1

73.9 74.8 75.7 76.6 77.5

77.2 78.2 79.1 80.0 80.9

80.3 81.2 82.2 83.1 84.1

84.1 85.1 86.0 87.0 88.0

91.1 92.2 93.2 94.2 95.2

99.7 100.8 101.9 103.0 104.1

78.4 79.3 80.2 81.1 81.9 82.9 83.8 84.7 85.6 86.5

8~.8

82.8 83.7 84.6 85.5 86.4 87.4 88.3 89.2 90.1

85.0 85.9 86.9 87.8 88.8 89.7 90.7 91.6 92.5 93.5

88.9 89.9 90.9 91.9 92.8 93.8 94.8 95.7 96.7 97.7

96.3 97.3 98.3 99.3 100.4 101.4 102.4 103.4 104.5 105.5

105.2 106.3 107.4 108.5 109.6 110.7 111.8 112.9 114.0 115.1

111 112 113 114 115 116 117 118 119 120

85.0 85.9 86.7 87.6 88.5 89.4 90.3 91.2 92.1 93.0

87.4 88.3 89.2 90.1 91.0 91.9 92.8 93.7 94.6 95.5

91.0 92.0 92.9 93.8 94.7 95.7 96.6 97.5 98.5 99.4

94.4 95.4 96.3 97.3 98.2 99.2 100.1 101.1 102.0 103.0

98.7 99.6 100.6 101.6 102.5 103.5 104.5 105.5 106.4 107.4

106.5 107.5 108.6 109.6 110.6 111.7 112.7 113.7 114.7 115.8

116.2 117.3 118.4 119.5 12d.6 121.7 122.8 123.9 125.0 126.1

(table continues)

Chapter 3


39

Table 3-1. Erlang B Traffic Capacity in Erlangs (Continued) Traffic (A) in Erlangs for B =

No. of Trunks

(N)

.001

.002

.005

.010

.020

.050

.100

121 122 123 124 125

93.8 94.7 95.6 96.5 97.4

96.4 97.3 98.2 99.1 100.0

100.3 101.2 102.2 103.1 104.0

103.9 104.9 105.8 106.8 107.7

108.4 109.4 110.3 111.3 112.3

116.8 117.8 118.9 119.9 120.9

127.2 128.3 129.4 130.5 131.6

126 127 128 129 130

98.3 99.2 100.1 101.0 101.9

100.9 101.8 102.7 103.7 104.6

105.0 105.9 106.8 107.8 108.7

108.7 109.6 110.6 111.5 112.5

113.3 114.2 115.2 116.2 117.2

122.0 123.0 124.0 125.1 126.1

132.7 133.8 134.9 136.0 137.1

131 132 133 134 135

102.8 103.7 104.6 105.5 106.4

105.5 106.4 107.3 108.2 109.1

109.6 110.5 111.5 112.4 113.3

113.4 114.4 115.3 116.3 117.2

118.2 119.1 120.1 121.1 122.1

127.1 128.2 129.2 130.2 131.3

138.2 139.3 140.4 141.5 142.6

136 137 138 139 140

107.3 108.2 109.1 110.0 110.9

110.0 111.0 111.9 112.8 113.7

114.3 115.2 116.2 117.1 118.0

118.2 119.1 120.1 121.0 122.0

123.1 124.0 125.0 126.0 127.0

132.3 133.3 134.3 135.4 136.4

143.7 144.8 145.9 147.0 148.1

141 142 143 144 145

111.8 112.7 113.6 114.5 115.4

114.6 115.5 116.4 117.4 118.3

119.0 119.9 120.8 121.8 122.7

122.9 123.9 124.8 125.8 126.7

128.0 128.9 129.9 130.9 131.9

137.4 138.5 139.5 140.5 141.6

149.2 150.3 151.4 152.5 153.6

146 147 148 149 150

116.3 117.2 118.1 119.0 119.9

119.2 120.1 121.0 121.9 122.9

123.6 124.6 125.5 126.5 127.4

127.7 128.6 129.6 130.6 131.6

132.9 133.8 134.8 135.8 136.8

142.6 143.6 144.7 145.7 146.7

154.7 155.8 156.9 158.0 159.1

151 152 153 154 155

120.8 121.7 122.6 123.5 124.4

123.8 124.7 125.6 126.5 127.5

128.3 129.3 130.2 131.2 132.1

132.5 133.5 134.5 135.4 136.4

137.8 138.7 139.7 140.7 141.7

147.8 148.8 149.8 150.9 150.9

160.3 161.4 162.5 163.6 164.7

156 157 158 159 160

125.3 126.2 127.1 128.0 129.0

128.4 129.3 130.2 131.1 132.1

133.0 134.0 134.9 135.9 136.8

137.4 138.3 139.3 140.3 141.2

142.7 143.6 144.6 145.6 146.6

152.9 153.9 154.9 156.0 157.0

165.8 166.9 168.0 169.1 170.2

(table continues)

40

Chapter3


Table 3-1. Erlang 8 Traffic Capacity in Erlangs (Continued) Traffic (A) in Erlangs for B =

No. of Trunks (N)

.001

.002

.005

.010

.020

.050

.100

161 162 163 164 165

129.9 130.8 131.7 132.6 133.5

133.0 133.9 134.8 135.8 136.7

137.7 138.7 139.6 140.6 141.5

142.2 143.2 144.1 145.1 146.1

147.6 148.6 149.6 150.5 151.5

158.0 159.1 160.1 161.1 162.2

171.3 172.4 173.5 174.6 175.7

166 167 168 169 170

134.4 135.3 136.2 137.1 138.1

137.6 138.5 139.4 140.4 141.3

147.0 148.0 148.9 149.9 150.8

176.8 177.9 179.0 180.1 181.2

139.0 139.9 140.8 141.7 142.7 143.6 144.5 145.5 146.3 147.3

151.8 152.8 153.7 154.7 155.6 156.6 157.5 158.5 159.4 160.4 161.4 162.3 163.3 164.3 165.2 166.2 167.2 168.1 169.1 170.1 171.0 172.0 173.0 173.9 174.5

168.4 169.4 170.5 171.5 172.6 173.6 174.7 175.7 176.8 177.8

182.3 183.4 184.5 185.6 186.7

148.2 149.1 150.0 150.9 151.8 152.8 153.7 154.6 155.5 156.4 157.4 158.3 159.2 160.2 161.1

142.2 143.1 144.1 145.0 145.9 146.9 147.8 148.7 149.6 150.6 151.5 152.4 153.3 154.3 155.2 156.1 157.1 158.0 158.9 159.8 160.7 161.7 162.6 163.6 164.5

152.5 153.5 154.5 155.5 156.5 157.5 158.5 159.4 160.4 161.4 162.4 163.4 164.4 165.4 166.4 167.4 168.4 169.3 170.3 171.3 172.3 173.3 174.3 175.3 176.3 177.3 178.3 179.3 180.3 181.2

163.2 164.2 165.3 166.3 167.4

171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195

142.4 143.4 144.3 145.3 146.2 147.2 148.1 149.1 150.0 150.9 151.9 152.8 153.8 154.7 155.7 156.6 157.6 158.5 159.5 160.4 161.4 162.3 163.3 164.2 165.2 166.1 167.0 167.9 168.9 169.8

196 197 198 199 200

162.0 162.9 163.8 164.7 165.6

165.4 166.3 167.3 168.2 169.2

170.8 171.7 172.7 173.6 174.6

175.9 176.8 177.8 178.8 179.7

182.2 183.2 184.2 185.2 186.2

178.8 179.9 180.9 181.9 183.0 184.0 185.0 186.1 187.1 188.1 189.2 190.2 191.2 192.3 193.3

187.8 188.9 190.0 191.1 192.2 193.4 194.5 195.6 196.7 197.8 198.9 200.0 201.1 202.2 203.3 204.4 205.5 206.6 207.7 208.8

194.3 195.4 196.4 197.4 198.5

209.9 211.0 212.1 213.2 214.3

(table continues)

Chapter 3

41


Table 3-1. Erlang B Traffic Capacity in Erlangs (Continued) Traffic (A) in Erlangs for B =

No. of Trunks

(N)

.001

.002

.005

.010

.020

.050

.100

202 204 206 208 210

167.5 169.3 171.2 173.0 174.8

171.0 172.9 174.8 176.6 178.5

176.5 178.4 180.4 182.3 184.2

181.7 183.6 185.5 187.5 189.4

188.1 190.1 192.1 194.1 196.1

200.6 202.7 204.7 206.8 208.9

216.5 218.7 221.0 223.2 225.4

212 214 216 218 220

176.7 178.5 180.4 182.2 184.1

180.4 182.2 184.1 186.0 187.8

186.1 188.0 189.9 191.8 193.7

191.4 193.3 195.2 197.2 199.1

198.1 200.0 202.0 204.0 206.0

211.0 213.0 215.1 217.2 219.3

227.6 229.8 232.0 234.2 236.4

222 224 226 228 230

185.9 187.8 189.6 191.5 193.3

189.7 191.6 193.5 195.3 197.2

195.6 197.5 199.4 201.3 203.2

201.1 203.0 204.9 206.9 208.8

208.0 210.0 212.0 213.9 215.9

221.4 223.4 225.5 227.6 229.7

238.6 240.9 243.1 245.3 247.5

232 234 236 238 240

195.2 197.1 198.9 200.8 202.6

199.1 201.0 202.8 204.7 206.6

205.1 207.1 209.0 210.9 212.8

210.8 212.7 214.7 216.6 218.6

217.9 219.9 221.9 223.9 225.9

231.8 233.8 235.9 238.0 240.1

249.7 251.9 254.1 256.3 258.6

242 244 246 248 250

204.5 206.3 208.2 210.1 211.9

208.5 210.4 212.2 214.1 216.0

214.7 216.6 218.5 220.4 222.4

220.5 222.5 224.4 226.3 228.3

227.9 229.9 231.8 233.9 235.8

242.2 244.3 246.3 248.4 250.5

260.8 263.0 265.2 267.4 269.6

300 350 400 450 500

258.6 305.7 353.0 400.5 448.2

263.2 310.8 358.5 406.4 454.5

270.4 318.7 367.2 415.8 464.5

277.1 326.2 375.3 424.6 474.0

285.7 335.7 385.9 436.1 486.4

302.6 354.8 407.1 459.4 511.8

325.0 380.4 435.8 491.3 546.7

550 600 650 700 750

496.1 543.9 592.0 640.1 688.4

502.8 551.0 599.6 647.9 696.5

513.4 562.3 611.4 660.4 709.6

523.6 573.1 622.8 672.4 721.6

536.8 587.2 637.7 688.2 738.8

564.2 616.5 669.0 721.4 773.9

602.2 657.7 713.2 768.7 824.2

800 850 900 950 1000

736.6 785.0 833.3 881.8 930.3

741.5 790.2 842.5 891.3 940.1

758.7 808.7 857.2 906.6 955.9

771.8 821.7 871.5 921.4 971.2

789.3 840.0 890.6 941.3 991.9

826.4 878.9 931.4 983.7 1036.0

879.7 935.3 990.8 1046.4 1102.0

42

Chapter 3


Table 3-2. Erlang 8 Traffic Capacity in CCS Traffic (A) in CCS for B

=

No. of Trunks (N)

.001

.002

.005

.010

.020

.050

.100

1 2 3 4 5

.036 1.66 6.98 15.8 27.5

.072 2.38 8.96 19.3 32.4

.180 3.82 12.6 25.3 40.7

.396 5.51 16.4 31.3 49.0

.756 8.06 21.7 39.2 59.8

1.91 13.8 32.4 54.7 79.9

4.00 21.4 45.7 73.8 103.7

6 7 8 9 10

41.4 56.9 73.8 92.2 111.2

47.9 64.8 83.2 103.0 123.5

58.3 77.8 98.3 119.9 142.6

68.8 90.0 112.7 136.1 160.6

82.1 105.8 130.7 156.2 182.9

106.6 134.6 163.4 193.3 223.9

135.4 168.1 201.6 235.8 270.4

11 12 13 14 15

131.4 152.3 173.9 196.2 218.9

144.7 167.0 189.7 213.1 236.9

166.0 190.1 214.6 239.8 265.7

185.8 211.7 238.0 264.6 292.0

210.2 238.3 266.8 295.2 324.4

254.9 286.2 317.9 350.3 381.6

305.6 340.9 378.0 414.0 450.0

16 17 18 19 20

241.9 265.7 289.8 313.9 338.8

261.4 286.2 311.0 336.6 363.6

291.6 317.9 344.9 370.8 399.6

319.3 347.4 374.4 403.2 432.0

353.9 385.2 414.0 442.8 475.2

414.0 450.0 482.4 514.8 550.8

486.0 522.0 561.6 597.6 633.6

21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

363.6 388.8 414.0 439.2 468.0 493.2 518.4 547.2 572.4 601.2 626.4 655.2 684.0 709.2 738.0

388.8 414.0 442.8 468.0 496.8 522.0 550.8 579.6 604.8 633.6 662.4 691.2 720.0 748.8 777.6

428.4 453.6 482.4 511.2 540.0 568.8 597.6 626.4 655.2 684.0 716.4 745.2 774.0 802.8 835.2

460.8 493.2 522.0 550.8 579.6 612.0 640.8 669.6 702.0 730.8

504.0 536.4 568.8 597.6 630.0 662.4 694.8 727.2 756.0 788.4

583.2 615.6 651.6 684.0 720.0 752.4 788.4 824.4 856.8 892.8

673.2 709.2 745.2 784.8 820.8 860.4 896.4 936.0 975.6 1011.6

763.2 795.6 824.4 856.8 885.6

820.8 853.2 885.6 918.0 950.4

928.8 964.8 997.2 1033.2 1069.2

1051.2 1087.2 1126.8 1166.4 1202.4

36 37 38 39 40

766.8 795.6 824.4 853.2 878.4

806.4 835.2 864.0 892.8 921.6

864.0 896.4 925.2 954.0 986.4

918.0 950.4 982.8 1011.6 1044.0

982.8 1018.8 1051.2 1083.6 1116.0

1105.2 1137.6 1173.6 1209.6 1245.6

1242.0 1281.6 1317.6 1357.2 1396.8

(table continues)

Chapter 3


43

Table 3-2. Erlang B Traffic Capacity in CCS (Continued) Traffic (A) in CCS for B =

No. of Trunks

(N)

.001

.002

.005

.010

.020

.050

.100

41 42 43 44 45

907 936 965 994 1026

950 979 1012 1040 1069

1015 1047 1076 1109 1141

1076 1109 1141 1170 1202

1148 1181 1217 1249 1282

1282 1318 1354 1390 1426

1436 1472 1512 1552 1591

46 47 48 49 50 51 52 53 54 55

1055 1084 1112 1141 1170 1199 1231 1260 1289 1318

1098 1130 1159 1188 1220 1249 1282 1310 1343 1372

1170 1202 1231 1264 1296 1328 1357 1390 1422 1454

1235 1267 1300 1332 1364 1397 1429 1462 1494 1526

1314 1350 1382 1415 1451 1483 1516 1552 1584 1616

1458 1494 1530 1566 1602 1638 1674 1710 1746 1782

1627 1667 1706 1746 1786 1822 1861 1901 1940 1980

56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80

1350 1379 1408 1440 1469 1498 1530 1559 1591 1620 1649 1681 1710 1742 1771 1804 1832 1865 1897 1926 1958 1987 2020 2052 2081

1400 1433 1462 1494 1526 1555 1588 1616 1649 1681 1710 1742 1771 1804 1836 1868 1897 1930 1962 1991 2023 2056 2088 2117 2149

1483 1516 1548 1580 1613 1642 1674 1717 1739 1771 1804 1836 1868 1901 1933 1966 1998 2030 2063 2095 2128 2160 2192 2225 2257

1559 1591 1624 1656 1688 1724 1757 1789 1822 1854 1886 1922 1955 1987 2020 2052 2088 2120 2153 2185 2221 2254 2286 2318 2354

1652 1685 1721 1753 1786 1822 1854 1890 1922 1958 1991 2027 2059 2095 2128 2164 2196 2232 2264 2300 2336 2369 2405 2437 2473

1818 1854 1894 1930 1966 2002 2038 2074 2110 2146 2182 2218 2254 2293 2329 2365 2401 2437 2473 2509 2549 2585 2621 2657 2693

2020 2056 2095 2135 2174 2214 2254 2293 2333 2369 2408 2448 2488 2527 2567 2606 2646 2686 2722 2761 2801 2840 2880 2920 2959

(table continues)

Chapter 3

44


Table 3-2. Erlang 8 Traffic Capacity in CCS (Continued) Traffic (A) in CCS for B

No. of Trunks

=

(N)

.001

.002

.005

.010

.020

.050

.100

81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120

2113 2142 2174 2207 2236 2268 2300 2329 2362 2394 2426 2455 2488 2520 2552 2581 2914 2646 2678 2707 2740 2772 2804 2837 2869 2898 2930 2963 2995 3028 3060 3092 3121 3154 3186 3218 3251 3283 3316 3348

2182 2214 2246 2275 2308 2340 2372 2405 2437 2470 2498 2531 2563 2596 2628 2660 2693 2725 2758 2790 2822 2855 2887 2920 2948 2984 3017 3049 3082 3114 3146 3179 3211 3244 3276 3304 3341 3373 3406 3438

2290 2322 2354 2387 2419 2452 2484 2516 2549 2585 2617 2650 2682 2714 2747 2779 2815 2848 2880 2912 2945 2981 3013 3046 3078 3110 3146 3179 3211 3244 3276 3312 3344 3377 3409 3445 3478 3510 3546' 3578

2387 2419 2455 2488 2520 2552 2588 2621 2653 2689 2722 2758 2790 2822 2858 2891 2923 2959 2992 3028 3060 3092 3128 3161 3197 3229 3265 3298 3330 3366 3398 3434 3467 3503 3535 3571 3604 3640 3672 3708

2506 2542 2578 2610 2646 2682 2714 2750 2783 2819 2855 2887 2923 2959 2992 3028 3064 3096 3132 3168 3200 3236 3272 3308 3341 3377 3413 3445 3481 3517 3553 3586 3622 3658 3690 3726 3762 3798 3830 3866

2729 2768 2804 2840 2876 2912 2952 2988 3024 3060 3096 3136 3172 3208 3244 3280 3319 3355 3391 3427 3467 3503 3539 3575 3614 3650 3686 3722 3762 3798 3834 3870 3910 3946 3982 4021 4057 4093 4129 4169

2999 3038 3078 3118 3157 3197 3236 3276 3316 3352 3391 3431 3470 3510 3550 3589 3629 3668 3708 3748 3787 3827 3866 3906 3946 3985 4025 4064 4104 4144 4183 4223 4262 4302 4342 4381 4421 4460 4500 4540

(table continues)

Chapter 3


45


No. of Trunks (N)

.001

.002

.005

.010

.020

.050

.100

121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146

3377 3409 3442 3474 3506 3539 3571 3604 3636 3668 3701 3733 3766 3798 3830 3863 3895 3928 3960 3992 4025 4057 4090 4122 4154 4187

3470 3503 3535 3568 3600 3632 3665 3697 3733 3766 3798 3830 3863 3895 3928 3960 3996 4028 4061 4093 4126 4158 4190 4226 4259 4291

3611 3643 3679 3712 3744 3780 3812 3845 3881 3913 3946 3978 4014 4046 4079 4115 4147 4183 4216 4248 4284 4316 4349 4385 4417 4450

3740 3776 3809 3845 3877 3913 3946 3982 4014 4050 4082 4118 4151 4187 4219 4255 4288 4324 4356 4392 4424 4460 4493 4529 4561 4597

3902 3938 3971 4007 4043 4079 4111 4147 4183 4219 4255 4288 4324 4360 4396 4432 4464 4500 4536 4572 4608 4640 4676 4712 4748 4784

4205 4241 4280 4316 4352 4392 4428 4464 4504 4540 4576 4615 4651 4687 4727 4763 4799 4835 4874 4910 4946 4986 5022 5058 5097 5134

147

4579 4619 4658 4698 4738 4777 4617 4856 4896 4936 4975 5015 5054 5094 5134 5173 5213 5252 5292 5332 5371 5411 5450 5490 5530 5569

4219

4324

4486

148 149 150 151 152 153 154 155 156 157 158 159 160

4252 4284 4316 4349 4381 4414 4446 4478 4511 4543 4576 4608 4644

4518 4554 4586 4619 4655 4687 4723 4756 4788 4824 4856 4892 4925

4630

4817

4356 4388 4424 4457 4489 4522 4554 4590 4622 4655 4687 4720 4756

4666 4702 4738 4770 4806 4842 4874 4910 4946 4979 5015 5051 5083

4853 4889 4925 4961 4993 5029 5065 5101 5137 5170 5206 5242 5277

5170

5609

5209 5245 5281 5321 5357 5393 5432 5432 5504 5540 5576 5616 5652

5648 5688 5728 5771 5810 5850 5890 5929 5969 6008 6048 6088 6127

(table continues)

46

Chapter 3



No. of Trunks (N)

.001

.002

.005

.010

.020

.050

.100

161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200

4676 4709 4741 4774 4806 4838 4871 4903 4936 4972 5004 5036 5069 5101 5137 5170 5202 5238 5267 5303 5335 5368 5400 5432 5465 5501 5533 5566 5598 5630 5666 5699 5731 5767 5800 5832 5864 5897 5929 5962

4788 4820 4853 4889 4921 4954 4986 5018 5054 5087 5119 5152 5188 5220 5252 5288 5321 5353 5386 5422 5454 5486 5519 5555 5587 5620 5656 5688 5720 5753 5785 5821 5854 5890 5922 5954 5987 6023 6055 6091

4957 4993 5026 5062 5094 5126 5162 5195 5231 5263 5299 5332 5368 5400 5432 5468 5501 5537 5569 5605 5638 5674 5706 5742 5774 5810 5843 5879 5911 5947 5980 6012 6044 6080 6113 6149 6181 6217 6250 6286

5119 5155 5188 5224 5260 5292 5328 5360 5396 5429 5465 5501 5533 5569 5602 5638 5670 5706 5738 5774 5810 5843 5879 5915 5947 5983 6019 6052 6088 6124 6156 6192 6228 6260 6282 6332 6365 6401 6437 6469

5314 5350 5386 5418 5454 5490 5526 5562 5598 5634 5670 5706 5738 5774 5810 5846 5882 5918 5954 5990 6026 6062 6095 6131 6167 6203 6239 6275 6311 6347 6383 6419 6455 6491 6523 6559 6595 6631 6667 6703

5688 5728 5764 5800 5839 5875 5911 5951 5987 6026 6062 6098 6138 6174 6214 6250 6289 6325 6365 6401 6437 6476 6512 6548 6588 6624 6660 6700 6736 6772 6811 6847 6883 6923 6959 6995 7034 7070 7106 7146

6167 6206 6246 6286 6325 6365 6404 6444 6484 6523 6563 6602 6642 6682 6721 6761 6800 6840 6880 6919 6962 7002 7042 7181 7121 7160 7200 7240 7279 7319 7358 7398 7438 7477 7517 7556 7596 7636 7675 7715

(table continues)

Chapter 3

47


Table 3..2. Erlang B Traffic Capacity in CCS (Continued) Traffic (A) in CCS for B =

No. of Trunks (N)

.001

.002

.005

.010

.020

.050

.100

202 204 206 208 210

6030 6095 6163 6228 6293

6156 6224 6293 6358 6426

6354 6422 6494 6563 6631

6541 6610 6678 6750 6818

6772 6844 6916 6988 7060

7222 7297 7369 7445 7520

7794 7873 7956 8035 8114

212 214 216 218 220 222 224 226 228 230 232 234 236 238 240 242 244 246 248 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000

6361 6426 6494 6559 6628 6692 6761 6826 6894 6959 7027 7096 7160 7229 7294 7362 7427 7495 7564 7628 9310 11005 12708 14418 16135 17860 19580 21312 23044 24782 26518 28260 29999 31745 33491

6494 6559 6628 6696 6761 6829 6898 6966 7031 7099 7168 7236 7301 7369 7438 7506 7574 7639 7708 7776 9475

6700 6768 6836 6905 6973 7042 7110 7178 7247 7315 7384 7456 7524 7592 7661 7729 7798 7866 7934 8006 9734 11473 13219 14969 16722 18482 20243 22010 23774 25546 27313 29113 30859 32638 34412

6890 6959 7027 7099 7168 7240 7308 7376 7448 7517 7589 7657 7729 7798 7870 7938 8010 8078 8145 8219 9976 11743 13511 15286 17064 18850 20632 22421 24206 25978 27784 29581 31374 33170 34963

7132 7200 7272 7344 7416 7488 7560 7632 7700 7772 7844 7916 7988 8060 8132 8204 8276 8345 8420 8489 10285 12085 13892 15700 17510 19325 21139 22957 24775 26597 28415 30240 32062 33887 35708

7596 7668 7744 7819 7895 7970 8042 8118 8194 8269 8345 8417 8492 8568 8644 8719 8795 8867 8942 9018 10894 12773 14656 16538 18425 20311 22194 24084 25970 27860 29750 31640 33530 35413 37296

8194 8273 8352 8431 8510 8590 8672 8752 8831 8910 8989 9068 9148 9227 9310 9389 9468 9547 9626 2706 11700 13694 15689 17687 19681 21679 23677 25675 27673 29671 31669 33671 35669 37670 39672

11189

12906 14630 16362 18101 19836 21586 23324 25074 26694 28447 30330 32087 33844

48

3.4

Chapter 3


HIGH-USAGE TRAFFIC CAPACITY TABLES

High-usage traffic capacity tables, also known as alternate route trunking tables, are based on the Erlang B distribution. They are used to determine the traffic carried by each trunk in a trunk group, the cumulative traffic carried by the trunks, and the overflow traffic to the next trunk or to an alternate route. In this application, each added trunk carries less traffic than the preceding trunk; that is, it is less efficient insofar as traffic-carryingcapacity is concerned. The traffic load carried by the last trunk is important in determining the number of trunks required for a particular high-usage trunk group. The arrangementof data in high-usage trafficcapacity tables implies that traffic is always offered first to trunk I, then to trunk 2, and so forth. Modern switching systems, however, use trunk selection techniques that attempt to equalize the loadingon the individualtrunksin the trunkgroups. Nevertheless, the effect on the traffic-carrying capacity of a trunk group by the addition of another trunk will always be the same. Tables 3-3 through 3-8 are high-usage traffic capacity tables for trunk groups containing up to 24 trunks. Telephone operating companies may use tables applicable to trunk groups containing over a hundred trunks. The following examples illustrate typical table usage: Example 3-4 Consider a traffic load of 100 CCS offered to a group of 5 high-usage trunks. Table 3-3 (pages 50-55) indicates that trunk 1 will carry 26 CCS and overflow 74 CCS to trunk 2. Trunk 2 will carry 23 CCS and overflow 51 CCS to trunk 3. Trunk 3 will carry 19 CCS and overflow 32 CCS to trunk 4. Trunk 4 will carry 14 CCS and overflow 18 CCS to trunk 5. Referring to Table 3-4 (pages 56-60), for an offered trunk group traffic load of 100 CCS, trunk 5 will carry 9 CCS and 9 CCS will overflow to an alternate route.

Example 3-5 Determine the traffic load carried by the last trunk, total carried traffic, trunkgroup overflow traffic, and grade of service for a 10-channel trunk group offered 300 CCS of traffic. In Table 3-5 (pages 61-67), select the A row for 300 CCS and read over to the trunk 10 column to determine that the traffic carried by trunk lOis 16 CCS, total carried traffic is 259 CCS, and overflow traffic is 41 CCS. Grade of service (GOS) = 41 CCS/300 CCS = 0.137.

Chapter 3


49

Example 3-6 For the trunk group of Example 3-5, determine how many trunks must be added to obtain a grade of service of B.1 or better (overflow traffic equals 30 CCS or less). In Table 3-5, select the A row for 300 CCS and read over until 28 CCS overflow traffic is found (28 CCS/300 CCS = 0.093). Read up that column to determine that 11 trunks are required.

SO

Chapter 3


Table 3-3. High-Usage Traffic Capacity in CCS (Trunks 1 to 4) Trunk 1 Offered Traffic (A)

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44

Carried Trunk Total 4 5 6 6 7 8 8 9 10 10 11 11 11 12 12 13 13

14 14 14 15 15 15 16 16 16 17 17 17 18 18 18 18 19 19 19 19 19 19 20

4 5 6 6 7 8 8 9 10 10 11 11 11 12 12 13 13 14 14 14 15 15 15 16 16 16 17 17 17 18 18 18 18 19 19 19 19 19 19 20

Trunk 2 Overflow 1 1 1 2 2 2 3 3 3 4 4 5 6 6 7 7 8 8 9 10 10 11 12 12 13 14 14 15 16 16 17 18 19 19 20 21 22 23 24 24

Carried Trunk Total

3 4 5 5 5 5 6 6 6 7 7

7 8 8 9 9 9 9 10 10 10 11 11 11 11

12 13 13 13 13

14 15 16 17 17 18 19 20 20 21 22 22 23 24 25 25 26 26 27 28 28 29 29 30 30 31 32 32 32 33

Trunk 3 Overflow

1 1 1 1 2 2 2 2 3 3 3 4 4 4 4 5 5 6 6 6 7 7 8 8 9 9 9 10 11 11

Carried Trunk Total

4 4 4 4 4 5 5 5 5 6 6

6 6 7 7

29 30 30 31 32 33 34 34 35 36 37 38 38 39 40

Trunk 4 Overflow

Carried Trunk Total

Overflow

1

1 2 2 2 2 2

3 3 3 3 3 4 4 4 (table continues)


Chapter 3

51

Table 3-3. High-Usage Traffic Capacity in CCS (Trunks 1 to 4 Continued) Trunk 2

Trunk 1 Offered Traffic (A)

45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84

Carried Trunk Total 20 20 20 21 21 21 21 21 21 22 22 22 22 22 23 23 23 23 23 23 23 23 23 23 24 24 24 24 24 24 24 24 25 25 25 25 25 25 25 25

20 20 20 21 21 21 21 21 21 22 22 22 22 22 23 23 23 23 23 23 23 23 23 23 24 24 24 24 24 24 24 24 25 25 25 25 25 25 25 25

Overflow 25 26 27 27 28 29 30 31 32 32 33 34 35 36 36 37 38 39 40 41 42 43 44 45 45 46 47 48 49 50 51 52 52 53 54 55 56 57 58 59

Carried Trunk Total 14 14 14 14 14

34 34 34 35 35

15 15 15 16 16 16 16 17 17 17 17 17 17 17 18 18 18 19 19 19 19 19 19 20 20 20 20 20 20 20 20 21 21 21 21

36 36 36 37 37 38 38 39 39 40 40 40 40 40 41 41 41 42 42 43 43 43 43 44 44 44 44 45 45 45 45 46 46 46 46

Trunk 4

Trunk3 Overflow 11 12 13 13 14 14 15 16 16 17 17 18 18 19 19 20 21 22 23 23 24 25 25 26 26 27 28 29 29 30 31 32 32 53 34 35 35 36 37 38

Carried Trunk Total 7 7 8 8 8 8 9 9

9 10 10 10 10 10 10 11 11 12 12 12 12 13 13 13 13 13 13 14 14 14 14 15 15 15 15 15 15 16 16 16

41 41 42 43 43 44 45 45 46 47 48 48 49 49 50 51 51 52 52 53 53 54 55 55 56 56 56 57 58 58 58 59 60 60 60 60 61 62 62 62

Overflow 4 5 5 5 6 6 6 7 7 7 7 8 8 9 9 9 10 10 11 11 12 12 12 13 13 14 15 15 15 16 17 17 17 18 19 20 20 20 21 22

Carried Trunk Total

4 5 5 5 5 5 5 6 6 6 6 6 7 7 7 7 7 7 7

8 8 8 8 9 9 9 9 9 10 10 10 10 11 11

49 50 51 52 53 53 54 55 56 57 57 58 59 60 60 61 62 62 63 64 64 65 66 67 67 68 69 69 70 70 71 72 72 73

Overflow

2 2 2 2 2 3 3 3 3 3 4 4 4 4 5 5 5 6 6 6 7 7 7 7 8 8 8 9 9 10 10 10 11 11

(table continues)

52

Chapter 3




85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 102 104 106 108 110 112 114 116 118 120 122 124 126 128 130 132 134 136 138 140 142 144 146 148


25 25 26 26 26 26 26 26 26 26 26 26 26 26 26 26 27 27 27 27 27 27 27 27 27 28 28 28 28 28 28 28 28 28 29 29 29 29 29 29

Overflow 60 61 61 62 33 64 65 66 67 68 69 70 71 72 73 74 75 77 79 81 83 85 87 89 91 92 94 96 98 100 102 104 106 108 109 111 113 115 117 119


46 46 47 47 48 48 48 48 48 48 49 49 49 49 49 49 50 50 50 51 51 51 51 52 52 53 53 53 53 54 54 54 54 54 55 55 55 55 56 56

Trunk 4

Trunk 3 Overflow 39 40 40 41 41 42 43 44 45 46 46 47 48 49 50 51 52 54 56 57 59 61 63 64 66 67 69 71 73 74 76 78 80 82 83 85 87 89 90 92


63 63 64 64 65 65 65 65 66 66 67 67 67 68 69 68 69 69 70 71 71 72 72 73 73 74 75 75 75 76 76 77 77 77 78 78 79 79 80 80

Overflow 22 23 23 24 24 25 26 27 27 28 28 29 30 30 31 32 23 35 36 37 39 40 42 43 45 46 47 49 51 52 54 55 57 59 60 62 63 65 66 68


74 74 75 75 76 77 77 77 78 79 80 80 80 81 81 82 83 84 85 86 87 88 88 89 90 91 92 92 93 94 95 96 96 97 98 98 99 99 100 100

Overflow 11 12 12 13 13 13 14 15 15 15 15 16 17 17 18 18 19 20 21 22 23 24 26 27 28 29 30 32 33 34 35 36 38 39 40 42 43 45 46 48

(table continues)

Chapter 3

53


Table 3-3. High-Usage Traffic Capacity in CCS (Trunks 1 to 4 Continued) Trunk 1 Offered Traffic (A)

Carried Trunk Total

Trunk 2 Overflow

Carried Trunk Total

Trunk 3 Overflow

Carried Trunk Total

Trunk 4 Overflow

Carried Trunk Total

Overflow

150 152 154 156 158

29 29 29 29 30

29 29 29 29 30

121 123 125 127 128

27 27 27 27 27

56 56 56 56 57

94 96 98 100 101

24 25 25 25 25

80 81 81 81 82

70 71 73 75 76

21 21 21 21 21

101 102 102 102 103

49 50 52 54 55

160 162 164 166 168

30 30 30 30 30

30 30 30 30 30

130 132 134 136 138

27 27 27 27 27

57 57 57 57 57

103 105 107 109 111

25 25 25 26 26

82 82 82 83 83

78 80 82 83 85

22 24 22 22 23

104 104 104 105 106

56 58 60 61 62

170 172 174 176 178 180 182 184 186 188

30 30 30 30 30

30 30 30 30 30

113 114 116 118 120

26 26 26 26 26

83 84 84 84 84

87 88 90 92 94

23 23 23 24 24

106 107 107 108 108

64 65 67 68 70

30 30 30 30 30

27 28 28 28 28 29 29 29 29 29

57 58 58 58 58

30 30 30 30 30

140 142 144 146 148 150 152 154 156 158

59 59 59 59 59

121 123 125 127 129

26 26 26 26 27

85 85 85 85 86

95 97 99 101 102

24 24 24 24 24

109 109 109 109 110

71 73 75 77 78

190 192 194 196 198 200 202 204 206 208 210 212 214 216 218 220 222 224 226 228

30 30 30 30 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31

30 30 30 30 31 31 31 31 31 31 31 31 31 31 31

160 162 164 166 167 169 171 173 174 177 179 181 183 185 187

29 29 29 29 29 29 29 29 29 29 29 29 29 29 30

59 59 59 59

131 133 135 137 138 140 142 144 146 148 150 152 154 156 157

27 27 27 27 27 27 27 28 28 28 28 28 28 28 28

86 86 86 86 87 87 87 88 88 88 88 88 88 88 89

104 106 108 110 111 113 115 116 118 120 122 124 126 128 129

24 24 25 25 25 25 25 25 25 25 26 26 26 26 26

110 110 111 111 112 112 112 113 113 113 114 114 114 114 115

80 82 83 85 88 90 91 93 95 96 98 100 102 103

31 31 31 31 31

189 191 193 195 197

30 30 30 30 30

159 161 163 165 167

28 28 28 28 29

89 89 89 89 90

131 133 135 137 138

26 26 27 27 27

115 115 116 116 117

105 107 108 110 111

60

60 60 60 60 60 60 60 60 60 61 61 61 61 61 61

86

(table continues)

Chapter 3

54


Table 3-3. High-Usage Traffic Capacity in CCS (Trunks 1 to 4 Continued)

Offered Traffic

Carried

(A)

Trunk Total

230 232 234 236 238 240 242 244 246 248 250 252 254 256 258 260 262 264 266 268 270 272 274 276 278 280 282 284 286 288 290 292 294 296 298 300 305 310 315 320

31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32

31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32

Trunk 3

Trunk 2

Trunk 1 Overflow 199 201 203 205 207 209 211 213 215 217 219 221 223 225 227 229 231 233 234 236 238 240 242 244 246 248 250 252 254 256 258 260 262 264 266 268 273 278 283 288


61 61 61 61 61 61 61 62 62 62 62 62 62 62 62 62 62 62 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 64 64 64 64

Overflow 169 171 173 175 177 179 181 182 184 186 188 190 192 194 196 198 200 202 203 205 207 209 211 213 215 217 219 221 223 225 227 229 231 233 235 237 241 246 251 256


90 90 90 90 90 90 90 91 91 91 91 91 91 91 92 92 92 92 93 93 93 93 93 93 93 93 93 93 93 93 94 94 94 94 94 94 95 95 95 95

Trunk 4 Overflow 140 142 144 146 148 150 152 153 155 157 159 161 163 165 166 168 170 172 173 175 177 179 181 183 185 187 189 191 193 195 196 198 200 202 204 206 210 215 220 225


117 117 117 117 118 118 118 119 119 119 119 119 119 119 120 120 120 120 121 121 121 121 121 121 122 122 122 122 122 122 122 122 122 122 122 122 124 124 125 125

Overflow 113 115 117 119 120 122 124 125 127 129 131 133 135 137 138 140 142 144 145 147 149 151 153 155 156 158 160 162 164 166 167 169 171 173 175 177 181 186 190 195

(table continues)


Chapter 3

55



325 330 335 340 345 350 355 360 365 370 375 380 385 390 395 400 405 410 415 420 425 430 435 440 445 450 455 460 465 470

Carried Trunk Total 32 32 33 33 33 33

32 32 33 33 33 33

Overflow 293 298 302 307 312 317

Carried Trunk Total

Trunk 3 Overflow

Carried Trunk Total

32 32 32 32 32 32 32 32 32 32

64 64 65 65 65 65 65 65 65 65

261 266 270 275 280

31

285 290 295 300 305

32 32 32 32 33 33

65 65 65 65 66 66

310 315 320 325 329 334

31 31 31 31 32 32 32 32 32 32 32 32 32 32 32 32 32 33 33 33

31

31 31 31

95 95 96 96 96 96 96 96 96 97 97 97 97 97 98 98 98 98 98 98 98 98 99 99 99

Trunk 4 Overflow 230 235 239 244 249 254 259 264 269 273 278 283 288 293 297 302 307 312 317 322 327 332 336 341 346

Carried Trunk Total 30 30 30 30 30 31 31 31 31 31 31 31 31 31 31 31 31 31 32 32 32 32 32 32 32 32 32 32 32 32

125 125 126 126 126 127 127 127 127 128 128 128 128 128 129 129 129 129 130 130 130 130 131 131 131 131 131 131 131 131

Overflow 200 205 209 214 219 223 228 233 238 242 247 252 257 262 266 271 276 281 285 290 295 300 304 309 314 319 324 329 334 339

56

Chapter 3


Table 3-4. High-Usage Traffic Capacity in CCS (Trunks 5 to 8) Trunk 5 Offered Traffic

Carried

(A)

Trunk Total

Trunk 6 Overflow

Carried Trunk Total

Trunk 7 Overflow

67 68 69 70 71

3 4 4 4 4

65 66 67 68 68

3 2 2 2 3

72 73 74 75 76

4 4 4 5 5

69 70 71 72 73

3 3 3

77 78 79 80 81

5 5 5 6 6

74 74 75 76 77

3 4 4 4 4

82 83 84 85 86

6 6 6 6 7

78 78 79 80 81

4 5 5 5 5

87 88 89 90 91

7 7 7 7 7

82 82 83 84 84

5 6 6 6 7

4 4 4 4

86 87 88 88

2 2 2 3

92 93 94 95 96

8 8 8 8 8

85 86 87 88 88

7 7 7 7 8

4 4 4 4 5

89 90 91 92 93

3 3 3 3

Carried Trunk Total

Trunk 8 Overflow

Carried Trunk Total

Overflow

3 3

3

97 98 99 100 102

8 8 8 9 9

88 89 90 91 92

9 9 9 9 10

5 5 5 5 6

93 94 95 96 98

4 4 4 4 4

104 106 108 110 112

9 10 10 10 11

93 95 96 97 99

11 11 12 13 13

6 6 6

99 101 102 104 106

5 5 6 6 6

7 7

3

3

107 109

3

3 (table continues)

57

Erlang 8 Distribution

Chapter 3


Offered Traffic (A)

Carried Trunk Total

Trunk7

Trunk6

Trunk 5 Overflow

Carried Trunk Total

Overflow

9 9

114 116 117 118 120

10 10 11 12 12

5 5 6 6 6

119 121 123 124 126

5 5 5 6 6

13 14 14 15 16 17 18 19 19 20

6 7 7 7 8 8 8 9 9 9

127 129 131 132 134

7 7 7 8 8

4 4 4 4 4

12 12 12 13 13

121 122 124 125 126 127 128 129 131 132

135 136 138 140 141

9 10 10 10 11

5 5 5 5 5

13 14 14 14

133 134 135 136

21 22 23 24

9 9 10 10

142 143 145 146

12 13 13 14

148 149 151 153 154

8

156 157 159 160 162 163 165 166 167 169

8 9 9 10 10 11 11 12 13

170 172 173 174 175

14 14 15 16 17

14 15 16 17 17

7 7 8 8 8

107 108 110 111 113

7 8 8

124 126 128 130 132

13 14 14 14 14

105 107 108 109 110

19 19 20 21 22

9 9 9 9 10

134 136 138 140 142

15 15 15 16 16

111 112 113 114 115

23 24 25 26 27

10 10 11 11 11

144 146 148 150 152

16 16 17 17 17

115 116 117 118 119

29 30 31 32 33

154 156 158 160

17 18 18 18

119 120 121 122

35 36 37 38 39 41 42 43 45 46 47 48 50 51 53 54 56 57 58 60

19

123

123 124 125 125 126 127 128 128 129 129 130 130 131 132 132

Trunk Total

Overflow

3 4 4 4 4

100 101 102 103 105

19 19 19 19 19 20 20 20 20 20 21 21 21 22 22

Carried

111 112 114 116 118

12 12 12 12 13

162

Trunk Total

Overflow

4 4 4 5 5

114 116 118 120 122

164 166 168 170 172 174 176 178 180 182 184 186 188 190 192

Carried

Trunk8

14

137

25

10

147

15

15 15 15 16 16 16 16 17 17 17

138 139 140 141 142 143 144 145 146 146

26 27 28 29 30 31 32 33 34 36

11 10 11 11 12 12 12 12 12 13

149 150 151 152 154 155 156 157 158 159

15 16 17 18 18 19 20 21 22 23

6 6 6 7 7 7 7 8 8 8 8 9 9 9 10

17 18 18 18 18

147 1.48 149 150 150

37 38 39 40 42

13 14 14 14 15

160 162 163 164 165

24 24 25 26 27

10 10 10 10 10

131 133 135 136 138 140 141 143 145 146

3 3 3 4 4 4 5 5 5 6 6 7 7 7

13

(table continues)

Chapter 3

58



Offered Traffic (A)

194 196 198 200 202 204 206 208 210 212 214 216 218 220 222 224 226 228 230 232 234 236 238 240 242 244 246 248 250 252 254 256 258 260 262 264 266 268 270 272

Carried Trunk Total

Trunk 7

Trunk6

Trunk5 Overflow

22 22 22 22 23

133 133 134 134 135

61 63 64 66 67

23 23 23 23 23 24 24 24 24 24 24 24 24 24 24 25 25 25 25 25 25 25 25 25 25 26 26 26 26 26 26 26 26 26 27

136 136 136 137 137 138 138 139 139 139 140 140 141 141 141 142 142 143 143 143 144 144 144 144 144 145 145 146 146 146 146 147 147 147 148

68 70 72 73 75 76 78 79 81 83 84 86 87 89 91 92 94 95 97 99 100 102 104 106 108 109 111 112 114 116 118 119 121 123 124


Overflow

Carried Trunk Total

151 152 153 153 154

43 44 45 47 48

15 15 15 16 16

155 155 156 157 157 158 159 160 160 160 161 162 163 163 163 164 164 165 165 166 167 167 168 168 168 169 169 170 170 171 171 172 172 172 173

49 51 52 53 55 56 57 58 60 62 63 64 65 67 69 70 72 73 75 76 77 79 80 82 84 85 87 88 90 91 93 94 96 98 99

16 16 16 16 17 17 17 17 18 18 18 18 18 19 19 19 20 20 20 20 20 20 20 21 21 21 21 21 22 22 22 22 22 22 22

166 167 168 169 170 171 171 172 173 174 175 176 177 178 178 179 180 181 182 182 183 184 185 185 186 187 187 188 189 189 190 190 191 192 193 193 194 194 194 195

Trunk8 Overflow 28 29 30 31 32 33 35 36 37 38 39 40 41 42 44 45 46 47 48 50 51 52 53 55 56 57 59 60 61 63 64 66 67 68 69 71 72 74 76 77


177 178 179 181 182 183 184 185 186 187 188 190 191 192 193 194 195 196 197 198 199 200 201 201 203 204 204 205 206 207 208 208 209 210 211 212 213 213 214 215

Overflow 17 18 19 19 20 21 22 23 24 25 26 26 27 28 29 30 31 32 33 34 35 36 37 39 39 40 42 43 44 45 46 48 49 50 51 52 53 55 56 57

(table continues)


Chapter 3

59


Offered Traffic (A)

274 276 278 280

282 284 286 288 290 292 294 296 298 300 305 310 315 320 325 330 335 340 345 350 355 360 365 370 375 380 385 390 395 400 405 410 415 420 425 430

Carried Trunk Total

Trunk7

Trunk6

Trunk 5 Overflow

Carried Trunk Total

Overflow

Carried Trunk Total

Trunk8 Overflow

Carried Trunk Total

Overflow

27 27 27 27 27

148 148 149 149 149

126 128 129 131 133

25 25 25 25 26

173 173 174 174 175

101 103 104 106 107

23 23 23 23 23

196 196 197 197 198

78 80 81 83 84

20 20 20 21 21

216 216 217 218 219

58 60 61 62 63

27 27 27 27 27 28 28 28 28 28 28 28 29 29 29 29 29 29 29 30 30 30 30 30 30 31 31 31 31 31

149 149 149 150 150 151 151 151 151 152 152 153 154 154 154

26 26 26 26 26 26 26 26 27 27 27 27 27 27 28 28 28 28 28 28 29 29 29 29 29 29 29 29 29 29

175 175 175 176 176 177 177 177 178 179 179 180 181 181 182 183 183 183 184 185 186 186 187 187 187 188 188 189 189 189

109 111 113 114 116 117 119 121 122 126 131 135 139 144 148

155 155 155 156 157 157 157 158 158 158 159 159 160 160 160

135 137 139 140 142 143 145 147 149 153 158 162 166 171 176 180 185 190 194 198 203 208 212 217 222 226 231 235 240 245

152 157 162 166 170 164 169 183 188 193 197 202 206 211 216

23 23 24 24 24 24 24 24 24 24 25 25 25 26 26 26 27 27 27 27 27 27 27 28 28 28 28 28 29 29

198 198 199 200 200 201 201 201 202 203 204 205 206 207 208 209 210 210 211 212 213 213 214 215 215 216 216 217 218 218

86 88 89 90 92 93 95 97 98 102 106 110 114 118 122 126 130 135 139 143 147 152 156 160 165 169 174 178 182 187

219 219 220 221 221 222 222 223 224 225 227 228 229 230 232 233 234 235 236 237 238 239 240 241 242 243 243 244 245 246

65 67 68 69 71 72 74 75 76 80 83 87 91 95 98 102 106 110 114 118 122 126 130 134 138 142 147 151 155 159

31 31 31 31 31

160 161 161 161 161

250 254 259 264 269

30 30

190 191 191 191 191

220 224 229 234 239

29 29 29 29 30

219 220 220 220 221

191 195 200 205 209

21 21 21 21 21 21 21 22 22 22 23 23 23 23 24 24 24 25 25 25 25 26 26 26 27 27 27 27 27 28 28 28 28 28 28

247 248 248 248 249

163 167 172 177 181

30

30 30

(table continues)

60

Chapter 3



435 440 445 450 455 460 465 470 475 480 485 490 495 500 505 510 515 520 525 530 535 540 545 550 555 560 565 570 575 580

Carried Trunk Total 31 31 31 31 31 32 32 32 32 32 32 32 32 32 32 32 32 32 32

162 162 162 162 162 163 163 163 163 164 165 165 165 165 165 165 165 165 165

Trunk 6

Overflow 273 278 283 288 293 297 302 307 312 316 320 325 330 335 340 345 350 355 360

Carried Trunk Total 30 30 30 31 31 31 31 31 31 31 31 31 31 31 31 31 31 32 32 32 32

192 192 192 193 193 194 194 194 194 195 196 196 196 196 196 196 196 197 197 197 197

Trunk 7 Overflow 243 248 253 257 262 266 271 276 281 285 289 294 299 304 309 314 319 323 328 333 338

Carried Trunk Total 30 30 30 30 30 30 30 30 30 30 30 30 30 30 31 31 31 31 31 31 31 31 31 31

222 222 222 223 223 224 224 224 224 225 226 226 226 226 227 227 227 228 228 228 228 229 229 229

Trunk 8 Overflow 213 218 223 227 232 236 241 246 251 255 259 264 269 274 278 283 288 292 297 302 307 311 316 321

Carried Trunk Total 28 29 29 29 29 29 29 30 30 30 30 30 30 30 30 30 30 30 30 31 31

31 31 31 31 31 31 31 31 31

250 251 251 252 252 253 253 254 254 255 256 256 256 256 257 257 257 258 258 259 259 260 260 260 261 261 261 262 262 262

Overflow 185 189 194 198 203 207 212 216 221 225 229 234 239 244 248 253 258 262 267 271 276 280 285 290 294 299 304 308 313 318


Chapter 3

61

Table 3-5. High-Usage Traffic Capacity in CCS (Trunks 9 to 12)

Offered Traffic (A)

Carried Trunk Total

156 158 160 162 164

4 4 4

166 168 170 172 174 176 178 180 182 184 186 188 190 192 194 196 198 200 202 204 206 208 210 212 214 216 218 220 222 224 226 228 230 232 234

4 4 5

4 4

5

5 5

6 6 6 6 6 7 7 7 7 8 8 8 8 9 9 9 10 10 10 10 10 10 11 11 11 12 12 12 12

153 155 157 158 160 161 163 165 167 168 170 172 173 175 176 178 180 181 182 184 186 187 189 190 192 193 194 196 197 198 200 201 202 204 205 206 208 209 210 211

Trunk 11

Trunk 10

Trunk9 Overflow

Carried Trunk Total

Overflow

Carried Trunk Total

Trunk 12 Overflow

Carried Trunk Total

Overflow

3 3 3 4 4

5 5 5 5 6 6 6 7 7 8 8 8 9 10 10 10 11 11 12 12 13 14 14 15 16 16 17 18 18 19 20 20 21 22 23

4 4 4

4 4 5

5 5 5

5 5 5

6 6 6 6 7

7 7 7

7 8

8 8 8

9 9

179 180 182 184 185 187 189 191 192 194 195 197 199 200 202 203 205 207 208 209 211 213 214 216 217 219 220

3 4 4 4 5 5 5 5

6 6 7 7 7 8

3 4

8

4

9 9

4 4

9

4

10 11 11 11 12 12 13 13 14

5

5 5

5 5

5 6 6 6

202 204 206 207 209 211 213 214 216 218 219 221 223 225 226

4

4 4

5 5 5 5

6 6 6

7 7 7 7 8

3

4

228 230

4 4

(table continues)

Chapter 3

62



236 238 240 242 244 246 248 250 252 254 256 258 260 262 264 266 268 270 272 274 276 278 280 282 284 286 288 290 292 294 296 298 300 305 310 315 320 325 330 335


212 214 214 216 217 218 219 220 221 222 223 224 225 226 227 229 229 230 231 232 233 234 235 236 236 237 238 239 238 241 241 242 243 245 247 248 250 251 253 255

Trunk 11

Trunk10 Overflow 24 24 26 26 27 28 29 30 31 32 33 34 35 36 37 37 39 40 41 42 43 44 45 46 48 49 50 51 52 53 55 56 57 60 63 67 70 74 77 80

Carried Trunk Total 9

9 10 10 10 10 11 11 11 11 12 12 12 12 12 12 13 13 13 14 14 14 14 14 15 15 15 15 15 15 16 16 16 16 17 18 18 19 19 19

221 223 224 226 227 228 230 231 232 233 235 236 237 238 239 241 242 243 244 246 247 248 249 250 251 252 253 254 255 256 257 258 259 261 264 266 268 270 272 274

Overflow 15 15 16 16 17 18 18 19 20 21 21 22 23 24 25 25 26 27 28 28 29 30 31 32 33 34 35 36 37 38 39 40 41 44 46 49 52 55 58 61

Carried Trunk Total 227 229 231 233 234 7 235 7 237 7 238 8 240 8 241 8 243 8 244 9 246 9 247 9 248 9 250 9 251 10 253 10 254 10 256 10 257 10 258 10 259 11 261 11 262 11 263 11 264 12 266 12 267 12 268 12 269 12 270 13 272 14 275 14 278 14 280 15 283 16 286 16 288 16 290 6 6 7 7 7

Trunk 12

Overflow 9 9 9 9 10 11 11 12 12 13 13 14 14 15 16 16 17 17 18 18 19 20 21 21 22 23 24 24 25 26 27 28 28 30 32 35 37 39 42 45


231 233 235 237 238 240 242 243 245 247 249 250 252 253 255 257 258 260 261 263 264 266 267 269 270 271 273 275 276 278 279 280 282 285 288 291 294 298 301 303

Overflow 5 5 5 5 6 6 6 7 7 7 7 8 8 9 9 9 10 10 11 11 12 12 13 13 14 15 15 15 16 16 17 18 18 20 22 24 26 27 29 32

(table continues)

63


Chapter 3


Trunk 9 Offered

Carried

Traffic (A)

Trunk Total

Overflow

340 345 350 355 360 365 370 375 380 385 390 395 400 405 410 415 420 425 430 435 440 445 450 455 460 465 470 475 480 485 490 495 500 505 510

22 23 23 23 24 24 24 24 25 25 26 26 26 26 26 26 26 27 27 27 27 27 27 27 28 28 28 28 28 28 28 29 29 29 30

256 258 259 260 262 263 264 265 267 268 269 270 271 272 273 274 274 275 276 277

284 285 285 286 287

121 125 129 133 137 141 146 150 154 158 162 167 171 175 179 184 188 193 197 201 206 210 215 219 223

515 520 525 530 535

30 30 30 30 30

287 288 288 289 289

228 232 237 241 246

278 278 279 280 281 281 282 282 283 284

84 87 91 95 98 102 106 110 113 117

Carried Trunk Total

Trunk 11 Overflow

Carried Trunk Total

19 20 20 21 21 21 22 22 22 22 23 23 23 23 24 24 25 25 25 25 25 26 26 26 26 27 27 27 27 27

275 278 279 281 283 284 286 287 289 290 292 293 294 295 297 298 299 300 301 302 303 304 305 306 307 308 309 309 310 311

65 67 71 74 77 81 84 88 91 95 98 102 106 110 113 117 121 125 129 133 137 141 145 149 153 157 161 166 170 174

28 28 28 28 28

312 313 313 314 315

178 182 187 191 195

20 21 22 22 22 22 23 23 24 24 24 24 24 24 25 25 25 26 26 26 26 26 27 27 27

28 28 28 28 28

315 316 316 317 317

200 204 209 213 218

27 27 28 28 28

17 17 18 18 18 19 19 20 20 20

Trunk 12

Overflow

Trunk Total

95 98 102 105 109

14 14 14 15 16 16 17 17 17 18 18 19 19 20 20 20 20 21 21 21

338 339 340 341 342

113 117 121 125 128 132 136 140 144 148 152 156 160 164 168

22 22 23 23 23 23 23 23 24 24 24 24 25 25 25

342 343 344 345 345

173 177 181 185 190

26 26 26 26 27

292 295 297 299 301 303 305 307 309 310 312 314 316 317 319 320 322 323 325 326 327 328 329 330 332 333 334 335 336 337

48 50 53 56 59 62 65 68 71 75

Carried

78 81 84 88 91

306 309 311 314 317 319 322 324 326 328 330 333 335 337 339 340 342 344 346 347 349 350 352 353 355 356 357 358 360 361 362 363 365 366 367 368 369 370 371 372

Overflow 34 36 39 41 43 46 48 51 54 57 60 62 65 68 71 75 78 81 84 88 91 95 98 102 105 109 113 117 120 124 128 132 135 139 143 147 151 155 159 163

(table continues)

64

Chapter3


Table 3-5. High-Usage Traffic Capacity in CCS (Trunks 9 to 12 Continued) Trunk 9 Offered Traffic

Carried

(A)

Trunk Total

540 545 550 555 560 565 570 575 580 585 590 595 600 605 610 615 620 625 630 635 640 645 650 655 660 665 670 675 680 685 690 695 700 705 710

30 30 30 30 30 30 30 30 30 30 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31

290 290 290 291 291 291 292 292 292 293 294 294 294 294 295 296 296 296 296 296 297 297 297 298 298

Trunk 10 Overflow 250 255 260 264 269 274 278 283 288 292 296 301 306 311 315 319 324 329 334 339 343 348 353 357 362

Carried Trunk Total 28 29 29 29 29 30 30 30 30 30 30 30 30 30 30 30 30 30 31 31 31 31 31 31 31 31 31 31 31 31

318 319 319 320 320 321 322 322 322 323 324 324 324 324 325 326 326 326 327 327 328 328 328 329 329 330 330 330 330 330

Trunk 12

Trunk 11 Overflow 222 226 231 235 240 244 248 253 258 262 266 271 276 281 285 289 294 299 303 308 312 317 322 326 331 335 340 345 350 355

Carried Trunk Total 28 28 28 28 28 28 28 29 29 29 29 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 31 31 31 31 31 31 31

346 347 347 348 348 349 350 351 351 352 353 354 354 354 355 356 356 356 357 357 358 358 358 359 359 360 360 361 361 361 362 362 363 363

Overflow 194 198 203 207 212 216 220 224 229 233 237 241 246 251 255 259 264 269 273 278 282 287 292 296 301 305 310 314 319 324 328 333 337 342

Carried Trunk Total 27 27 27 27 28 28 28 28 28 28 28 28 28 29 29 29 29 29 29 29 29 30 30 30 30 30 30 30 30 30 30 30 30 30 30

373 374 374 375 376 377 378 379 379 380 381 382 382 383 384 385 385 385 386 386 387 388 388 389 389 390 390 391 391 391 392 392 393 393 394

Overflow 167 171 176 180 184 188 192 196 201 205 209 213 218 222 226 230 235 240 244 249 253 257 262 266 271 275 280 284 289 294 298 303 307 312 316


Chapter 3

65

Table 3-6. High-Usage Traffic Capacity in CCS (Trunks 13 to 16) Trunk 14


258 260 262 264 266 268 270 272 274 276 278 280 282 284 286 288 290 292 294 296 298 300 305 310 315 320 325 330 335 340 345 350 355 360 365 370 375 380 385 390

Carried Trunk Total 4 4 4 4 4 4 4 5 5 5 5 5 5 6 6 6 6 6 6 6 7 7 7 8 9

9 9 10 11 11 11 12 12 13 13 13 14 15 15 16

254 256 257 259 261 262 264 266 268 269 271 272 274 276 277 279 281 282 284 285 287 289 292 296 300 303 307 311 314 317 320 323 326 330 332 335 338 341 343 346

Overflow 4 4 5 5 5 6 6 6 6 7 7 8 8 8 9 9 9 10 10 11 11 11 13 14 15 17 18 19 21 23 25 27 29 30 33 35 37 39 42 44

Carried Trunk Total

4 4 4 4 4 4 4 5 5 5 5 6 6 7 7 7

8 8

9 9 10 10 11 11 11 12 13 13

278 280 281 283 285 286 288 290 292 294 297 302 306 310 314 318 322 325 329 332 336 340 343 346 349 353 356 359

Trunk 15 Overflow

4 4 5 5 5 6 6 6 6 6 8 8 9

10 11 12 13 15 16 18 19 20 22 24 26 27 29 31

Carried Trunk Total

4 4 4 4 5 5 6 6 7

7 7 7 8

9 9 9 9

306 310 314 318 323 327 331 335 339 343 347 350 354 358 362 365 368

Trunk 16 Overflow

Carried Trunk Total

Overflow

4 5

6 7 7 8 9 10 11 12 13 15 16 17 18 20 22

4 4 5 5 5 6 6 6 6

7 8

335 339 344 348 352 356 360 364 368 372 376

5 6 6 7 8 9 10 11 12 13 14

(table continues)

66

Chapter 3



395 400 405 410 415 420 425 430 435 440 445 450 455 460 465 470 475 480 485 490 495 500 505 510 515 520 525 530 535 540 545 550 555 560 565 570 575 580 585 590


349 351 353 356 358 360 362 364 366 368 370 372 374 376 377 379 380 382 384 385 386 388 389 390 392 393 394 395 397 398 399 400 401 402 403 404 405 406 407 408

Trunk 14 Overflow 46 49 52 54 57 60 63 66 69 72 75 78 81 84 88 91 95 98 101 105 109 112 106 120 123 127 131 135 138 142 146 150 154 158 162 166 170 174 178 182


362 364 367 370 373 375 378 380 383 385 387 390 392 394 396 398 400 402 404 406 407 409 411 412 414 416 417 418 420 421 422 424 425 426 427 429 430 431 432 433

Trunk 15 Overflow 33 36 38 40 42 45 47 50 52 55 58 60 63 66 69 72 75 78 81 84 88 91 94 98 101 104 108 112 115 119 123 126 130 134 138 141 145 149 153 157


372 375 378 382 385 388 391 394 397 399 402 405 407 410 412 415 417 419 422 424 426 428 430 432 434 436 438 439 441 443 444 446 447 448 450 452 453 454 456 457

Trunk 16 Overflow 23 25 27 28 30 32 34 36 38 41 43 45 48 50 53 55 58 61 63 66 69 72 75 78 81 84 87 91 94 97 101 104 108 112 115 118 122 126 129 133


380 384 387 391 394 398 401 405 408 411 414 417 420 423 426 429 431 434 437 440 442 444 447 449 451 454 456 458 460 462 464 466 468 469 471 473 475 476 478 480

Overflow 15 16 18 19 21 22 24 25 27 29 31 33 35 37 39 41 44 46 48 50 53 56 58 61 64 66 69 72 75 78 81 84 87 91 94 97 100 104 107 110

(table continues)

67


Chapter 3



595 600 605 610 615 620 625 630 635 640 645 650 655 660 665 670 675 680 685 690 695 700 705 710 715 720 725 730 735 740 745 750 755 760 765 770 775 780 785 790

Carried Trunk Total 27 27 27 27 27 28 28 28 29 29 29 29 29 29 29 29 29 29 29 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30

409 409 410 411 412 413 413 414 415 416 417 417 418 418 419 419 420 421 421 422 422 423 423 424 425 425 425 426 426 427 427 428 428 428

Overflow 186 191 195 199 203 207 212 216 220 224 228 233 237 242 246 251 255 259 264 268 273 277 282 286 290 295 300 304 309 313 318 322 327 332

Carried Trunk Total 25 26 26 26 26 26 27 27 27 27 27 27 27 28 28 28 28 28 28 28 29 29 29 29 29 29 29 29 29 29 30 30 30 30 30 30 30 30 30

434 435 436 437 438 439 440 441 442 443 444 444 445 446 447 447 448 449 449 450 451 452 452 453 454

454 454 455 455 456 457 458 458 458 459 459 460 460 460

Trunk 15 Overflow 161 165 169 173 177 181 185 189 193 197 201 206 210 214 218 223 227 231 236 240 244 248 253 257 261 266 271 275 280 284 288 292 297 302 306 311 315 320 325


458 459 461 462 463 465 466 467 468 469 470 471 472 473 474 474 475 476 477 478 479 480 480 481 482 482 483 484 484 485 486 487 487 487 488 488 489 490 490 491

Trunk 16

Overflow 137 141 144 148 152 155 159 163 167 171 175 179 183 187 191 196 200 204 208 212 216 220 225 229 233 238 242 246 251 255 259 263 268 273 277 282 286 290 295 299


481 482 484 485 486 488 490 491 492 493 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 509 510 511 512 513 514 515 515 516 517 517 518 519 519 520

Overflow 114 118 121 125 129 132 135 139 143 147 150 154 158 162 166 170 174 178 182 186 190 194 198 202 206 211 215 219 223 227 231 235 240 244 248 253 257 261 266 270

68

Chapter 3


Table 3-7. High-Usage Traffic Capacity in CCS (Trunks 17 to 20) Trunk 17 Offered Traffic (A)

365 370 375 380 385 390 395 400 405 410 415 420 425 430 435 440 445 450 455 460 465 470 475 480 485 490 495 500 505 510 515 520 525 530 535 540 545 550 555 560


360 364 368 373 377 381 385 390 393 397 401 405 409 413 417 420 424 427 430 434 437 440 443 446 450 453 455 458 461 464 466 469 472 474 477 479 481 484 486 488

Trunk 18 Overflow 5 6 7 7 8 9 10 10 12 13 14 15 16 17 18 20 21 23 25 26 28 30 32 34 35 37 40 42 44 46 49 51 53 56 58 61 64 66 69 72

Carried Trunk Total

4 4 4 5 5 5 6 6 6 6 7 7 8 8 8 9 9 10 10 10 11 11 12 12 12 13 13 13 14 14 15 15 15 16 16

385 389 394 398 402 406 411 415 419 423 427 431 435 438 442 446 449 453 456 460 464 466 470 473 476 479 482 485 488 491 494 496 499 502 504

Trunk 19

Overflow

5 6 6 7 8 9 9 10 11 12 13 14 15 17 18 19 21 22 24 25 26 29 30 32 34 36 38 40 42 44 46 49 51 53 56

Carried Trunk Total

4 4 4 5 5 5 5 6 6 6 7 7 8 8 8 9 9

10 10 10 11 11 11 12 12 13 13 13 14

415 419 423 428 432 436 440 444 448 452 456 460 464 468 472 475 479 483 486 489 493 496 499 503 506 509 512 515 518

Trunk 20 Overflow

5 6 7 7 8 9 10 11 12 13 14 15 16 17 18 20 21 22 24 26 27 29 31 32 34 36 38 40 42

Carried Trunk Total

4 4 4 4 5 5 5 5 6 6 7 7 7 7 8 8 9 9 9 10 10 10 11 11

440 444 448 452 457 461 465 469 474 478 482 486 490 493 497 501 505 508 512 516 519 522 526 529

Overflow

5 6 7 8 8 9 10 11 11 12 13 14 15 17 18 19 20 22 23 24 26 28 29 31

(table continues)

69


Chapter 3



565 570 575 580 585 590 595 600 605 610 615 620 625 630 635 640 645 650 655 660 665 670 675 680 685

690 695 700 705 710 715 720 725 730 735 740 745 750 755 760


490 492 494 496 498 500 502 503 505 507 508 510 512 513 515 516 518 519 520 522 523 524 526 527 528 529 531 532 533 534 535 536 537 538 539 540 541 542 543 544

Overflow 75 78 81 84 87 90 93 97 100 103 107 110 113 117 120 124 127 131 135 138 142 146 149 153 157 161 164 168 172 176 180 184 188 192 196 200 204 208 212 216


506 509 511 513 516 518 520 522 524 526 528 530 532 534 536 537 539 540 542 544 546 547 549 550 552 553 555 556 557 558 560 561 562 563 564 565 567 568 569 570

Trunk 19 Overflow 59 61 64 67 69 72 75 78 81 84 87 90 93 96 99 103 106 110 113 116 119 123 126 130 133 137 140 144 148 152 155 159 163 167 171 175 178 182 186 190


520 523 526 528 531 534 536 539 541 543 546 548 550 552 554 556 558 560 562 564 566 568 570 571 573 575 577 578 579 581 583 584 585 587 588 589 591 592 593 594

Trunk 20 Overflow 45 47 49 52 54 56 59 61 64 67 69 72 75 78 81 84 87 90 93 96 99 102 105 109 112 115 118 122 126 129 132 136 140 143 147 151 154 158 162 166


532 535 538 541 544 547 550 553 556 558 561 564 566 569 571 573 576 578 580 583 585 587 589 591 593 595 597 598 600 602 604 605 607 609 610 612 614 615 616 618

Overflow 33 35 37 39 41 43 45 47 49 52 54 56 59 61 64 67 69 72 75 77 80 83 86 89 92 95 98 102 105 108 111 115 118 121 125 128 131 135 139 142

(table continues)

70

Chapter3


Table 3-7. High-Usage Traffic Capacity in CCS (Trunks 17to 20 Continued) Trunk 17 Offered Traffic (A)

765 770 775 780 785 790 795 800 810 820 830 840 850 860 870 880 890 900 910 920

Carried Trunk Total 28 28 28 28 28 28 28 29 29 29 29 29 29 29 30 30 30 30

545 545 546 547 547 548 549 550 551 552 554 555 556 557 559 560 561 562

Overflow 220 225 229 233 238 242 246 250 259 268 276 285 294 303 311 320 329 338

Trunk 19

Trunk 18 Carried Trunk Total 26 27 27 27 27 27 27 27 28 28 28 28 29 29 29 29 29 29 30 30

571 572 573 574 574 575 576 577 579 580 582 583 585 586 588 589 590 591 593 594

Overflow 194 198 202 206 211 215 219 223 231 240 248 257 265 274 282 291 300 309 317 326

Carried Trunk Total 25 25 25 25 26 26 26 26 26 27 27 28 28 28 28 28 28 29 29 29

596 597 598 599 600 601 602 603 605 607 609 611 613 614 616 617 618 620 622 623

Trunk 20

Overflow 169 173 177 181 185 189 193 197 205 213 221 229 237 246 254 263 272 280 288 297

Carried Trunk Total 24 24 24 24 25 25 25 25 25 26 26 26 26 27 27 28 28 28 28 28

620 621 622 623 625 626 627 628 631 633 635 637 632 641 643 645 646 648 650 651

Overflow 145 149 153 157 160 164 168 172 179 187 195 203 211 219 227 235 244 252 260 269

Chapter 3

Erlang B DIstribution

71

Table 3-8. High-Usage Traffic Capacity in CCS (Trunks 21 to 24) Trunk 22


470 475 480 485 490 495 500 505 510 515 520 525 530 535 540 545 550 555 560 565 570 575 580 585 590

595 600 605 610 615 620 625 630 635 640 645 650 655 660 665

Carried Trunk Total 4 4 4 4 4 4 5 5 6

465 469 473 478 482 486 491 495 499 6 503 6 507 6 511 7 515 7 519 7 523 8 527 8 530 8 534 9 538 9 541 10 545 10 548 11 552 11 555 11 558 12 562 12 565 12 568 13 571 13 574 13 577 14 580 14 583 15 586 15 588 15 591 16 594 16 596 16 599 16 601

Over-

flow 5 6 7 7 8 9 9 10 11 12 13 14 15 16 17 18 20 21 22 24

Carried Trunk Total

4 4 4 4 5 5 5 5 6

Trunk 23 Overflow

Carried Trunk Total

Trunk 24 Overflow

9

495 499 503 507 512 516 520 524 529 533 537 541 545 548 552 556 560 564

6

558 562 566 570

32

9

567

23

7

574

16

33 35 37 39 41 43 45 47 49 52 54 56 59 61 64

9 10 10 11 11 11 12 12 12 13 13 13 14 14 14

571 575 578 582 585 588 592 595 598 601 604 607 610 613 615

24 25 27 28 30 32 33 35 37 39 41 43 45 47 47

7 7 8 8 8 9 9 9 10 10 10 11 11 11 12

578 582 586 590 593 597 601 604 608 611 614 618 621 624 627

17 18 19 20 22 23 24 26 27 29 31 32 34 36 36

25 27 28 30

6

7 7 7 7 7 8 8

Carried Trunk Total

Overflow

5 6 7 8 8

9 10 11 11 12 13 14

4 4 4 4 4 5

15

5

17 18 19 20 21

6 6 6 6

524 528 533 537 541 546 550 554

6 7 7 8 9 9 10 11 12 13 14 15

4 4 4 5 5 5

554 558 562 567 571 575

6 7 8 8 9 10

5

579

11

584 6 588 6 592 6 596 7 600 7 604 7 608 8 612 8 616 8 619 9 623 9 627 9 630 10 634 10 637 6

11 12 13 14 15 16 17 18 19 21 22 23 25 26 28

(table continues)

Chapter3

72




670 675 680 685 690 695 700 705 710 715 720 725 730 735 740 745 750 755 760 765 770 775 780 785 790 795 800 810 820 830 840 850 860 870 880 890 900 910 920 930


604 606 608 611 613 615 617 619 622 624 625 627 629 631 633 635 636 638 640 642 643 644 646 648 649 651 652 655 657 660 662 665 667 669 671 673 675 677 679 681

Overflow 66 69 72 74 77 80 83 86 88 91 95 98 101 104 107 110 114 117 120 123 127 131 134 137 141 144 148 155 163 170 178 185 193 201 209 217 225 233 241 249


618 621 623 627 629 631 634 636 639 641 643 645 648 650 652 654 656 658 660 662 663 665 667 669 670 672 674 677 680 683 686 689 692 694 696 699 701 703 705 707

Trunk 23 Overflow 52 54 57 58 61 64 66 69 71 74 77 80 82 85 88 91 94 97 100 103 107 110 113 116 120 123 126 133 140 147 154 161 168 176 184 191 199 207 215 223


631 634 636 640 642 645 648 651 654 656 659 661 664 667 669 671 674 676 678 680 682 684 686 688 690 692 694 698 702 705 708 712 715 718 720 723 726 728 730 733

Trunk 24 Overflow 39 41 44 45 48 50 52 54 56 59 61 64 66 68 71 74 76 79 82 85 88 91 94 97 100 103 106 112 118 125 132 138 145 152 160 167 174 182 190 197


641 644 647 651 654 657 660 663 667 669 673 675 678 681 684 686 689 692 694 697 699 701 704 706 708 711 713 717 721 725 729 733 736 740 743 746 749 752 754 757

Overflow 29 31 33 34 36 38 40 42 43 46 47 50 52 54 56 59 61 63 66 68 71 74 76 79 82 84 87 93 99 105 111 117 124 130 137 144 151 158 166 173

Chapter3

3.5

73


PEAKED TRAFFIC CAPACITY TABLES

Peaked traffic capacity tables, also known as Wilkinson or Neal-Wilkinson high-usagetrunk capacity tables, are based on the ErlangB distribution and the Neal-Wilkinson equivalent random theory. They are used in the North AmericanPSTN in lieu of the Poisson traffic capacitytables to compensate for system-induced peakedness caused by alternate-routing overflow traffic. Peakedness in offered traffic can also be caused by abnormally high traffic, suchas on Mother's Day and Christmas,or due to a disaster,stock marketcrash, or similar occurrence. The peakedness factor (2) for any trunk group is obtained by calculating the variance-to-mean ratio of the busy-hour traffic, as expressed in Equation 3-2. Peakedness factors of unity (one) indicate random (Poisson) traffic, whereas factors greater than unity indicate peaked traffic. Traffic with a peakedness factor less than unity indicates smooth traffic, which is treated as random traffic. (3.2) Z=~

a

where Z = peakedness factor v

= variance of the traffic data =L

2

(x- a)

(N - 1)

LX

a =arithmetic mean (average) of the sample values =N = individual values in the sample N = number of values in the sample X

Tables 3-9 (page 75) and 3.10 (page 76) are used for trunk groups engineered fora grade of service ofB.O1. Tables 3-11 (page 77) and 3-12 (page 78) are used for trunk groups engineered for a grade of service of B.02. The following example illustrates typical peakedness-factor determination and peaked traffic capacity table usage. Example 3-6 Determine the peakedness factor and the number of trunks required for a final trunkgroup in an alternateroutingsystemengineeredfor B.02service,wherethe busy-hour traffic statistics are as follows:

74

Chapter 3

Sample

x (CCS)

I 2 3 4 5 6 7 8

115 103 145 140 131 147 142

ill

Lx = 1035

a=

N:Ex

v=

L

Z

v


x- a

(x - a)2

-14.4 -26.4 +15.6 +10.6 +1.6 +17.6 +12.6 -17.4

207.4 697.0 243.4 112.4 2.6 309.8 158.8 302.8 L(x - a)2 =2034.2

1035 = -8- = 129.4CCS (x - a)

2

(N - 1)

= (2034.2) 7

= 290.6

290.6

= (i = 129.4 =2.2

Referringto Table 3-11,determinethat 12trunksare requiredto handleup to 139 CCS of offered traffic for a peakedness factor of 2.2.

Chapter 3

75


Table 3-9. 8.01 Peaked Traffic Capacity in CCS (Z No. of Trunks (N)

= 1.0 to 2.4)

Traffic (A) in CCS for Z =

1.0

1.2

1.4

1.6

2 4 6 8 10

5 31 69 113 161

19 55 97 142

8 41 81 125

29 69 111

17 55 97

4 43 83

12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

212 265 319 376 433 491 551 611 671 732 794 856 918 981 1044

191 242 295 350 406 463 521 580 638 698

173 223 275 327 382 437 494 551 610 668 727 787 847 908 969

156 204 255 306 360 414 470 526 582 639

140 188 237 287 339 392 447 501 557 613

698 757 816 875 936

670 728 787 845 905

126 171 219 269 320 371 425 478 534 589 645 702 759 817 875

42 44 46 48 50 52 54 56 58 60 64 68 72 76 80 84 88 92 96 100

1108 1172 1236 1300 1364 1429 1494 1559 1625 1690 1822 1954 2086 2220 2353

1030 1092 1154 1216 1279 1342 1405 1468 1532 1595 1723 1853 1982 2112 2243

996 1057 1118 1179 1241 1303 1365 1428 1491 1554 1681 1808 1936 2065 2194

964 1024 1085 1145 1206 1267 1329 1390 1452 1515 1640 1766 1893 2020 2148

934 993 1053 1113 1173 1233 1294 1355 1416 1478 1602 1727 1852 1979 2105

906 964 1023 1082 1141 1201 1262 1322 1383 1443 1566 1689 1814 1939 2065

879 936 994 1053 1112 1171 1230 1290 1350 1410 1532 1654 1777 1901 2025

2373 2505 2637 2770 2902

2324 2454 2585 2716 2847

2277 2406 2535 2665 2796

2233 2360 2489 2618 2747

2191 2317 2445 2573 2701

2151 2277 2403 2530 2657

2487 2621 2756 2891 3026

759 819 881 943 1004 1067 1130 1192 1256 1319 1383 1447 1511 1576 1640 1770 1901 2032 2163 2295 2428 2561 2694 2828 2962

1.8

2.0

2.2

29 68 111 156 203 250 301 351 404 457 510 565 621 677 733 790 848

2.4

15 55 97 140 186 234 282 332 384 436 489 543 597 652 708 764 821

76

Chapter 3


Table 3-10. 8.01 Peaked Traffic Capacity in CCS (Z = 2.6 to 4.0) No. of Trunks (N)

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 64 68 72 76 80 84 88 92 96 100

Traffic (A) in CCS for Z

2.6

2.8

3.0

42

28 69 111 155 201 248 296 346 396 448 500 554 607 661 716 772 828 884 941 998 1055 1113 1171 1230 1289 1348 1467 1587 1708 1830 1953 2076 2200 2324 2450 2575

55 97 141 185 231 279 328 378 428 480 532 586 639 694 748 804 859 916 972 1029 1086 1144 1202 1260 1319 1437 1556 1676 1797 1918 2041 2163 2287 2411 2536

83 126 170 216 265 314 364 416 468 521 575 629 685 740 796 852 909 967 1025 1083 1141 1201 1259 1319 1379 1499 1620 1742 1865 1988 2112 2237 2363 2489 2615

=

3.2

3.4

3.6

3.8

4.0

40 83 126 170 215 262 310 360 410 461 512 565 618 671 725 780 836 891 947 1003 1060 1117 1175 1233 1291 1408 1526 1645 1764 1885 2007 2129 2252 2375 2499

26 68 110 154 200 246 293 342 392 442 493 545 597 650 704 757 812 867 923 979 1035 1091 1149 1206 1264 1380 1498 1614 1733 1853 1974 2095 2217 2339 2462

53 96 140 184 230 277 325 373 423 473 525 576 629 682 736 790 845 899 955 1010 1066 1123 1180 1237 1352 1468 1585 1703 1822 1942 2062 2183 2305 2427

39 82 125 169 214 260 308 356 405 455 506 557 609 661 715 768 822 876 931 987 1042 1098 1154 1211 1325 1440 1557 1674 1792 1911 2030 2151 2272 2393

22 67 110 154 199 245 293 339 388 437 487 538 589 641 694 747 800 854 908 963 1018 1074 1130 1186 1299 1414 1529 1645 1763 1881 1999 2119 2239 2360

12

Chapter 3


77


Traffic (A) in CCS for Z

1.0

1.2

1.4

1.6

2 4 6 8 10

8 39 82 131 183

27 67 114 165

17 55 100 148

7 43 87 134

12 14 16 18 20

238 295 354 414 475

219 274 330 389 449

184 237 292 347 404

22 24 26 28 30

536 599 662 725 790

509 570 632 694 758

32 34 36 38 40

854 919 984 1050 1116

42 44 46 48 50 52 54 56 58 60 64 68 72 76 80 84 88 92 96 100

1182 1249 1315 1382 1449 1516 1584 1652 1719 1787 1923 2060 2197 2335 2473

821 885 949 1013 1079 1144 1201 1275 1341 1407 1474 1540 1607 1674 1741

201 254 310 367 426 484 545 606 667 728 791 854 917 981 1045

2542 2750 2888 3028 3167

1876 2011 2147 2284 2420

1109 1174 1239 1304 1370 1435 1501 1567 1633 1700 1833 1967 2102 2236 2372

2557 2695 2832 2970 3109

2508 2644 2781 2917 3055

462 521 581 640 702 763 825 888 950 1014 1077 1141 1205 1270 1334 1400 1464 1530 1595 1661 1793 1926 2059 2193 2327 2462 2597 2732 2868 3004

=:

1.8

2.0

2.2

2.4

32 74 120

20 62 106

7 48 92

36 80

168 220 273 328 384 440 498 557 616 676 737 798 869 922 984

154 204 256 310 364

125 173 223 275 328

420 477 535 593 652

139 188 239 292 346 401 457 514 571 630

1047 1110 1174 1237 1301 1365 1430 1495 1560 1625 1755 1887 2019 2151 2285

1018 1081 1144 1206 1270 1333 1397 1461 1525 1590 1720 1850 1981 2113 2245

689 748 808 869 930 991 1053 1115 1177 1240 1303 1366 1430 1493 1557 1686 1815 1945 2076 2207

666 725 784 844 905 965 1026 1088 1150 1212 1274 1337 1400 1463 1526 1653 1782 1911 2040 2171

2418 2553 2687 2822 2957

2378 2511 2644 2778 2913

2339 2471 2604 2737 2870

2302 2433 2564 2697 2829

713 772 833 895 956

382 437 494 550 608

78

Chapter 3



Traffic (A) in

ecs for Z =

2.6

2.8

3.0

3.2

3.4

3.6

3.8

4.0

2 4 6 8 10

23 66

8 54

40

27

12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 64 68 72 76 80 84 88 92 96 100

111 159 208 259 311 364 418 474 531 587 644 703 762 821 880 940 1001 1061 1123 1184 1246 1308 1370 1433 1496 1622 1750 1878 2006 2136 2266 2396 2527 2658 2790

98 144 193 243 294 347 401 455 510 567

84 130 178 227 278 330 383 437 492 547 603 661 718 776 834 893 953 1012 1072 1133 1193 1254 1316 1377 1439 1564 1689 1815 1942 2070 2198 2327 2456 2586 2716

45 90 135 183 232 282 333 385 438 492 546 601 657 714 770 827 885 944 1002 1061 1120 1180 1240 1300 1361 1483 1605 1729 1854 1979 2105 2231 2358 2486 2614

30 76 122 169 217 267 318 369 421 474

624 681 739 798 857 916 976 1037 1097 1158 1219 1281 1343 1405 1467 1592 1719 1846 1974 2102 2231 2361 2491 2622 2753

71 116 163 212 262 314 366 419 473 528 584 640 697 755 812 871 930 989 1048 1108 1168 1229 1290 1351 1412 1536 1660 1786 1912 2038 2166 2294 2423 2552 2681

9 58 103 150 198 247 297 349 402 456 510 565 620 677 734 791 849 907 966 1025 1085 1144 1204 1265 1325 1386 1509 1632 1757 1882 2008 2135 2262 2390 2518 2647

12 62 109 155 203 251 301 353 404 457 511 564 619 675 730 787 843 901 958 1016 1075 1134 1193 1252 1312 1432 1553 1675 1799 1922 2047 2172 2298 2424 2551

528 583 638 694 750 807 864 922 980 1039 1097 1157 1216 1276 1336 1457 1579 1702 1826 1950 2075 2201 2328 2455 2582

4 Erlang C Distribution

The Erlang C distribution is used for dimensioning common-equipment server pools where call attempts wait in a first-in, first-out (FIFO) queue until an idle server is available. It is based on the following assumptions: • • • • 4.1

Calls are served in order of arrival. There are an infinite number of sources. Blocked calls are delayed. Holding times are exponential. ERlANG C FORMULA

The Erlang C Formula given in Equation 4.1, also known as the Erlangformula ofthe second kind [E2(N, A)], is used to predict the probability that a call will be delayed. Equation 4.2 is used to predict the probability that a call will be delayed longer than a given time. Equations 4.3 and 4.4 are used to predict the average delay on all calls and the average delay on calls delayed, respectively.

79

Chapter 4

80

P(>O) = £2 (N, A)

=


N! (N-I) N-I

A;

(4.1)

L-., +--I. ;=0

T

P(>t)

=P(>O)£

-(N-A) -

1

T2

(4.2)

(4.3)

T2

D=--2

(N-A)

(4.4)

where P(>O) = Probability of delay greater than zero P(>t) = Probability of delay greater than T 1 N = Number of servers in full-availability group A = Traffic offered to group in Erlangs T 1 =Acceptable delay time in seconds T2 = Mean server-holding time in seconds D I =Average delay on all calls in seconds D2 =Average delay on calls delayed in seconds E = Natural logarithm base (2.71828 ... ) 4.2

ERLANG C COMPUTER PROGRAM

The following computer program can be used to calculate Equations 4.1 through 4.4 to determine the Erlang C delay loss probabilities and delays. Required inputs are the number of servers in the group, the traffic offered to the group expressed in Erlangs, the acceptable delay time expressed in seconds, and server-holding time expressed in seconds. 100 110 120 130

REM ERLANG C INPUT "ENTER INPUT "ENTER INPUT "ENTER

DELAY LOSS CALCULATIONS NUMBER OF SERVERS (N) ";N OFFERED TRAFFIC IN ERLANGS (A) ";A ACCEPTABLE DELAY (T1)IN SEC.";T1

Chapter 4

140 150 160 170 180 190 200 210 220 230 240 250 260 270

4.3

81


INPUT lIENTER SERVER-HOLDING TIME (T2)IN SEC.";T2 LET X=l LET Y=O FOR 1=1 TO N LET Y=X+Y LET X=X*A/I NEXT I LET PO=( (N/(N-A) )*X)/(Y+( (N/(N-A) )*X)) LET Pl=PO*EXP(-(N-A)*T1/T2) PRINT USING "P(>O) = #.#####";PO PRINT USING "P(>T) = #.#####";P1 PRINT USING "D1 ##.#### SEC.1I;PO*T2/(N-A) PRINT USING "02 = ##.#### SEC.";T2/(N-A) END

ERLANG C DELAY LOSS PROBABILITY TABLES

Erlang C delay loss probability tables are used to determine the probability that a call will be delayed and the probability that a call will be delayed longer than a given time (expressed as a ratio of delay time to server-holding time). Tables 4-1 through 4-24 present delay loss probabilities for common-equipment groups containing 2 to 25 servers and delay-to-holding time ratios of 0.2 to 2.0. Blanks (no data) in the tables indicate that the delay loss probability is less than 0.000005 (i.e., not necessarily zero, but essentially nonblocking for practical applications). The following examples illustrate typical table usage: Example 4-1 Determine the delay loss probabilities and delays for a 24-server commonequipment pool if the busy-hour traffic is 20 Erlangs, the acceptable delay time is 4 seconds, and the mean server-holding time (negative exponential distribution) is 4 seconds. Using Table 4-23 (pages 127-128) (24 servers), read down the P(>O) column to the A row for 20 Erlangs and read. 29807 (probability the call will be delayed) at the intersection. Then read down the 1.0 column ( 4 sec/4 sec) to the A row for 20 Erlangs and read .00546 (probability the call will be delayed longer than 4 seconds) at the intersection. D

1

=P(>O) (

(T z

N-A)

) = (0.29807)(4 seconds) (2~20)

=0.29807 seconds

82

Chapter 4

D = _T_2_ 2

(N-A)

=


(4 seconds) = 1 second (24-20)

Example 4-2 For the common-equipment pool of Example 4-1, determine the offered traffic in Erlangs and CCS to achieve a grade of service of 0.002 or better. Using Table 4-23 (24 servers), read down the 1.0column until .00179 is found. Read across that A row to determine that 19.2 Erlangs of traffic can be offered. (19.2 Erl)(36 CCSlErI) = 691.2 CCS

Example 4-3 Determine the number of servers required to handle a busy-hour traffic level of 10Erlangs at a grade of service of 0.00 I or better, given that the acceptable delay time is 4 seconds and the mean server-holding time (negative exponential distribution) is 2 seconds. Usinga trial-and-error method, find that Table 4-12 (pages 105-106) (13 servers) yields a grade of service of 0.00071 at the intersection of the 2.0 column and the A row for 10 Erlangs. Therefore, at least 13 servers are required.

Chapter 4

83


Table 4-1. Erlang C Delay Loss Probability (N Offered Traffic (A in Erl)

= 2 Servers)

Delay Loss Probability P (>t) for T,fT2 =

P(>O)

0.2

0.4

0.6

0.8

1.0

0.11 0.12 0.13 0.14 0.15

.00573 .00679 .00793 .00916 .01047

.00393 .00466 .00546 .00631 .00723

.00269 .00320 .00376 .00435 .00499

.00185 .00220 .00258 .00300 .00345

.00126 .00151 .00178 .00207 .00238

.00087 .00104 .00122 .00143 .00165

0.16 0.17 0.18 0.19 0.20

.01185 .01332 .01486 .01648 .01818

.00820 .00924 .01033 .01148 .01269

.00568 .00641 .00718 .00799 .00885

.00393 .00444 .00499 .00556 .00617

.00272 .00308 .00347 .00387 .00431

.00188 .00214 .00241 .00270 .00301

0.21 0.22 0.23 0.24 0.25

.01995 .02180 .02372 .02571 .02778

.01395 .01527 .01665 .01808 .01957

.00975 .01070 .01169 .01272 .01379

.00682 .00749 .00820 .00894 .00972

.00477 .00525 .00576 .00629 .00685

.00333 .00368 .00404 .00442 .00483

0.26 0.27 0.28 0.29 0.30

.02991 .03211 .03439 .03672 .03913

.02112 .02272 .02438 .02609 .02785

.01491 .01608 .01728 .01853 .01982

.01053 .01137 .01225 .01316 .01411

.00744 .00805 .00869 .00935 .01004

.00525 .00569 .00616 .00664 .00715

0.31 0.32 0.33 0.34 0.35

.04160 .04414 .04674 .04940 .05213

.02967 .03154 .03347 .03545 .03748

.02116 .02254 .02396 .02543 .02694

.01509 .01611 .01716 .01825 .01937

.01076 .01151 .01229 .01309 .01393

.00768 .00823 .00880 .00939 .01001

0.36 0.37 0.38 0.39 0.40

.05492 .05776 .06067 .06364 .06667

.03956 .04169 .04388 .04612 .04841

.02850 .03010 .03174 .03342 .03515

.02053 .02172 .02295 .02422 .02553

.01479 .01568 .01660 .01755 .01854

.01065 .01132 .01201 .01272 .01346

0.41 0.42 0.43 0.44 0.45

.06975 .07289 .07609 .07934 .08265

.05075 .05314 .05559 .05808 .06062

.03693 .03874 .04061 .04251 .04446

.02687 .02825 .02966 .03112 .03261

.01955 .02059 .02167 .02278 .02392

.01422 .01501 .01583 .01667 .01754

0.46 0.47 0.48 0.49 0.50

.08602 .08943 .09290 .09643 .10000

.06321 .06586 .06855 .07129 .07408

.04646 .04850 .05058 .05271 .05488

.03414 .03571 .03732 .03897 .04066

.02509 .02630 .02754 .02881 .03012

.01844 .01937 .02032 .02130 .02231

(table continues)

84

Chapter 4



=2 Servers

Continued)

Delay Loss Probability P (>t) for T/T2 =

P(>O)

1.2

1.4

1.6

1.8

2.0

0.11 0.12 0.13 0.14 0.15

.00573 .00679 .00793 .00916 .01047

.00059 .00071 .00084 .00098 .00114

.00041 .00049 .00058 .00068 .00079

.00028 .00034 .00040 .00047 .00054

.00019 .00023 .00027 .00032 .00037

.00013 .00016 .00019 .00022 .00026

0.16 0.17 0.18 0.19 0.20

.01185 .01332 .01486 .01648 .01818

.00130 .00148 .00167 .00188 .00210

.00090 .00103 .00116 .00131 .00146

.00062 .00071 .00081 .00091 .00102

.00043 .00049 .00056 .00063 .00071

.00030 .00034 .00039 .00044 .00050

0.21 0.22 0.23 0.24 0.25

.01995 .02180 .02372 .02571 .02778

.00233 .00258 .00284 .00311 .00340

.00163 .00180 .00199 .00219 .00240

.00114 .00126 .00140 .00154 .00169

.00080 .00089 .00098 .00108 .00119

.00056 .00062 .00069 .00076 .00084

0.26 0.27 0.28 0.29 0.30

.02991 .03211 .03439 .03672 .03913

.00371 .00403 .00437 .00472 .00509

.00262 .00285 .00309 .00335 .00362

.00185 .00202 .00219 .00238 .00258

.00131 .00143 .00156 .00169 .00183

.00092 .00101 .00110 .00120 .00131

0.31 0.32 0.33 0.34 0.35

.04160 .04414 .04674 .04940 .05213

.00547 .00588 .00630 .00674 .00720

.00390 .00420 .00451 .00484 .00517

.00278 .00300 .00323 .00347 .00372

.00199 .00215 .00231 .00249 .00267

.00142 .00153 .00166 .00179 .00192

0.36 0.37 0.38 0.39 0.40

.05492 .05776 .06067 .06364 .06667

.00767 .00817 .00868 .00922 .00977

.00553 .00590 .00628 .00668 .00710

.00398 .00426 .00454 .00484 .00515

.00287 .00307 .00329 .00351 .00374

.00207 .00222 .00238 .00254 .00272

0.41 0.42 0.43 0.44 0.45

.06975 .07289 .07609 .07934 .08265

.01035 .01095 .01156 .01220 .01287

.00753 .00798 .00845 .00893 .00944

.00548 .00582 .00617 .00654 .00692

.00399 .00424 .00451 .00479 .00508

.00290 .00309 .00329 .00350 .00372

0.46 0.47 0.48 0.49 0.50

.08602 .08943 .09290 .09643 .10000

.01355 .01426 .01499 .01575 .01653

.00996 .01050 .01106 .01164 .01225

.00732 .00773 .00816 .00861 .00907

.00538 .00569 .00602 .00636 .00672

.00395 .00419 .00444 .00471 .00498

Chapter 4

85


Table 4-2. Erlang C Delay Loss Probability (N = 3 Servers) Offered Traffic (A in Erl)

Delay Loss Probability P (>t) for T,IT2

P(>O)

=

0.2

0.4

0.6

0.8

1.0

0.13 0.16 0.19 0.22 0.25

.00034 .00061 .00101 .00154 .00221

.00019 .00035 .00058 .00088 .00128

.00011 .00020 .00033 .00051 .00074

.00006 .00011 .00019 .00029 .00042

.00003 .00006 .00011 .00017 .00025

.00002 .00003 .00006 .00010 .00014

0.28 0.31 0.34 0.37 0.40

.00305 .00406 .00526 .00665 .00825

.00177 .00237 .00309 .00393 .00490

.00103 .00138 .00181 .00232 .00292

.00060 .00081 .00107 .00137 .00173

.00035 .00047 .00063 .00081 .00103

.00020 .00028 .00037 .00048 .00061

0.43 0.46 0.49 0.52 0.55

.01006 .01209 .01435 .01684 .01957

.00602 .00727 .00868 .01025 .01199

.00360 .00438 .00526 .00624 .00734

.00215 .00263 .00318 .00380 .00450

.00129 .00158 .00193 .00232 .00276

.00077 .00095 .00117 .00141 .00169

0.58 0.61 0.64 0.67 0.70

.02254 .02576 .02923 .03295 .03692

.01389 .01597 .01823 .02067 .02331

.00856 .00990 .01137 .01297 .01471

.00528 .00614 .00709 .00814 .00929

.00325 .00381 .00442 .00511 .00586

.00200 .00236 .00276 .00321 .00370

0.73 0.76 0.79 0.82 0.85

.04115 .04564 .05039 .05540 .06067

.02614 .02916 .03239 .03582 .03947

.01660 .01863 .02082 .02316 .02567

.01054 .01190 .01338 .01498 .01670

.00669 .00761 .00860 .00969 .01086

.00425 .00486 .00553 .00626 .00707

0.88 0.91 0.94 0.97 1.00

.06620 .07199 .07804 .08434 .09091

.04332 .04739 .05169 .05620 .06094

.02835 .03120 .03423 .03745 .04085

.01855 .02054 .02267 .02495 .02738

.01214 .01352 .01502 .01662 .01835

.00795 .00890 .00995 .01108 .01230

1.03 1.06 1.09 1.12 1.15

.09773 .10481 .11214 .11973 .12756

.06591 .07110 .07654 .08220 .08811

.04444 .04824 .05224 .05644 .06086

.02997 .03273 .03565 .03875 .04204

.02021 .02220 .02433 .02661 .02904

.01363 .01506 .01661 .01927 .02006

1.18 1.21 1.24 1.27 1.30

.13565 .14398 .15256 .16138 .17044

.09426 .10065 .10729 .11418 .12132

.06550 .07036 .07546 .08078 .08635

.04552 .04919 .05307 .05715 .06146

.03163 .03439 .03732 .04044 .04375

.02198 .02404 .02625 .02861 .03114

(table continues)

86

Chapter 4


Table 4-2. Erlang C Delay Loss Probability (N = 3 Servers Continued) Offered Traffic (A in Erl)


P(>O) 1.2

1.4

1.6

1.8

2.0

0.13 0.16 0.19 0.22 0.25

.00034 .00061 .00101 .00154 .00221

.00001 .00002 .00003 .00005 .00008

.00001 .00001 .00002 .00003 .00005

.00001 .00001 .00002 .00003

.00001 .00001 .00002

.00001 .00001

0.28 0.31 0.34 0.37 0.40

.00305 .00406 .00526 .00665 .00825

.00012 .00016 .00022 .00028 .00036

.00007 .00009 .00013 .00017 .00022

.00004 .00005 .00007 .00010 .00013

.00002 .00003 .00004 .00006 .00008

.00001 .00002 .00003 .00003 .00005

0.43 0.46 0.49 0.52 0.55

.01006 .01209 .01435 .01684 .01957

.00046 .00057 .00071 .00086 .00103

.00028 .00035 .00043 .00052 .00063

.00016 .00021 .00026 .00032 .00039

.00010 .00012 .00016 .00019 .00024

.00006 .00008 .00009 .00012 .00015

0.58 0.61 0.64 0.67 0.70

.02254 .02576 .02923 .03295 .03692

.00124 .00146 .00172 .00201 .00234

.00076 .00091 .00107 .00126 .00148

.00047 .00056 .00067 .00079 .00093

.00029 .00035 .00042 .00050 .00059

.00018 .00022 .00026 .00031 .00037

0.73 0.76 0.79 0.82 0.85

.04115 .04564 .05039 .05540 .06067

.00270 .00310 .00355 .00405 .00460

.00171 .00198 .00228 .00262 .00299

.00109 .00127 .00147 .00169 .00195

.00069 .00081 .00094 .00109 .00127

.00044 .00052 .00061 .00071 .00082

0.88 0.91 0.94 0.97 1.00

.06620 .07199 .07804 .08434 .09091

.00520 .00586 .00659 .00738 .00825

.00340 .00386 .00436 .00492 .00553

.00223 .00254 .00289 .00328 .00371

.00146 .00167 .00191 .00218 .00248

.00095 .00110 .00127 .00145 .00167

1.03 1.06 1.09 1.12 1.15

.09773 .10481 .11214 .11973 .12756

.00919 .01022 .01133 .01254 .01385

.00620 .00693 .00774 .00861 .00957

.00418 .00470 .00528 .00591 .00661

.00282 .00319 .00360 .00406 .00457

.00190 .00216 .00246 .00279 .00315

1.18 1.21 1.24 1.27 1.30

.13565 .14398 .15256 .16138 .17044

.01527 .01681 .01846 .02024 .02216

.01061 .01175 .01298 .01432 .01577

.00737 .00821 .00913 .01013 .01123

.00512 .00574 .00642 .00717 .00799

.00356 .00401 .00452 .00507 .00569

Chapter 4

87


Table 4-3. Erlang C Delay Loss Probability (N =4 Servers) Offered Traffic (A in Erl)


P(>O) 0.2

0.4

0.6

0.8

1.0

0.54 0.58 0.62 0.66 0.70

.00239 .00309 .00392 .00489 .00602

.00119 .00156 .00199 .00251 .00311

.00060 .00079 .00101 .00129 .00161

.00030 .00040 .00052 .00066 .00083

.00015 .00020 .00026 .00034 .00043

.00008 .00010 .00013 .00017 .00022

0.74 0.78 0.82 0.86 0.90

.00731 .00878 .01043 .01228 .01433

.00381 .00461 .00552 .00655 .00771

.00198 .00242 .00292 .00350 .00415

.00103 .00127 .00155 .00187 .00223

.00054 .00067 .00082 .00100 .00120

.00028 .00035 .00043 .00053 .00065

0.94 0.98 1.02 1.06 1.10

.01659 .01908 .02180 .02475 .02795

.00900 .01043 .01201 .01375 .01565

.00488 .00570 .00662 .00764 .00876

.00265 .00312 .00365 .00424 .00491

.00143 .00170 .00201 .00236 .00275

.00078 .00093 .00111 .00131 .00154

1.14 1.18 1.22 1.26 1.30

.03139 .03510 .03907 .04330 .04781

.01772 .01977 .02240 .02503 .02786

.01000 .01136 .01285 .01447 .01624

.00564 .00646 .00737 .00837 .00946

.00319 .00368 .00423 .00484 .00551

.00180 .00209 .00242 .00280 .00321

1.34 1.38 1.42 1.46 1.50

.05260 .05766 .06302 .06866 .07459

.03090 .03415 .03761 .04131 .04524

.01815 .02022 .02245 .02486 .02744

.01066 .01197 .01340 .01496 .01664

.00626 .00709 .00800 .00900 .01009

.00368 .00420 .00477 .00541 .00612

1.54 1.58 1.62 1.66 1.70

.08081 .08733 .09415 .10126 .10868

.04941 .05382 .05849 .06341 .06861

.03021 .03317 .03634 .03971 .04331

.01847 .02044 .02257 .02487 .02734

.01129 .01260 .01402 .01558 .01726

.00690 .00777 .00871 .00975 .01090

1.74 1.78 1.82 1.86 1.90

.11639 .12441 .13273 .14135 .15028

.07407 .07981 .08583 .09214 .09874

.04713 .05119 .05550 .06006 .06488

.02999 .03284 .03589 .03914 .04263

.01909 .02106 .02320 .02552 .02801

.01215 .01351 .01500 .01663 .01840

1.94 1.98 2.02 2.06 2.10

.15951 .16904 .17887 .18900 .19942

.10565 .11286 .12038 .12822 .13638

.06997 .07535 .08101 .08698 .09326

.04634 .05031 .05452 .05901 .06378

.03069 .03359 .03669 .04003 .04362

.02033 .02242 .02470 .02716 .02983

(table continues)

88

Chapter 4



Delay Loss Probability P (>t) for T,IT2 =

P(>O)

1.2

1.4

1.6

1.8

2.0

0.54 0.58 0.62 0.66 0.70

.00239 .00309 .00392 .00489 .00602

.00004 .00005 .00007 .00009 .00011

.00002 .00003 .00003 .00005 .00006

.00001 .00001 .00002 .00002 .00003

.00001 .00001 .00001 .00002

.00001 .00001

0.74 0.78 0.82 0.86 0.90

.00731 .00878 .01043 .01228 .01433

.00015 .00018 .00023 .00028 .00035

.00008 .00010 .00012 .00015 .00019

.00004 .00005 .00006 .00008 .00010

.00002 .00003 .00003 .00004 .00005

.00001 .00001 .00002 .00002 .00003

0.94 0.98 1.02 1.06 1.10

.01659 .01908 .02180 .02475 .02795

.00042 .00051 .00061 .00073 .00086

.00023 .00028 .00034 .00040 .00048

.00012 .00015 .00019 .00022 .00027

.00007 .00008 .00010 .00012 .00015

.00004 .00005 .00006 .00007 .00008

1.14 1.18 1.22 1.26 1.30

.03139 .03510 .03907 .04330 .04781

.00101 .00119 .00139 .00162 .00187

.00057 .00068 .00080 .00093 .00109

.00032 .00039 .00046 .00054 .00064

.00018 .00022 .00026 .00031 .00037

.00010 .00012 .00015 .00018 .00022

1.34 1.38 1.42 1.46 1.50

.05260 .05766 .06302 .06866 .07459

.00216 .00249 .00285 .00326 .00371

.00127 .00147 .00170 .00196 .00225

.00075 .00087 .00102 .00118 .00137

.00044 .00052 .00061 .00071 .00083

.00026 .00031 .00036 .00043 .00050

1.54 1.58 1.62 1.66 1.70

.08081 .08733 .09415 .10126 .10868

.00422 .00479 .00541 .00611 .00688

.00258 .00295 .00336 .00383 .00434

.00158 .00182 .00209 .00240 .00274

.00096 .00112 .00130 .00150 .00173

.00059 .00069 .00081 .00094 .00109

1.74 1.78 1.82 1.86 1.90

.11639 .12441 .13273 .14135 .15028

.00773 .00867 .00970 .01084 .01209

.00492 .00556 .00627 .00707 .00794

.00313 .00357 .00406 .00461 .00522

.00199 .00229 .00262 .00300 .00343

.00127 .00147 .00170 .00196 .00225

1.94 1.98 2.02 2.06 2.10

.15951 .16904 .17887 .18900 .19942

.01346 .01497 .01662 .01843 .02040

.00892 .01000 .01119 .01250 .01395

.00591 .00667 .00753 .00848 .00954

.00391 .00446 .00507 .00575 .00652

.00259 .00297 .00341 .00390 .00446

Chapter 4

89


Table 4-4. Erlang C Delay Loss Probability (N = 5 Servers) Offered Traffic (Ain Erl)


P(>O) 0.2

0.4

0.6

0.8

1.0

1.05 1.10 1.15 1.20 1.25

.00471 .00573 .00689 .00821 .00971

.00214 .00262 .00319 .00384 .00459

.00097 .00120 .00148 .00180 .00217

.00044 .00055 .00068 .00084 .00102

.00020 .00025 .00032 .00039 .00048

.00009 .00012 .00015 .00018 .00023

1.30 1.35 1.40 1.45 1.50

.01139 .01326 .01533 .01762 .02014

.00543 .00639 .00746 .00866 .01000

.00259 .00308 .00363 .00426 .00497

.00124 .00148 .00177 .00209 .00247

.00059 .00072 .00086 .00103 .00122

.00028 .00034 .00042 .00051 .00061

1.55 1.60 1.65 1.70 1.75

.02289 .02589 .02914 .03265 .03643

.01148 .01311 .01491 .01687 .01902

.00576 .00664 .00763 .00872 .00993

.00289 .00337 .00390 .00451 .00518

.00145 .00171 .00200 .00233 .00271

.00073 .00086 .00102 .00120 .00141

1.80 1.85 1.90 1.95 2.00

.04050 .04485 .04950 .05444 .05970

.02135 .02389 .02663 .02958 .03276

.01126 .01272 .01432 .01607 .01798

.00594 .00678 .00771 .00873 .00987

.00313 .00361 .00414 .00475 .00542

.00165 .00192 .00223 .00258 .00297

2.05 2.10 2.15 2.20 2.25

.06527 .07116 .07738 .08393 .09081

.03618 .03984 .04376 .04794 .05239

.02006 .02231 .02475 .02738 .03023

.01112 .01249 .01400 .01564 .01744

.00616 .00699 .00791 .00893 .01006

.00342 .00392 .00448 .00510 .00581

2.30 2.35 2.40 2.45 2.50

.09803 .10559 .11350 .12176 .13037

.05713 .06215 .06748 .07312 .07907

.03329 .03658 .04012 .04391 .04796

.01940 .02153 .02385 .02637 .02909

.01131 .01267 .01418 .01583 .01764

.00659 .00746 .00843 .00951 .01070

2.55 2.60 2.65 2.70 2.75

.13993 .14865 .15833 .16837 .17876

.08536 .09198 .09896 .10629 .11398

.05229 .05692 .06185 .06710 .07268

.03204 .03522 .03866 .04236 .04634

.01963 .02179 .02416 .02674 .02955

.01202 .01349 .01510 .01688 .01884

2.80 2.85 2.90 2.95 3.00

.18952 .20063 .21211 .22393 .23615

.12206 .13051 .13937 .14862 .15830

.07861 .08490 .09157 .09863 .10611

.05063 .05523 .06017 .06546 .07113

.03261 .03593 .03953 .04344 .04768

.02100 .02337 .02597 .02883 .03196

(table continues)

90

Chapter 4




P(>O)

=

1.2

1.4

1.6

1.8

2.0

.00002 .00002 .00003 .00004 .00005

.00001 .00001 .00001 .00002 .00002

.00001 .00001 .00001 .00001

.00001 .00001 .00001 .00001 .00001 .00002

1.05 1.10 1.15 1.20 1.25

.00471 .00573 .00689 .00821 .00971

.00004 .00005 .00007 .00009 .00011

1.30 1.35 1.40 1.45 1.50

.01139 .01326 .01533 .01762 .02014

.00013 .00017 .00020 .00025 .00030

.00006 .00008 .00010 .00012 .00015

.00003 .00004 .00005 .00006 .00007

.00001 .00002 .00002 .00003 .00004

1.55 1.60 1.65 1.70 1.75

.02289 .02589 .02914 .03265 .03643

.00036 .00044 .00052 .00062 .00074

.00018 .00022 .00027 .00032 .00038

.00009 .00011 .00014 .00017 .00020

.00005 .00006 .00007 .00009 .00010

.00002 .00003 .00004 .00004 .00005

1.80 1.85 1.90 1.95 2.00

.04050 .04485 .04950 .05444 .05970

.00087 .00102 .00120 .00140 .00163

.00046 .00055 .00065 .00076 .00090

.00024 .00029 .00035 .00041 .00049

.00013 .00015 .00019 .00022 .00027

.00007 .00008 .00010 .00012 .00015

2.05 2.10 2.15 2.20 2.25

.06527 .07116 .07738 .08393 .09081

.00189 .00219 .00253 .00292 .00335

.00105 .00123 .00143 .00167 .00193

.00058 .00069 .00081 .00095 .00111

.00032 .00038 .00046 .00054 .00064

.00018 .00022 .00026 .00031 .00037

2.30 2.35 2.40 2.45 2.50

.09803 .10559 .11350 .12176 .13037

.00384 .00439 .00501 .00571 .00649

.00224 .00258 .00298 .00343 .00394

.00130 .00152 .00177 .00206 .00239

.00076 .00090 .00105 .00124 .00145

.00044 .00053 .00063 .00074 .00088

2.55 2.60 2.65 2.70 2.75

.13993 .14865 .15833 .16837 .17876

.00737 .00834 .00944 .01066 .01201

.00451 .00516 .00590 .00673 .00766

.00276 .00320 .00369 .00425 .00488

.00169 .00198 .00230 .00268 .00311

.00104 .00122 .00144 .00169 .00199

2.80 2.85 2.90 2.95 3.00

.18952 .20063 .21211 .22393 .23615

.01352 .01520 .01707 .01913 .02142

.00871 .00989 .01121 .01270 .01436

.00561 .00643 .00737 .00843 .00963

.00361 .00418 .00484 .00559 .00645

.00233 .00272 .00318 .00371 .00433

Chapter 4


91



P(>O)

0.2

0.4

0.6

0.8

1.0

1.80 1.85 1.90 1.95 2.00

.01115 .01265 .01429 .01607 .01802

.00481 .00551 .00629 .00715 .00810

.00208 .00240 .00277 .00318 .00364

.00090 .00105 .00122 .00142 .00163

.00039 .00046 .00054 .00063 .00073

.00017 .00020 .00024 .00028 .00033

2.05 2.10 2.15 2.20 2.25

.02012 .02240 .02485 .02748 .03030

.00913 .01027 .01150 .01285 .01431

.00414 .00471 .00533 .00601 .00676

.00188 .00216 .00247 .00281 .00319

.00085 .00099 .00114 .00131 .00151

.00039 .00045 .00053 .00061 .00071

2.30 2.35 2.40 2.45 2.50

.03331 .03653 .03995 .04359 .04744

.01589 .01760 .01945 .02143 .02356

.00758 .00848 .00947 .01054 .01170

.00362 .00409 .00461 .00518 .00581

.00173 .00197 .00224 .00255 .00289

.00082 .00095 .00109 .00125 .00143

2.55 2.60 2.65 2.70 2.75

.05152 .05583 .06038 .06516 .07019

.02584 .02829 .03090 .03368 .03664

.01296 .01433 .01581 .01741 .01913

.00650 .00726 .00809 .00900 .00999

.00326 .00368 .00414 .00465 .00521

.00164 .00186 .00212 .00240 .00272

2.80 2.85 2.90 2.95 3.00

.07547 .08100 .08678 .09283 .09914

.03979 .04314 .04668 .05044 .05441

.02098 .02297 .02511 .02741 .02986

.01106 .01224 .01351 .01489 .01639

.00583 .00652 .00727 .00809 .00899

.00308 .00347 .00391 .00440 .00494

3.05 3.10 3.15 3.20 3.25

.10572 .11257 .11970 .12710 .13479

.05861 .06303 .06769 .07260 .07776

.03249 .03529 .03828 .04147 .04487

.01801 .01976 .02165 .02369 .02589

.00998 .01106 .01224 .01353 .01493

.00553 .00619 .00692 .00773 .00862

3.30 3.35 3.40 3.45 3.50

.14275 .15100 .15953 .16835 .17747

.08319 .08888 .09485 .10110 .10764

.04848 .05231 .05639 .06071 .06529

.02825 .03079 .03352 .03654 .03960

.01646 .01812 .01993 .02189 .02402

.00959 .01067 .01185 .01315 .01457

3.55 3.60 3.65 3.70 3.75

.18687 .19657 .20656 .21684 .22742

.11448 .12163 .12910 .13689 .14501

.07013 .07526 .08069 .08641 .09246

.04297 .04657 .05043 .05455 .05896

.02632 .02882 .03152 .03444 .03759

.01613 .01783 .01970 .02174 .02397

(table continues)

92

Chapter 4



Delay Loss Probability P (>t) for T,JT2 =

P(>O)

1.2

1.4

1.6

1.8

2.0

1.80 1.85 1.90 1.95 2.00

.01115 .01265 .01429 .01607 .01802

.00007 .00009 .00010 .00012 .00015

.00003 .00004 .00005 .00006 .00007

.00001 .00002 .00002 .00002 .00003

.00001 .00001 .00001 .00001 .00001

.00001

2.05 2.10 2.15 2.20 2.25

.02012 .02240 .02485 .02748 .03030

.00018 .00021 .00024 .00029 .00034

.00008 .00010 .00011 .00013 .00016

.00004 .00004 .00005 .00006 .00008

.00002 .00002 .00002 .00003 .00004

.00001 .00001 .00001 .00001 .00002

2.30 2.35 2.40 2.45 2.50

.03331 .03653 .03995 .04359 .04744

.00039 .00046 .00053 .00062 .00071

.00019 .00022 .00026 .00030 .00035

.00009 .00011 .00013 .CJ0015 .00018

.00004 .00005 .00006 .00007 .00009

.00002 .00002 .00003 .00004 .00004

2.55 2.60 2.65 2.70 2.75

.05152 .05583 .06038 .06516 .07019

.00082 .00094 .00108 .00124 .00142

.00041 .00048 .00055 .00064 .00074

.00021 .00024 .00028 .00033 .00039

.00010 .00012 .00015 .00017 .00020

.00005 .00006 .00007 .00009 .00011

2.80 2.85 2.90 2.95 3.00

.07547 .08100 .08678 .09283 .09914

.00162 .00185 .00210 .00239 .00271

.00086 .00098 .00113 .00130 .00149

.00045 .00052 .00061 .00071 .00082

.00024 .00028 .00033 .00038 .00045

.00013 .00015 .00018 .00021 .00025

3.05 3.10 3.15 3.20 3.25

.10572 .11257 .11970 .12710 .13479

.00307 .00347 .00392 .00441 .00497

.00170 .00194 .00221 .00252 .00287

.00094 .00109 .00125 .00144 .00165

.00052 .00061 .00071 .00082 .00095

.00029 .00034 .00040 .00047 .00055

3.30 3.35 3.40 3.45 3.50

.14275 .15100 .15953 .16835 .17747

.00559 .00628 .00704 .00789 .00884

.00326 .00370 .00419 .00474 .00536

.00190 .00218 .00249 .00285 .00325

.00111 .00128 .00148 .00171 .00197

.00064 .00075 .00088 .00103 .00120

3.55 3.60 3.65 3.70 3.75

.18687 .19657 .20656 .21684 .22742

.00988 .01103 .01231 .01372 .01528

.00605 .00683 .00769 .00866 .00975

.00371 .00422 .00481 .00547 .00621

.00227 .00261 .00301 .00345 .00396

.00139 .00162 .00188 .00218 .00253

Chapter 4

93



Delay Loss Probability P (>t) for T,JT2 =

P(>O) 0.2

0.4

0.6

0.8

1.0

1.88 1.96 2.04 2.12 2.20

.00344 .00431 .00535 .00657 .00799

.00123 .00157 .00198 .00248 .00306

.00044 .00057 .00074 .00093 .00117

.00016 .00021 .00027 .00035 .00045

.00006 .00008 .00010 .00013 .00017

.00002 .00003 .00004 .00005 .00007

2.28 2.36 2.44 2.52 2.60

.00963 .01151 .01365 .01608 .01880

.00375 .00455 .00549 .00656 .00780

.00146 .00180 .00220 .00268 .00323

.00057 .00071 .00089 .00109 .00134

.00022 .00028 .00036 .00045 .00056

.00009 .00011 .00014 .00018 .00023

2.68 2.74 2.86 2.92 3.00

.02184 .02523 .02898 .03311 .03765

.00921 .01081 .01261 .01464 .01692

.00388 .00463 .00549 .00647 .00760

.00164 .00198 .00239 .00286 .00342

.00069 .00085 .00104 .00127 .00153

.00029 .00036 .00045 .00056 .00069

3.08 3.16 3.24 3.32 3.40

.04260 .04799 .05384 .06016 .06697

.01945 .02226 .02538 .02882 .03260

.00888 .01033 .01197 .01380 .01587

.00405 .00479 .00564 .00661 .00772

.00185 .00222 .00266 .00317 .00376

.00085 .00103 .00125 .00152 .00183

3.48 3.56 3.64 3.72 3.80

.07429 .08213 .09050 .09941 .10889

.03674 .04127 .04622 .05159 .05742

.01817 .02074 .02360 .02677 .03028

.00899 .01043 .01205 .01389 .01596

.00445 .00524 .00616 .00721 .00842

.00220 .00263 .00314 .00374 .00444

3.88 3.96 4.04 4.12 4.20

.11894 .12957 .14080 .15262 .16505

.06373 .07054 .07789 .08579 .09428

.03415 .03841 .04309 .04823 .05385

.01830 .02091 .02384 .02711 .03076

.00980 .01138 .01319 .01524 .01757

.00525 .00620 .00730 .00857 .01004

4.28 4.36 4.44 4.52 4.60

.17810 .19178 .20608 .22101 .23659

.10337 .11311 .12350 .13459 .14640

.06000 .06671 .07401 .08196 .09059

.03483 .03934 .04436 .04991 .05605

.02021 .02320 .02658 .03039 .03469

.01173 .01369 .01593 .01851 .02146

4.68 4.76 4.84 4.92 5.00

.25281 .26967 .28718 .30534 .32415

.15896 .17229 .18644 .20143 .21728

.09995 .11008 .12104 .13288 .14565

.06284 .07033 .07858 .08766 .09763

.03951 .04493 .05101 .05782 .06544

.02484 .02871 .03312 .03815 .04387

(table continues)

94

Chapter 4




P(>O)

1.2

1.4

1.6

1.8

2.0

1.88 1.96 2.04 2.12 2.20

.00344 .00431 .00535 .00657 .00799

.00001 .00001 .00001 .00002 .00003

.00001 .00001 .00001

2.28 2.36 2.44 2.52 2.60

.00963 .01151 .01365 .01608 .01880

.00003 .00004 .00006 .00007 .00010

.00001 .00002 .00002 .00003 .00004

.00001 .00001 .00001 .00001 .00002

.00001 .00001

2.68 2.74 2.86 2.92 3.00

.02184 .02523 .02898 .03311 .03765

.00012 .00016 .00020 .00025 .00031

.00005 .00007 .00009 .00011 .00014

.00002 .00003 .00004 .00005 .00006

.00001 .00001 .00002 .00002 .00003

.00001 .00001 .00001 .00001

3.08 3.16 3.24 3.32 3.40

.04260 .04799 .05384 .06016 .06697

.00039 .00048 .00059 .00073 .00089

.00018 .00022 .00028 .00035 .00043

.00008 .00010 .00013 .00017 .00021

.00004 .00005 .00006 .00008 .00010

.00002 .00002 .00003 .00004 .00005

3.48 3.56 3.64 3.72 3.80

.07429 .08213 .09050 .09941 .10889

.00109 .00132 .00161 .00194 .00234

.00054 .00067 .00082 .00101 .00123

.00027 .00033 .00042 .00052 .00065

.00013 .00017 .00021 .00027 .00034

.00007 .00008 .00011 .00014 .00018

3.88 3.96 4.04 4.12 4.20

.11894 .12957 .14080 .15262 .16505

.00281 .00337 .00404 .00482 .00573

.00151 .00184 .00223 .00271 .00327

.00081 .00100 .00124 .00152 .00187

.00043 .00054 .00068 .00086 .00107

.00023 .00030 .00038 .00048 .00061

4.28 4.36 4.44 4.52 4.60

.17810 .19178 .20608 .22101 .23659

.00681 .00807 .00955 .01127 .01328

.00395 .00476 .00572 .00686 .00822

.00229 .00281 .00343 .00418 .00509

.00133 .00166 .00205 .00255 .00315

.00077 .00098 .00123 .00155 .00195

4.68 4.76 4.84 4.92 5.00

.25281 .26967 .28718 .30534 .32415

.01562 .01834 .02150 .02516 .02941

.00982 .01172 .01396 .01660 .01971

.00618 .00749 .00906 .01095 .01321

.00388 .00478 .00588 .00722 .00886

.00244 .00306 .00382 .00477 .00594

Chapter 4

95



P(>O)

2.1 2.2 2.3 2.4 2.5

Delay Loss Probability P (>t) for TJT2

=

0.2

0.4

0.6

0.8

1.0

.00156 .00208 .00273 .00354 .00452

.00048 .00065 .00087 .00115 .00150

.00015 .00020 .00028 .00038 .00050

.00005 .00006 .00009 .00012 .00017

.00001 .00002 .00003 .00004 .00006

.00001 .00001 .00001 .00002

2.6 2.7 2.8 2.9 3.0

.00570 .00710 .00876 .01070 .01295

.00193 .00246 .00310 .00386 .00476

.00066 .00085 .00109 .00139 .00175

.00022 .00030 .00039 .00050 .00064

.00008 .00010 .00014 .00018 .00024

.00003 .00004 .00005 .00007 .00009

3.1 3.2 3.3 3.4 3.5

.01554 .01849 .02185 .02564 .02989

.00583 .00708 .00854 .01022 .01215

.00219 .00271 .00333 .00407 .00494

.00082 .00104 .00130 .00162 .00201

.00031 .00040 .00051 .00065 .00082

.00012 .00015 .00020 .00026 .00033

3.6 3.7 3.8 3.9 4.0

.03462 .03988 .04568 .05206 .05904

.01436 .01687 .01972 .02293 .02653

.00596 .00714 .00851 .01010 .01192

.00247 .00302 .00368 .00445 .00536

.00102 .00128 .00159 .00196 .00241

.00043 .00054 .00069 .00086 .00108

4.1 4.2 4.3 4.4 4.5

.06666 .07492 .08387 .09352 .10389

.03056 .03504 .04002 .04452 .05159

.01401 .01639 .01909 .02216 .02652

.00642 .00766 .00911 .01078 .01272

.00294 .00358 .00435 .00525 .00632

.00135 .00168 .00207 .00256 .00314

4.6 4.7 4.8 4.9 5.0

.11500 .12688 .13954 .15300 .16727

.05826 .06558 .07358 .08230 .09180

.02952 .03389 .03880 .04428 .05038

.01495 .01752 .02046 .02382 .02765

.00758 .00905 .01079 .01281 .01517

.00384 .00468 .00569 .00689 .00833

5.1 5.2 5.3 5.4 5.5

.18236 .19829 .21507 .23271 .25122

.10210 .11327 .12533 .13835 .15237

.05717 .06470 .07304 .08225 .09242

.03201 .03696 .04256 .04890 .05605

.01792 .02111 .02480 .02907 .03400

.01003 .01206 .01445 .01728 .02062

5.6 5.7 5.8 5.9 6.0

.27060 .29086 .31201 .33405 .35698

.16744 .18362 .20095 .21949 .23929

.10361 .11592 .12942 .14421 .16040

.06411 .07318 .08335 .09475 .10752

.03967 .04619 .05368 .06226 .07207

.02455 .02916 .03457 .04091 .04831

(table continues)

96

Chapter 4




P(>O)

1.2

1.4

1.6

1.8

2.0

2.1 2.2 2.3 2.4 2.5

.00156 .00208 .00273 .00354 .00452

.00001

2.6 2.7 2.8 2.9 3.0

.00570 .00710 .00876 .01070 .01295

.00001 .00001 .00002 .00002 .00003

.00001 .00001 .00001

3.1 3.2 3.3 3.4 3.5

.01554 .01849 .02185 .02564 .02989

.00004 .00006 .00008 .00010 .00013

.00002 .00002 .00003 .00004 .00005

.00001 .00001 .00001 .00002 .00002

.00001 .00001

3.6 3.7 3.8 3.9 4.0

.03462 .03988 .04568 .05206 .05904

.00018 .00023 .00030 .00038 .00049

.00007 .00010 .00013 .00017 .00022

.00003 .00004 .00006 .00007 .00010

.00001 .00002 .00002 .00003 .00004

.00001 .00001 .00001 .00001 .00002

4.1 4.2 4.3 4.4 4.5

.06666 .07492 .08387 .09352 .10389

.00062 .00078 .00099 .00124 .00156

.00028 .00037 .00047 .00061 .00077

.00013 .00017 .00023 .00029 .00038

.00006 .00008 .00011 .00014 .00019

.00003 .00004 .00005 .00007 .00009

4.6 4.7 4.8 4.9 5.0

.11500 .12688 .13954 .15300 .16727

.00194 .00242 .00300 .00371 .00457

.00099 .00125 .00158 .00199 .00251

.00050 .00065 .00083 .00107 .00138

.00025 .00033 .00044 .00058 .00076

.00013 .00017 .00023 .00031 .00041

5.1 5.2 5.3 5.4 5.5

.18236 .19829 .21507 .23271 .25122

.00562 .00689 .00842 .01028 .01251

.00315 .00393 .00491 .00611 .00759

.00176 .00225 .00286 .00363 .00460

.00099 .00128 .00167 .00216 .00279

.00055 .00073 .00097 .00128 .00169

5.6 5.7 5.8 5.9 6.0

.27060 .29086 .31201 .33405 .35698

.01519 .01841 .02227 .02688 .03238

.00940 .01162 .01434 .01766 .02171

.00582 .00734 .00924 .01160 .01455

.00360 .00463 .00595 .00762 .00975

.00223 .00292 .00383 .00501 .00654

Chapter 4

97



P(>O)

3.1 3.2 3.3 3.4 3.5


=

0.2

0.4

0.6

0.8

.00501 .00613 .00744 .00897 .01072

.00154 .00192 .00238 .00293 .00357

.00047 .00060 .00076 .00095 .00119

.00015 .00019 .00024 .00031 .00040

.00004 .00006 .00008 .00010 .00013

.00001 .00002 .00002 .00003 .00004

3.6 3.7 3.8 3.9 4.0

.01273 .01502 .01760 .02051 .02376

.00432 .00520 .00622 .00739 .00874

.00147 .00180 .00220 .00267 .00322

.00050 .00062 .00078 .00096 .00118

.00017 .00022 .00027 .00035 .00044

.00006 .00007 .00010 .00013 .00016

4.1 4.2 4.3 4.4 4.5

.02738 .03140 .03583 .04071 .04605

.01028 .01202 .01400 .01622 .01872

.00386 .00460 .00547 .00647 .00761

.00145 .00176 .00214 .00258 .00309

.00054 .00067 .00083 .00103 .00126

.00020 .00026 .00033 .00041 .00051

4.6 4.7 4.8 4.9 5.0

.05188 .05822 .06509 .07251 .08051

.02152 .02464 .02810 .03194 .03625

.00893 .01042 .01213 .01407 .01625

.00370 .00441 .00524 .00620 .00730

.00154 .00187 .00226 .00273 .00328

.00064 .00079 .00098 .00120 .00147

5.1 5.2 5.3 5.4 5.5

.08910 .09831 .10814 .11862 .12977

.04084 .04597 .05160 .05774 .06444

.01872 .02150 .02462 .02811 .03200

.00858 .01006 .01175 .01368 .01589

.00393 .00470 .00560 .00666 .00789

.00180 .00220 .00267 .00324 .00392

5.6 5.7 5.8 5.9 6.0

.14160 .15412 .16735 .18130 .19598

.07174 .07966 .08824 .09753 .10756

.03634 .04117 .04653 .05246 .05903

.01841 .02128 .02453 .02822 .03240

.00933 .01100 .01294 .01518 .01778

.00473 .00458 .00682 .00817 .00976

6.1 6.2 6.3 6.4 6.5

.21140 .22758 .24452 .26223 .28071

.11837 .13000 .14249 .15590 .17026

.06627 .07426 .08304 .09269 .10327

.03711 .04242 .04839 .05510 .06263

.02078 .02423 .02820 .03276 .03799

.01163 .01384 .01643 .01948 .02304

6.6 6.7 6.8 6.9 7.0

.29997 .32003 .34087 .36251 .38495

.18562 .20203 .21953 .23819 .25804

.11486 .12754 .14139 .15650 .17297

.07107 .08051 .09106 .10283 .11594

.04398 .05083 .05864 .06756 .07772

.02721 .03209 .03777 .04439 .05210

1.0

(table continues)

98

Chapter 4


Table 4-8. Erlang C Delay Loss Probability (N = 9 Servers Continued) Delay Loss Probability P (>t) for TJT2 =

Offered Traffic (A in Erl)

P(>O)

3.1 3.2 3.3 3.4 3.5

.00501 .00613 .00744 .00897 .01072

.00001 .00001 .00001 .00001

3.6 3.7 3.8 3.9 4.0

.01273 .01502 .01760 .02051 .02376

4.1 4.2 4.3 4.4 4.5

1.2

1.4

1.6

.00002 .00003 .00003 .00005 .00006

.00001 .00001 .00001 .00002 .00002

.00001 .00001

.02738 .03140 .03583 .04071 .04605

.00008 .00010 .00013 .00016 .00021

.00003 .00004 .00005 .00006 .00008

.00003

.00001 .00001 .00001 .00001

4.6 4.7 4.8 4.9 5.0

.05188 .05822 .06509 .07251 .08051

.00026 .00033 .00042 .00053 .00066

.00011 .00014 .00018 .00023 .00030

.00005 .00006 .00008 .00010 .00013

.00002 .00003 .00003 .00005 .00006

.00001 .00001 .00001 .00002 .00003

5.1 5.2 5.3 5.4 5.5

.08910 .09831 .10814 .11862 .12977

.00083 .00103 .00128 .00158 .00195

.00038 .00048 .00061 .00077 .00097

.00017 .00022 .00029 .00037 .00048

.00008 .00011 .00014 .00018 .00024

.00004 .00005 .00007 .00009 .00012

5.6 5.7 5.8 5.9 6.0

.14160 .15412 .16735 .18130 .19598

.00239 .00294 .00360 .00439 .00535

.00121 .00152 .00190 .00236 .00294

.00061 .00078 .00100 .00127 .00161

.00031 .00041 .00053 .00068 .00089

.00016 .00021 .00028 .00037 .00049

6.1 6.2 6.3 6.4 6.5

.21140 .22758 .24452 .26223 .28071

.00651 .00791 .00958 .01158 .01398

.00365 .00452 .00558 .00688 .00848

.00204 .00258 .00325 .00409 .00514

.00114 .00147 .00190 .00243 .00312

.00064 .00084 .00110 .00145 .00189

6.6 6.7 6.8 6.9 7.0

.29997 .32003 .34087 .36251 .38495

.01684 .02025 .02432 .02917 .03492

.01042 .01279 .01567 .01916 .02341

.00645 .00807 .01009 .01259 .01569

.00399 .00510 .00650 .00827 .01052

.00247 .00322 .00418 .00544 .00705

.00001 .00001 .00002

.00003

1.8

2.0

Chapter 4

99



P(>O)

4.1 4.2 4.3 4.4 4.5

Delay Loss Probability P (>t) for T/T2

=

0.2

0.4

0.6

0.8

1.0

.01038 .01216 .01416 .01641 .01892

.00319 .00381 .00453 .00535 .00630

.00098 .00119 .00145 .00175 .00210

.00030 .00037 .00046 .00057 .00070

.00009 .00012 .00015 .00019 .00023

.00003 .00004 .00005 .00006 .00008

4.6 4.7 4.8 4.9 5.0

.02171 .02481 .02823 .03199 .03611

.00737 .00859 .00998 .01153 .01328

.00250 .00298 .00353 .00416 .00489

.00085 .00103 .00125 .00150 .00180

.00029 .00036 .00044 .00054 .00066

.00010 .00012 .00016 .00020 .00024

5.1 5.2 5.3 5.4 5.5

.04061 .04550 .05082 .05658 .06279

.01524 .01742 .01985 .02255 .02553

.00572 .00667 .00775 .00899 .01038

.00215 .00255 .00303 .00358 .00422

.00081 .00098 .00118 .00143 .00172

.00030 .00037 .00046 .00057 .00070

5.6 5.7 5.8 5.9 6.0

.06947 .07665 .08433 .09255 .10130

.02882 .03244 .03641 .04076 .04552

.01195 .01373 .01572 .01795 .02045

.00496 .00581 .00679 .00791 .00919

.00206 .00246 .00293 .00348 .00413

.00085 .00104 .00126 .00153 .00186

6.1 6.2 6.3 6.4 6.5

.11061 .12049 .13096 .14203 .15371

.05070 .05635 .06248 .06913 .07633

.02324 .02635 .02981 .03365 .03791

.01065 .01232 .01422 .01638 .01882

.00488 .00576 .00679 .00797 .00935

.00224 .00270 .00324 .00388 .00464

6.6 6.7 6.8 6.9 7.0

.16602 .17897 .19256 .20681 .22173

.08411 .09250 .10154 .11125 .12169

.04261 .04781 .05354 .05985 .06678

.02159 .02471 .02823 .03219 .03665

.01094 .01277 .01489 .01732 .02011

.00554 .00660 .00787 .00932 .01104

7.1 7.2 7.3 7.4 7.5

.23732 .25360 .27057 .28824 .30661

.13288 .14486 .15768 .17136 .18597

.07440 .08275 .09189 .10188 .11280

.04166 .04726 .05355 .06057 .06841

.02332 .02700 .03120 .03601 .04150

.01306 .01542 .01818 .02141 .02517

7.6 7.7 7.8 7.9 8.0

.32569 .34548 .36599 .38722 .40918

.20153 .21810 .23571 .25442 .27428

.12470 .13768 .15181 .16717 .18386

.07716 .08692 .09777 .10984 .12324

.04775 .05487 .06297 .07217 .08261

.02955 .03464 .04055 .04742 .05538

(table continues)

Chapter 4

100

Table 4-9. Erlang C Delay Loss Probability (N


=10 Servers Continued)



P(>O)

4.1 4.2 4.3 4.4 4.5

.01038 .01216 .01416 .01641 .01892

.00001 .00001 .00002 .00002 .00003

.00001 .00001

4.6 4.7 4.8 4.9 5.0

.02171 .02481 .02823 .03199 .03611

.00003 .00004 .00006 .00007 .00009

.00001 .00001 .00002 .00003 .00003

.00001 .00001 .00001 .00001

5.1 5.2 5.3 5.4 5.5

.04061 .04550 .05082 .05658 .06279

.00011 .00014 .00018 .00023 .00028

.00004 .00005 .00007 .00009 .00012

5.6 5.7 5.8 5.9 6.0

.06947 .07665 .08433 .09255 .10130

.00035 .00044 .00055 .00068 .00083

6.1 6.2 6.3 6.4 6.5

.11061 .12049 .13096 .14203 .15371

6.6 6.7 6.8 6.9 7.0

1.4

1.6

1.8

2.0

.00002 .00002 .00003 .00004 .00005

.00001 .00001 .00001 .00001 .00002

.00001

.00015 .00019 .00024 .00030 .00037

.00006 .00008 .00010 .00013 .00017

.00003 .00003 .00004 .00006 .00008

.00001 .00001 .00002 .00003 .00003

.00103 .00126 .00154 .00189 .00231

.00047 .00059 .00074 .00092 .00114

.00022 .00028 .00035 .00045 .00057

.00010 .00013 .00017 .00022 .00028

.00005 .00006 .00008 .00011 .00014

.16602 .17897 .19256 .20681 .22173

.00281 .00341 .00414 .00501 .00606

.00142 .00176 .00218 .00270 .00332

.00072 .00091 .00115 .00145 .00182

.00036 .00047 .00061 .00078 .00100

.00018 .00024 .00032 .00042 .00055

7.1 7.2 7.3 7.4 7.5

.23732 .25360 .27057 .28824 .30661

.00731 .00881 .01060 .01273 .01527

.00409 .00503 .00618 .00757 .00926

.00229 .00287 .00360 .00450 .00562

.00128 .00164 .00210 .00267 .00341

.00072 .00094 .00122 .00159 .00207

7.6 7.7 7.8 7.9 8.0

.32569 .34548 .36599 .38722 .40918

.01828 .02187 .02612 .03116 .03712

.01131 .01380 .01682 .02047 .02488

.00700 .00871 .01083 .01345 .01668

.00433 .00550 .00698 .00884 .01118

.00268 .00347 .00449 .00581 .00749

1.2

Chapter 4

101



P(>O)

5.1 5.2 5.3 5.4 5.5


=

0.2

0.4

0.6

0.8

1.0

.01726 .01966 .02232 .02523 .02843

.00530 .00616 .00714 .00823 .00946

.00163 .00193 .00228 .00269 .00315

.00050 .00061 .00073 .00088 .00105

.00015 .00019 .00023 .00029 .00035

.00005 .00006 .00007 .00009 .00012

5.6 5.7 5.8 5.9 6.0

.03193 .03574 .03988 .04437 .04922

.01084 .01238 .01410 .01600 .01811

.00368 .00429 .00498 .00577 .00666

.00125 .00149 .00176 .00208 .00245

.00042 .00051 .00062 .00075 .00090

.00014 .00018 .00022 .00027 .00033

6.1 6.2 6.3 6.4 6.5

.05445 .06008 .06611 .07257 .07948

.02044 .02300 .02583 .02892 .03231

.00767 .00881 .01009 .01153 .01314

.00288 .00337 .00394 .00459 .00534

.00108 .00129 .00154 .00183 .00217

.00041 .00049 .00060 .00073 .00088

6.6 6.7 6.8 6.9 7.0

.08683 .09466 .10298 .11179 .12111

.03602 .04006 .04446 .04923 .05442

.01494 .01695 .01919 .02168 .02445

.00620 .00717 .00829 .00955 .01099

.00257 .00304 .00358 .00421 .00494

.00107 .00128 .00154 .00185 .00222

7.1 7.2 7.3 7.4 7.5

.13095 .14133 .15226 .16375 .17581

.06003 .06610 .07265 .07971 .08730

.02752 .03091 .03466 .03880 .04335

.01261 .01446 .01654 .01888 .02153

.00578 .00676 .00789 .00919 .01069

.00265 .00316 .00376 .00447 .00531

7.6 7.7 7.8 7.9 8.0

.18844 .20167 .21549 .22992 .24496

.09547 .10423 .11363 .12368 .13444

.04837 .05387 .05991 .06653 .07378

.02450 .02784 .03159 .03579 .04049

.01241 .01439 .01666 .01923 .02222

.00629 .00744 .00878 .01036 .01220

8.1 8.2 8.3 8.4 8.5

.26062 .27691 .29384 .31140 .32961

.14592 .15817 .17123 .18513 .19992

.08170 .09035 .09979 .11007 .12126

.04574 .05161 .05815 .06544 .07355

.02561 .02948 .03389 .03890 .04461

.01434 .01684 .01975 .02313 .02706

8.6 8.7 8.8 8.9 9.0

.34847 .36798 .38815 .40898 .43047

.21563 .23230 .24998 .26872 .28855

.13343 .14665 .16100 .17656 .19342

.08256 .09258 .10369 .11601 .12966

.05109 .05844 .06678 .07622 .08691

.03161 .03689 .04301 .05008 .05826

(table continues)

Chapter 4

102


P(>O)

5.1 5.2 5.3 5.4 5.5


=11 Servers

Continued)

Delay Loss Probability P (>t) for T/T2 = 1.2

1.4

.01726 .01966 .02232 .02523 .02843

.00001 .00002 .00002 .00003 .00004

.00001 .00001 .00001 .00001

5.6 5.7 5.8 5.9 6.0

.03193 .03574 .03988 .04437 .04922

.00005 .00006 .00008 .00010 .00012

6.1 6.2 6.3 6.4 6.5

.05445 .06008 .06611 .07257 .07948

6.6 6.7 6.8 6.9 7.0

1.6

1.8

.00002 .00002 .00003 .00004 .00004

.00001 .00001 .00001 .00001 .00002

.00001

.00015 .00019 .00023 .00029 .00036

.00006 .00007 .00009 .00012 .00015

.00002 .00003 .00004 .00005 .00006

.00001 .00001 .00001 .00002 .00002

.00001 .00001 .00001

.08683 .09466 .10298 .11179 .12111

.00044 .00054 .00067 .00082 .00100

.00018 .00023 .00029 .00036 .00045

.00008 .00010 .00012 .00016 .00020

.00003 .00004 .00005 .00007 .00009

.00001 .00002 .00002 .00003 .00004

7.1 7.2 7.3 7.4 7.5

.13095 .14133 .15226 .16375 .17581

.00122 .00148 .00180 .00218 .00264

.00056 .00069 .00086 .00106 .00131

.00026 .00032 .00041 .00052 .00065

.00012 .00015 .00020 .00025 .00032

.00005 .00007 .00009 .00012 .00016

7.6 7.7 7.8 7.9 8.0

.18844 .20167 .21549 .22992 .24496

.00319 .00384 .00463 .00557 .00669

.00161 .00199 .00244 .00300 .00367

.00082 .00103 .00129 .00161 .00202

.00041 .00053 .00068 .00087 .00111

.00021 .00027 .00036 .00047 .00061

8.1 8.2 8.3 8.4 8.5

.26062 .27691 .29384 .31140 .32961

.00803 .00962 .01151 .01375 .01641

.00450 .00549 .00671 .00817 .00995

.00252 .00314 .00391 .00486 .00604

.00141 .00179 .00228 .00289 .00366

.00079 .00102 .00133 .00172 .00222

8.6 8.7 8.8 8.9 9.0

.34847 .36798 .38815 .40898 .43047

.01956 .02329 .02770 .03291 .03905

.01210 .01470 .01784 .02162 .02618

.00749 .00928 .01149 .01421 .01755

.00463 .00586 .00740 .00933 .01176

.00287 .00370 .00477 .00613 .00788

2.0

Chapter 4

103


Table 4-11. Erlang C Delay Loss Probability (N= 12 Servers) Delay Loss Probability P (>t) for T,IT2 =

Offered

Traffic

P(>O)

(A in Erl)

0.2

0.4

0.6

0.8

1.0

6.1 6.2 6.3 6.4 6.5

.02521 .02819 .03143 .03495 .03876

.00775 .00884 .01005 .01140 .01290

.00238 .00277 .00321 .00372 .00429

.00073 .00087 .00103 .00121 .00143

.00022 .00027 .00033 .00040 .00048

.00007 .00009 .00011 .00013 .00016

6.6 6.7 6.8 6.9 7.0

.04287 .04729 .05205 .05716 .06262

.01456 .01639 .01840 .02061 .02304

.00494 .00568 .00650 .00743 .00847

.00168 .00197 .00230 .00268 .00312

.00057 .00068 .00081 .00097 .00115

.00019 .00024 .00029 .00035 .00042

7.1 7.2 7.3 7.4 7.5

.06846 .07468 .08130 .08833 .09578

.02569 .02859 .03176 .03520 .03894

.00964 .01095 .01240 .01403 .01583

.00362 .00419 .00485 .00559 .00644

.00136 .00161 .00189 .00223 .00262

.00051 .00061 .00074 .00089 .00106

7.6 7.7 7.8 7.9 8.0

.10367 .11201 .12081 .13009 .13984

.04300 .04740 .05216 .05729 .06283

.01784 .02006 .02252 .02523 .02823

.00740 .00849 .00972 .01111 .01269

.00307 .00359 .00420 .00489 .00570

.00127 .00152 .00181 .00216 .00256

8.1 8.2 8.3 8.4 8.5

.15009 .16084 .17210 .18389 .19621

.06880 .07522 .08211 .08951 .09743

.03154 .03518 .03918 .04357 .04838

.01446 .01645 .01869 .02121 .02403

.00663 .00769 .00892 .01032 .01193

.00304 .00360 .00425 .00502 .00592

8.6 8.7 8.8 8.9 9.0

.20906 .22247 .23642 .25094 .26603

.10591 .11498 .12466 .13499 .14600

.05366 .05943 .06573 .07262 .08013

.02718 .03072 .03466 .03907 .04398

.01377 .01588 .01828 .02101 .02413

.00698 .00821 .00964 .01130 .01325

9.1 9.2 9.3 9.4 9.5

.28170 .29795 .31478 .33220 .35022

.15772 .17019 .18344 .19750 .21242

.08831 .09721 .10690 .11742 .12884

.04944 .05553 .06229 .06981 .07815

.02768 .03172 .03630 .04150 .04740

.01550 .01812 .02115 .02467 .02875

9.6 9.7 9.8 9.9 10.0

.36884 .38807 .40790 .42834 .44939

.22823 .24498 .26270 .28144 .30123

.14123 .15465 .16919 .18492 .20192

.08739 .09763 .10896 .12150 .13535

.05407 .06163 .07018 .07983 .09073

.03346 .03891 .04520 .05245 .06082

(table continues)

Chapter 4

104



Delay Loss Probability P (>t) for TJT2 =

P(>O)

1.2

1.4

1.6

1.8

2.0

6.1 6.2 6.3 6.4 6.5

.02521 .02819 .03143 .03495 .03876

.00002 .00003 .00003 .00004 .00005

.00001 .00001 .00001 .00001 .00002

.00001

6.6 6.7 6.8 6.9 7.0

.04287 .04729 .05205 .05716 .06262

.00007 .00008 .00010 .00013 .00016

.00002 .00003 .00004 .00005 .00006

.00001 .00001 .00001 .00002 .00002

.00001 .00001

7.1 7.2 7.3 7.4 7.5

.06846 .07468 .08130 .08833 .09578

.00019 .00024 .00029 .00035 .00043

.00007 .00009 .00011 .00014 .00018

.00003 .00003 .00004 .00006 .00007

.00001 .00001 .00002 .00002 .00003

.00001 .00001 .00001

7.6 7.7 7.8 7.9 8.0

.10367 .11201 .12081 .13009 .13984

.00053 .00064 .00078 .00095 .00115

.00022 .00027 .00034 .00042 .00052

.00009 .00012 .00015 .00018 .00023

.00004 .00005 .00006 .00008 .00010

.00001 .00002 .00003 .00004 .00005

8.1 8.2 8.3 8.4 8.5

.15009 .16084 .17210 .18389 .19621

.00139 .00168 .00203 .00245 .00294

.00064 .00079 .00097 .00119 .00146

.00029 .00037 .00046 .00058 .00073

.00013 .00017 .00022 .00028 .00036

.00006 .00008 .00011 .00014 .00018

8.6 8.7 8.8 8.9 9.0

.20906 .22247 .23642 .25094 .26603

.00353 .00424 .00508 .00608 .00727

.00179 .00219 .00268 .00327 .00399

.00091 .00113 .00141 .00176 .00219

.00046 .00059 .00074 .00095 .00120

.00023 .00030 .00039 .00051 .00066

9.1 9.2 9.3 9.4 9.5

.28170 .29795 .31478 .33220 .35022

.00868 .01035 .01233 .01467 .01744

.00486 .00591 .00718 .00872 .01058

.00272 .00338 .00419 .00518 .00641

.00152 .00193 .00244 .00308 .00389

.00085 .00110 .00142 .00183 .00236

9.6 9.7 9.8 9.9 10.0

.36884 .38807 .40790 .42834 .44939

.02070 .02456 .02911 .03446 .04077

.01281 .01551 .01875 .02264 .02733

.00793 .00979 .01207 .01488 .01832

.00491 .00618 .00778 .00978 .01228

.00304 .00390 .00501 .00642 .00823

Chapter 4


105

Table 4-12. Erlang C Delay Loss Probability (N= 13 Servers) Offered Traffic (A in Erl)


P(>O) 0.2

0.4

0.6

0.8

1.0

7.1 7.2 7.3 7.4 7.5

.03387 .03737 .04114 .04518 .04952

.01041 .01171 .01316 .01474 .01648

.00320 .00367 .00421 .00481 .00549

.00098 .00115 .00135 .00157 .00183

.00030 .00036 .00043 .00051 .00061

.00009 .00011 .00014 .00017 .00020

7.6 7.7 7.8 7.9 8.0

.05416 .05912 .06440 .07002 .07600

.01839 .02048 .02276 .02525 .02796

.00625 .00710 .00805 .00911 .01029

.00212 .00246 .00284 .00328 .00378

.00072 .00085 .00101 .00118 .00139

.00024 .00030 .00036 .00043 .00051

8.1 8.2 8.3 8.4 8.5

.08234 .08905 .09615 .10364 .11154

.03090 .03410 .03756 .04130 .04535

.01160 .01306 .01467 .01646 .01844

.00435 .00500 .00573 .00656 .00750

.00163 .00191 .00224 .00261 .00305

.00061 .00073 .00087 .00104 .00124

8.6 8.7 8.8 8.9 9.0

.11986 .12860 .13779 .14741 .15750

.04971 .05442 .05948 .06493 .07077

.02062 .02303 .02568 .02860 .03180

.00855 .00974 .01109 .01259 .01429

.00355 .00412 .00479 .00555 .00642

.00147 .00174 .00207 .00244 .00288

9.1 9.2 9.3 9.4 9.5

.16805 .17908 .19059 .20258 .21508

.07704 .08375 .09093 .09861 .10681

.03531 .03917 .04338 .04800 .05304

.01619 .01832 .02070 .02336 .02634

.00742 .00857 .00988 .01137 .01308

.00340 .00401 .00471 .00554 .00649

9.6 9.7 9.8 9.9 10.0

.22808 .24160 .25563 .27018 .28527

.11555 .12487 .13479 .14534 .15656

.05854 .06454 .07107 .07819 .08592

.02966 .03336 .03748 .04206 .04716

.01502 .01728 .01976 .02263 .02588

.00761 .00891 .01042 .01217 .01420

10.1 10.2 10.3 10.4 10.5

.30089 .31706 .33377 .35103 .36885

.16847 .18111 .19450 .20870 .22372

.09433 .10345 .11335 .12407 .13569

.05281 .05909 .06605 .07376 .08230

.02957 .03375 .03849 .04385 .04992

.01656 .01928 .02243 .02607 .03028

10.6 10.7 10.8 10.9 11.0

.38722 .40616 .42566 .44572 .46636

.23961 .25640 .27414 .29286 .31261

.14826 .16186 .17656 .19242 .20955

.09174 .10218 .11371 .12643 .14047

.05677 .06451 .07323 .08307 .09416

.03513 .04072 .04716 .05458 .06312

(table continues)

106

Chapter 4


P(>O)

7.1 7.2 7.3 7.4 7.5


= 13 Servers Continued)


1.2

1.4

1.6

.03387 .03737 .04114 .04518 .04952

.00003 .00004 .00004 .00005 .00007

.00001 .00001 .00001 .00002 .00002

.00001 .00001

7.6 7.7 7.8 7.9 8.0

.05416 .05912 .06440 .07002 .07600

.00008 .00010 .00013 .00015 .00019

.00003 .00004 .00004 .00006 .00007

.00001 .00001 .00002 .00002 .00003

.00001 .00001 .00001

8.1 8.2 8.3 8.4 8.5

.08234 .08905 .09615 .10364 .11154

.00023 .00028 .00034 .00042 .00050

.00009 .00011 .00013 .00017 .00020

.00003 .00004 .00005 .00007 .00008

.00001 .00002 .00002 .00003 .00003

.00001 .00001 .00001 .00001

8.6 8.7 8.8 8.9 9.0

.11986 .12860 .13779 .14741 .15750

.00061 .00074 .00089 .00108 .00130

.00025 .00031 .00039 .00047 .00058

.00011 .00013 .00017 .00021 .00026

.00004 .00006 .00007 .00009 .00012

.00002 .00002 .00003 .00004 .00005

9.1 9.2 9.3 9.4 9.5

.16805 .17908 .19059 .20258 .21508

.00156 .00187 .00225 .00269 .00323

.00071 .00088 .00107 .00131 .00160

.00033 .00041 .00051 .00064 .00080

.00015 .00019 .00024 .00031 .00039

.00007 .00009 .00012 .00015 .00020

9.6 9.7 9.8 9.9 10.0

.22808 .24160 .25563 .27018 .28527

.00386 .00461 .00549 .00655 .00779

.00195 .00238 .00290 .00352 .00428

.00099 .00123 .00153 .00189 .00235

.00050 .00064 .00081 .00102 .00129

.00025 .00033 .00042 .00055 .00071

10.1 10.2 10.3 10.4 10.5

.30089 .31706 .33377 .35103 .36885

.00927 .01101 .01307 .01550 .01836

.00519 .00629 .00762 .00922 .01114

.00291 .00359 .00444 .00548 .00676

.00163 .00205 .00259 .00326 .00410

.00091 .00117 .00151 .00194 .00249

10.6 10.7 10.8 10.9 11.0

.38722 .40616 .42566 .44572 .46636

.02174 .02571 .03038 .03586 .04231

.01345 .01623 .01956 .02356 .02836

.00832 .01024 .01260 .01548 .01901

.00515 .00647 .00811 .01017 .01274

.00319 .00408 .00523 .00668 .00854

1.8

2.0

Chapter 4


107


P(>O)

8.1 8.2 8.3 8.4 8.5

Delay Loss Probability P (>t) for T,fT2 =

0.2

0.4

0.6

0.8

1.0

.04298 .04694 .05117 .05567 .06047

.01321 .01471 .01636 .01817 .02013

.00406 .00461 .00523 .00593 .00670

.00125 .00145 .00167 .00193 .00223

.00038 .00045 .00054 .00063 .00074

.00012 .00014 .00017 .00021 .00025

8.6 8.7 8.8 8.9 9.0

.06557 .07098 .07671 .08277 .08918

.02227 .02459 .02711 .02985 .03281

.00756 .00852 .00958 .01076 .01207

.00257 .00295 .00339 .00388 .00444

.00087 .00102 .00120 .00140 .00163

.00030 .00035 .00042 .00050 .00060

9.1 9.2 9.3 9.4 9.5

.09593 .10305 .11054 .11841 .12667

.03600 .03946 .04318 .04719 .05150

.01351 .01511 .01687 .01881 .02094

.00507 .00578 .00659 .00749 .00851

.00190 .00221 .00257 .00299 .00346

.00071 .00085 .00101 .00119 .00141

9.6 9.7 9.8 9.9 10.0

.13533 .14440 .15388 .16379 .17413

.05613 .06110 .06643 .07214 .07824

.02328 .02586 .02868 .03177 .03516

.00966 .01094 .01238 .01399 .01580

.00401 .00463 .00535 .00616 .00710

.00166 .00196 .00231 .00271 .00319

10.1 10.2 10.3 10.4 10.5

.18491 .19614 .20783 .21998 .23259

.08477 .09173 .09916 .10708 .11550

.03886 .04290 .04731 .05212 .05736

.01781 .02006 .02257 .02537 .02848

.00817 .00938 .01077 .01235 .01414

.00374 .00439 .00514 .00601 .00702

10.6 10.7 10.8 10.9 11.0

.24569 .25926 .27332 .28787 .30292

.12447 .13400 .14412 .15486 .16625

.06306 .06926 .07599 .08331 .09124

.03195 .03580 .04007 .04481 .05007

.01618 .01850 .02113 .02411 .02748

.00820 .00956 .01114 .01297 .01508

11.1 11.2 11.3 11.4 11.5

.31847 .33453 .35110 .36819 .38579

.17831 .19109 .20460 .21890 .23400

.09984 .10915 .11923 .13014 .14193

.05590 .06235 .06948 .07737 .08608

.03130 .03561 .04049 .04600 .05221

.01752 .02034 .02360 .02735 .03167

11.6 11.7 11.8 11.9 12.0

.40392 .42257 .44175 .46146 .48171

.24994 .26679 .28451 .30320 .32290

.15466 .16840 .18323 .19922 .21645

.09570 .10631 .11801 .13090 .14509

.05922 .06711 .07600 .08601 .09726

.03664 .04237 .04895 .05651 .06519

(table continues)

Chapter 4

108


Table 4-13. Erlang C Delay Loss Probability (N = 14 Servers Continued) Delay Loss Probability P (>t) for T,JT2 =


P(>O)

8.1 8.2 8.3 8.4 8.5

.04298 .04694 .05117 .05567 .06047

.00004 .00004 .00005 .00007 .00008

.00001 .00001 .00002 .00002 .00003

.00001 .00001 .00001

8.6 8.7 8.8 &.9 9.0

.06557 .07098 .07671 .08277 .08918

.00010 .00012 .00015 .00018 .00022

.00003 .00004 .00005 .00007 .00008

.00001 .00001 .00002 .00002 .00003

.00001 .00001 .00001 .00001

9.1 9.2 9.3 9.4 9.5

.09593 .10305 .11054 .11841 .12667

.00027 .00032 .00039 .00047 .00057

.00010 .00012 .00015 .00019 .00023

.00004 .00005 .00006 .00008 .00009

.00001 .00002 .00002 .00003 .00004

.00001 .00001 .00001 .00001 .00002

9.6 9.7 9.8 9.9 10.0

.13533 .14440 .15388 .16379 .17413

.00069 .00083 .00100 .00120 .00143

.00029 .00035 .00043 .00053 .00064

.00012 .00015 .00019 .00023 .00029

.00005 .00006 .00008 .00010 .00013

.00002 .00003 .00003 .00004 .00006

10.1 10.2 10.3 10.4 10.5

.18491 .19614 .20783 .21998 .23259

.00172 .00205 .00245 .00293 .00349

.00079 .00096 .00117 .00142 .00173

.00036 .00045 .00056 .00069 .00086

.00017 .00021 .00027 .00034 .00043

.00008 .00010 .00013 .00016 .00021

10.6 10.7 10.8 10.9 11.0

.24569 .25926 .27332 .28787 .30292

.00415 .00494 .00587 .00698 .00828

.00210 .00255 .00310 .00375 .00454

.00107 .00132 .00163 .00202 .00249

.00054 .00068 .00086 .00109 .00137

.00027 .00035 .00045 .00058 .00075

11.1 11.2 11.3 11.4 11.5

.31847 .33453 .35110 .36819 .38579

.00981 .01162 .01375 .01626 .01921

.00549 .00664 .00801 .00967 .01165

.00308 .00379 .00467 .00575 .00707

.00172 .00217 .00272 .00342 .00429

.00096 .00124 .00159 .00203 .00260

11.6 11.7 11.8 11.9 12.0

.40392 .42257 .44175 .46146 .48171

.02267 .02675 .03152 .03713 .04370

.01403 .01688 .02030 .02440 .02929

.00868 .01066 .01308 .01603 .01964

.00537 .00673 .00842 .01053 .01316

.00332 .00425 .00542 .00692 .00882

1.2

1.4

1.6

1.8

2.0

Chapter 4

109



P(>O)

9.1 9.2 9.3 9.4 9.5


0.2

0.4

0.6

0.8

1.0

.05234 .05671 .06134 .06625 .07144

.01608 .01778 .01962 .02162 .02378

.00494 .00557 .00627 .00705 .00792

.00152 .00175 .00201 .00230 .00264

.00047 .00055 .00064 .00075 .00088

.00014 .00017 .00021 .00024 .00029

9.6 9.7 9.8 9.9 10.0

.07694 .08273 .08884 .09528 .10204

.02613 .02866 .03140 .03436 .03754

.00887 .00993 .01110 .01239 .01381

.00301 .00344 .00392 .00447 .00508

.00102 .00119 .00139 .00161 .00187

.00035 .00041 .00049 .00058 .00069

10.1 10.2 10.3 10.4 10.5

.10915 .11661 .12442 .13260 .14115

.04097 .04465 .04860 .05284 .05739

.01537 .01710 .01899 .02106 .02333

.00577 .00655 .00742 .00839 .00949

.00217 .00251 .00290 .00334 .00386

.00081 .00096 .00113 .00133 .00157

10.6 10.7 10.8 10.9 11.0

.15009 .15941 .16914 .17926 .18980

.06225 .06746 .07302 .07895 .08528

.02582 .02885 .03152 .03477 .03832

.01071 .01208 .01361 .01532 .01722

.00444 .00511 .00588 .00675 .00774

.00184 .00216 .00254 .00297 .00348

11.1 11.2 11.3 11.4 11.5

.20076 .21214 .22395 .23620 .24889

.09203 .09921 .10685 .11497 .12360

.04219 .04640 .05098 .05596 .06138

.01934 .02170 .02432 .02724 .03048

.00886 .01015 .01161 .01326 .01514

.00406 .00475 .00554 .00645 .00752

11.6 11.7 11.8 11.9 12.0

.26204 .27563 .28969 .30421 .31919

.13275 .14246 .15275 .16365 .17518

.06725 .07363 .08054 .08803 .09614

.03407 .03806 .04247 .04736 .05276

.01726 .01967 .02239 .02548 .02896

.00875 .01017 .01181 .01370 .01589

12.1 12.2 12.3 12.4 12.5

.33465 .35059 .36701 .38391 .40131

.18737 .20026 .21388 .22825 .24340

.10491 .11439 .12464 .13570 .14763

.05874 .06534 .07263 .08067 .08954

.03289 .03732 .04233 .04796 .05431

.01841 .02132 .02467 .02852 .03294

12.6 12.7 12.8 12.9 13.0

.41919 .43756 .45644 .47581 .49568

.25939 .27623 .29396 .31263 .33227

.16051 .17438 .18932 .20541 .22273

.09932 .11008 .12193 .13497 .14930

.06146 .06949 .07853 .08868 .10008

.03803 .04387 .05058 .05827 .06708

(table continues)

Chapter 4

110



Delay Loss Probability P (>t) for T,fT2

P(>O) 1.2

1.4

1.6

1.8

= 2.0

9.1 9.2 9.3 9.4 9.5

.05234 .05671 .06134 .06625 .07144

.00004 .00005 .00007 .00008 .00010

.00001 .00002 .00002 .00003 .00003

.00001 .00001 .00001 .00001

9.6 9.7 9.8 9.9 10.0

.07694 .08273 .08884 .09528 .10204

.00012 .00014 .00017 .00028 .00025

.00004 .00005 .00006 .00008 .00009

.00001 .00002 .00002 .00003 .00003

.00001 .00001 .00001 .00001

10.1 10.2 10.3 10.4 10.5

.10915 .11661 .12442 .13260 .14115

.00031 .00037 .00044 .00053 .00064

.00011 .00014 .00017 .00021 .00026

.00004 .00005 .00007 .00008 .00011

.00002 .00002 .00003 .00003 .00004

.00001 .00001 .00001 .00001 .00002

10.6 10.7 10.8 10.9 11.0

.15009 .15941 .16914 .17926 .18980

.00076 .00092 .00109 .00131 .00156

.00032 .00039 .00047 .00058 .00070

.00013 .00016 .00020 .00025 .00032

.00005 .00007 .00009 .00011 .00014

.00002 .00003 .00004 .00005 .00006

11.1 11.2 11.3 11.4 11.5

.20076 .21214 .22395 .23620 .24889

.00186 .00222 .00264 .00314 .00373

.00085 .00104 .00126 .00153 .00185

.00039 .00049 .00060 .00074 .00092

.00018 .00023 .00029 .00036 .00046

.00008 .00011 .00014 .00018 .00023

11.6 11.7 11.8 11.9 12.0

.26204 .27563 .28969 .30421 .31919

.00443 .00525 .00623 .00737 .00872

.00224 .00272 .00328 .00397 .00479

.00114 .00140 .00173 .00213 .00263

.00058 .00073 .00091 .00115 .00144

.00029 .00037 .00048 .00062 .00079

12.1 12.2 12.3 12.4 12.5

.33465 .35059 .36701 .38391 .40131

.01031 .01218 .01437 .01695 .01998

.00577 .00696 .00838 .01008 .01212

.00323 .00397 .00488 .00599 .00735

.00181 .00227 .00284 .00356 .00446

.00101 .00130 .00166 .00212 .00270

12.6 12.7 12.8 12.9 13.0

.41919 .43756 .45644 .47581 .49568

.02353 .02769 .03257 .03828 .04497

.01456 .01748 .02098 .02515 .03014

.00901 .01104 .01351 .01653 .02021

.00558 .00697 .00870 .01086 .01354

.00345 .00440 .00560 .00714 .00908

Chapter 4


111


P(>O)

10.1 10.2 10.3 10.4 10.5


=

0.2

0.4

0.6

0.8

1.0

.06181 .06653 .07152 .07678 .08232

.01899 .02086 .02287 .02505 .02740

.00584 .00654 .00732 .00817 .00912

.00179 .00205 .00234 .00267 .00304

.00055 .00064 .00075 .00087 .00101

.00017 .00020 .00024 .00028 .00034

10.6 10.7 10.8 10.9 11.0

.08815 .09428 .10072 .10747 .11454

.02994 .03266 .03560 .03875 .04214

.01017 .01132 .01258 .01397 .01550

.00345 .00392 .00445 .00504 .00570

.00117 .00136 .00157 .00182 .00210

.00040 .00047 .00056 .00066 .00077

11.1 11.2 11.3 11.4 11.5

.12194 .12968 .13776 .14619 .15499

.04576 .04965 .05381 .05826 .06301

.01718 .01901 .02102 .02322 .02562

.00645 .00728 .00821 .00925 .01042

.00242 .00279 .00321 .00369 .00423

.00091 .00107 .00125 .00147 .00172

11.6 11.7 11.8 11.9 12.0

.16415 .17368 .18359 .19389 .20458

.06808 .07349 .07926 .08539 .09192

.02824 .03110 .03422 .03761 .04130

.01171 .01316 .01477 .01656 .01856

.00486 .00557 .00638 .00730 .00834

.00202 .00236 .00275 .00321 .00375

12.1 12.2 12.3 12.4 12.5

.21566 .22715 .23905 .25137 .26411

.09886 .10623 .11406 .12236 .13115

.04532 .04968 .05442 .05956 .06513

.02077 .02323 .02596 .02899 .03234

.00952 .01087 .01239 .01411 .01606

.00437 .00508 .00591 .00678 .00798

12.6 12.7 12.8 12.9 13.0

.27727 .29086 .30489 .31935 .33426

.14047 .15033 .16076 .17179 .18345

.07116 .07770 .08477 .09242 .10068

.03605 .04016 .04470 .04971 .05525

.01827 .02076 .02357 .02674 .03032

.00925 .01073 .01243 .01439 .01664

13.1 13.2 13.3 13.4 13.5

.34962 .36542 .38168 .39840 .41558

.19575 .20873 .22243 .23686 .25206

.10960 .11923 .12962 .14082 .15288

.06137 .06811 .07554 .08372 .09273

.03436 .03890 .04402 .04977 .05624

.01924 .02222 .02565 .02959 .03411

13.6 13.7 13.8 13.9 14.0

.43322 .45133 .46991 .48896 .50849

.26807 .28492 .30264 .32127 .34085

.16588 .17987 .19491 .21109 .22848

.10264 .11355 .12553 .13870 .15315

.06351 .07168 .08085 .09113 .10266

.03930 .04525 .05207 .05988 .06882

(table continues)

112

Chapter 4



P(>O)

10.1 10.2 10.3 10.4 10.5


1.2

1.4

1.6

.06181 .06653 .07152 .07678 .08232

.00005 .00006 .00008 .00009 .00011

.00002 .00002 .00002 .00003 .00004

.00001 .00001 .00001 .00001

10.6 10.7 10.8 10.9 11.0

.08815 .09428 .10072 .10747 .11454

.00014 .00016 .00020 .00024 .00028

.00005 .00006 .00007 .00009 .00010

11.1 11.2 11.3 11.4 11.5

.12194 .12968 .13776 .14619 .15499

.00034 .00041 .00049 .00059 .00070

11.6 11.7 11.8 11.9 12.0

.16415 .17368 .18359 .19389 .20458

12.1 12.2 12.3 12.4 12.5

=

1.8

2.0

.00002 .00002 .00002 .00003 .00004

.00001 .00001 .00001 .00001 .00001

.00001

.00013 .00016 .00019 .00023 .00028

.00005 .00006 .00007 .00009 .00012

.00002 .00002 .00003 .00004 .00005

.00001 .00001 .00001 .00001 .00002

.00084 .00100 .00119 .00142 .00168

.00035 .00042 .00051 .00062 .00076

.00014 .00018 .00022 .00027 .00034

.00006 .00008 .00010 .00012 .00015

.00002 .00003 .00004 .00005 .00007

.21566 .22715 .23905 .25137 .26411

.00200 .00238 .00282 .00334 .00396

.00092 .00111 .00135 .00163 .00197

.00042 .00052 .00064 .00079 .00098

.00019 .00024 .00031 .00039 .00048

.00009 .00011 .00015 .00019 .00024

12.6 12.7 12.8 12.9 13.0

.27727 .29086 .30489 .31935 .33426

.00469 .00554 .00655 .00774 .00913

.00238 .00287 .00346 .00416 .00501

.00120 .00148 .00182 .00224 .00275

.00061 .00077 .00096 .00120 .00151

.00031 .00044 .00051 .00065 .00083

13.1 13.2 13.3 13.4 13.5

.34962 .36542 .38168 .39840 .41558

.01077 .01269 .01495 .01759 .02069

.00603 .00725 .00871 .01046 .01255

.00338 .00414 .00508 .00622 .00761

.00189 .00237 .00296 .00370 .00462

.00106 .00135 .00172 .00220 .00280

13.6 13.7 13.8 13.9 14.0

.43322 .45133 .46991 .48896 .50849

.02432 .02857 .03353 .03934 .04613

.01505 .01803 .02160 .02585 .03092

.00931 .01138 .01391 .01698 .02073

.00576 .00719 .00896 .01116 .01389

.00357 .00454 .00577 .00733 .00931

Chapter 4


113


P(>O)

11.1 11.2 11.3 11.4 11.5

.07128 .07632 .08162 .08718 .09303

11.6 11.7 11.8 11.9 12.0


0.4

0.6

0.8

1.0

.02190 .02392 .02610 .02845 .03097

.00673 .00750 .00835 .00928 .01031

.00207 .00235 .00267 .00303 .00343

.00064 .00074 .00085 .00099 .00114

.00020 .00023 .00027 .00032 .00038

.09915 .10557 .11228 .11930 .12663

.03367 .03657 .03969 .04302 .04658

.01143 .01267 .01403 .01551 .01714

.00388 .00439 .00496 .00559 .00630

.00132 .00152 .00175 .00202 .00232

.00045 .00053 .00062 .00073 .00085

12.1 12.2 12.3 12.4 12.5

.13428 .14225 .15056 .15920 .16819

.05040 .05447 .05881 .06345 .06838

.01891 .02086 .02297 .02528 .02780

.00710 .00799 .00897 .01008 .01130

.00266 .00306 .00351 .00402 .00460

.00100 .00117 .00137 .00160 .00187

12.6 12.7 12.8 12.9 13.0

.17753 .18723 .19729 .20772 .21853

.07364 .07923 .08517 .09149 .09819

.03054 .03353 .03677 .04029 .04412

.01267 .01419 .01587 .01775 .01982

.00525 .00600 .00685 .00782 .00891

.00218 .00254 .00296 .00344 .00400

13.1 13.2 13.3 13.4 13.5

.22971 .24128 .25324 .26559 .27835

.10530 .11284 .12082 .12928 .13822

.04827 .05277 .05765 .06293 .06864

.02213 .02468 .02750 .03063 .03409

.01014 .01154 .01312 .01491 .01693

.00465 .00540 .00626 .00726 .00841

13.6 13.7 13.8 13.9 14.0

.29151 .30507 .31905 .33345 .34827

.14768 .15768 .16823 .17938 .19113

.07482 .08150 .08871 .09650 .10490

.03790 .04212 .04678 .05191 .05757

.01920 .02177 .02466 .02792 .03159

.00973 .01125 .01301 .01502 .01734

14.1 14.2 14.3 14.4 14.5

.36351 .37918 .39528 .41181 .42878

.20353 .21659 .23035 .24483 .26007

.11396 .12372 .13424 .14556 .15774

.06380 .07067 .07823 .08654 .09568

.03572 .04037 .04559 .05145 .05803

.02000 .02306 .02657 .03059 .03520

14.6 14.7 14.8 14.9 15.0

.44619 .46404 .48234 .50108 .52028

.27610 .29294 .31065 .32924 .34875

.17085 .18493 .20007 .21632 .23378

.10572 .11674 .12885 .14214 .15671

.06542 .07370 .08299 .09339 .10504

.04048 .04653 .05345 .06136 .07041

(table continues)

Chapter 4

114

Table Offered Traffic (A in Erl)

4~16.

Erlang C Delay Loss Probability (N

ErJang C Distribution

= 17 Servers

Continued)


P(>O) 1.2

1.4

1.6

1.8

= 2.0

11.1 11.2 11.3 11.4 11.5

.07128 .07632 .08162 .08718 .09303

.00006 .00007 .00009 .00011 .00013

.00002 .00002 .00003 .00003 .00004

.00001 .00001 .00001 .00001 .00001

11.6 11.7 11.8 11.9 12.0

.09915 .10557 .11228 .11930 .12663

.00015 .00018 .00022 .00026 .00031

.00005 .00006 .00008 .00009 .00012

.00002 .00002 .00003 .00003 .00004

.00001 .00001 .00001 .00001 .00002

.00001

12.1 12.2 12.3 12.4 12.5

.13428 .14225 .15056 .15920 .16819

.00038 .00045 .00053 .00064 .00076

.00014 .00017 .00021 .00025 .00031

.00005 .00007 .00008 .00010 .00013

.00002 .00003 .00003 .00004 .00005

.00001 .00001 .00001 .00002 .00002

12.6 12.7 12.8 12.9 13.0

.17753 .18723 .19729 .20772 .21853

.00090 .00180 .00128 .00152 .00180

.00038 .00045 .00055 .00067 .00081

.00016 .00019 .00024 .00029 .00036

.00006 .00008 .00010 .00013 .00016

.00003 .00003 .00004 .00006 .00007

13.1 13.2 13.3 13.4 13.5

.22971 .24128 .25324 .26559 .27835

.00213 .00252 .00299 .00353 .00417

.00098 .00118 .00143 .00172 .00207

.00045 .00055 .00068 .00084 .00103

.00021 .00026 .00032 .00041 .00051

.00009 .00012 .00015 .00020 .00025

13.6 13.7 13.8 13.9 14.0

.29151 .30507 .31905 .33345 .34827

.00493 .00582 .00686 .00808 .00952

.00250 .00301 .00362 .00435 .00522

.00127 .00155 .00191 .00234 .00287

.00064 .00080 .00101 .00126 .00157

.00032 .00042 .00053 .00068 .00086

14.1 14.2 14.3 14.4 14.5

.36351 .37918 .39528 .41181 .42878

.01120 .01317 .01548 .01818 .02135

.00627 .00752 .00902 .01081 .01295

.00351 .00430 .00526 .00643 .00785

.00197 .00245 .00306 .00382 .00476

.00110 .00140 .00179 .00227 .00289

14.6 14.7 14.8 14.9 15.0

.44619 .46404 .48234 .50108 .52028

.02505 .02937 .03442 .04032 .04720

.01550 .01854 .02217 .02649 .03164

.00959 .01170 .01428 .01741 .02121

.00593 .00739 .00920 .01144 .01422

.00367 .00466 .00592 .00751 .00953

Chapter 4


115


P(>O)

12.1 12.2 12.3 12.4 12.5


=

0.2

0.4

0.6

0.8

1.0

.08068 .08600 .09157 .09741 .10351

.02479 .02696 .02929 .03178 .03446

.00762 .00845 .00937 .01037 .01147

.00234 .00265 .00300 .00338 .00382

.00072 .00083 .00096 .00110 .00127

.00022 .00026 .00031 .00036 .00042

12.6 12.7 12.8 12.9 13.0

.10989 .11656 .12351 .13076 .13831

.03732 .04038 .04366 .04715 .05088

.01267 .01399 .01543 .01700 .01872

.00430 .00485 .00545 .00613 .00689

.00146 .00168 .00193 .00221 .00253

.00050 .00058 .00068 .00080 .00093

13.1 13.2 13.3 13.4 13.5

.14617 .15434 .16283 .17165 .18080

.05486 .05910 .06361 .06841 .07351

.02059 .02263 .02485 .02726 .02989

.00773 .00866 .00971 .01086 .01215

.00290 .00332 .00379 .00433 .00494

.00109 .00127 .00148 .00173 .00201

13.6 13.7 13.8 13.9 14.0

.19029 .20012 .21030 .22083 .23172

.07893 .08468 .09079 .09726 .10412

.03274 .03583 .03919 .04284 .04678

.01358 .01516 .01692 .01887 .02102

.00563 .00642 .00730 .00831 .00945

.00234 .00272 .00315 .00366 .00424

14.1 14.2 14.3 14.4 14.5

.24297 .25460 .26659 .27896 .29171

.11138 .11907 .12719 .13579 .14486

.05106 .05568 .06069 .06609 .07194

.02341 .02604 .02895 .03217 .03572

.01073 .01218 .01381 .01566 .01774

.00492 .00570 .00659 .00762 .00881

14.6 14.7 14.8 14.9 15.0

.30485 .31838 .33230 .34662 .36134

.15444 .16456 .17522 .18646 .19831

.07824 .08505 .09239 .10031 .10883

.03964 .04396 .04872 .05396 .05973

.02008 .02272 .02569 .02903 .03278

.01017 .01174 .01355 .01562 .01799

15.1 15.2 15.3 15.4 15.5

.37664 .39199 .40792 .42428 .44104

.21078 .22391 .23772 .25224 .26751

.11802 .12790 .13853 .14996 .16225

.06608 .07306 .08073 .08916 .09841

.03700 .04173 .04704 .05301 .05969

.02071 .02384 .02742 .03151 .03620

15.6 15.7 15.8 15.9 16.0

.45823 .47583 .49386 .51231 .53118

.28354 .30038 .31806 .33661 .35607

.17545 .18963 .20484 .22117 .23868

.10857 .11971 .13193 .14532 .15999

.06718 .07557 .08497 .09548 .10723

.04154 .04771 .05472 .06274 .07189

(table continues)

116

Chapter 4




Delay Loss Probability P (>1) for T,JT2 =

P(>O)

1.2

1.4

1.6

1.8

2.0

12.1 12.2 12.3 12.4 12.5

.08068 .08600 .09157 .09741 .10351

.00007 .00008 .00010 .00012 .00014

.00002 .00003 .00003 .00004 .00005

.00001 .00001 .00001 .00001 .00002

.00001

12.6 12.7 12.8 12.9 13.0

.10989 .11656 .12351 .13076 .13831

.00017 .00020 .00024 .00029 .00034

.00006 .00007 .00009 .00010 .00013

.00002 .00002 .00003 .00004 .00005

.00001 .00001 .00001 .00001 .00002

.00001

13.1 13.2 13.3 13.4 13.5

.14617 .15434 .16283 .17165 .18080

.00041 .00049 .00058 .00069 .00082

.00015 .00019 .00023 .00027 .00033

.00006 .00007 .00009 .00011 .00013

.00002 .00003 .00003 .00004 .00005

.00001 .00001 .00001 .00002 .00002

13.6 13.7 13.8 13.9 14.0

.19029 .20012 .21030 .22083 .23172

.00097 .00115 .00136 .00161 .00191

.00040 .00049 .00059 .00071 .00086

.00017 .00021 .00025 .00031 .00039

.00007 .00009 .00011 .00014 .00017

.00003 .00004 .00005 .00006 .00008

14.1 14.2 14.3 14.4 14.5

.24297 .25460 .26659 .27896 .29171

.00225 .00266 .00314 .00371 .00437

.00103 .00125 .00150 .00181 .00217

.00047 .00058 .00072 .00088 .00108

.00022 .00027 .00034 .00043 .00054

.00010 .00013 .00016 .00021 .00027

14.6 14.7 14.8 14.9 15.0

.30485 .31838 .33230 .34662 .36134

.00515 .00607 .00714 .00840 .00987

.00261 .00314 .00377 .00452 .00542

.00132 .00162 .00199 .00243 .00297

.00067 .00084 .00105 .00131 .00163

.00034 .00043 .00055 .00070 .00090

15.1 15.2 15.3 15.4 15.5

.37664 .39199 .40792 .42428 .44104

.01160 .01362 .01598 .01874 .02196

.00649 .00778 .00931 .01114 .01332

.00364 .00444 .00543 .00662 .00808

.00204 .00254 .00316 .00394 .00490

.00114 .00145 .00184 .00234 .00297

15.6 15.7 15.8 15.9 16.0

.45823 .47583 .49386 .51231 .53118

.02572 .03012 .03524 .04122 .04819

.01592 .01901 .02270 .02708 .03230

.00985 .01200 .01462 .01780 .02165

.00609 .00758 .00941 .01169 .01451

.00377 .00478 .00606 .00768 .00973

Chapter 4


117

Table 4-18. Erlang C Delay Loss Probability (N = 19 Servers) Offered Traffic (Ain Erl)

P(>O)

13.1 13.2 13.3 13.4 13.5


0.4

0.6

0.8

1.0

.08997 .09553 .10135 .10742 .11375

.02765 .02995 .03241 .03505 .03786

.00849 .00939 .01037 .01144 .01260

.00261 .00294 .00332 .00373 .00420

.00080 .00092 .00106 .00122 .00140

.00025 .00029 .00034 .00040 .00046

13.6 13.7 13.8 13.9 14.0

.12035 .12723 .13439 .14184 .14957

.04087 .04408 .04750 .05115 .05502

.01388 .01527 .01679 .01844 .02024

.00471 .00529 .00593 .00665 .00745

.00160 .00183 .00210 .00240 .00274

.00054 .00064 .00074 .00086 .00101

14.1 14.2 14.3 14.4 14.5

.15761 .16595 .17460 .18356 .19284

.05915 .06354 .06820 .07315 .07840

.02220 .02433 .02664 .02915 .03188

.00833 .00932 .01041 .01162 .01296

.00313 .00357 .00407 .00463 .00527

.00117 .00137 .00159 .00185 .00214

14.6 14.7 14.8 14.9 15.0

.20245 .21239 .22266 .23327 .24422

.08397 .08987 .09612 .10274 .10974

.03483 .03803 .04150 .04525 .04931

.01445 .01609 .01791 .01993 .02216

.00599 .00681 .00773 .00878 .00996

.00249 .00288 .00334 .00387 .00447

15.1 15.2 15.3 15.4 15.5

.25552 .26718 .27919 .29156 .30429

.11713 .12495 .13320 .14192 .15111

.05369 .05843 .06355 .06908 .07504

.02461 .02733 .03032 .03362 .03726

.01128 .01278 .01447 .01637 .01850

.00517 .00598 .00690 .00797 .00919

15.6 15.7 15.8 15.9 16.0

.31740 .33088 .34473 .35896 .37357

.16080 .17101 .18177 .19310 .20502

.08146 .08839 .09585 .10388 .11252

.04127 .04568 .05054 .05588 .06175

.02091 .02361 .02665 .03006 .03389

.01059 .01220 .01405 .01617 .01860

16.1 16.2 16.3 16.4 16.5

.38857 .40396 .41973 .43590 .45247

.21756 .23074 .24460 .25915 .27444

.12181 .13180 .14254 .15407 .16646

.06820 .07529 .08307 .09160 .10096

.03819 .04301 .04841 .05446 .06124

.02138 .02456 .02821 .03238 .03714

16.6 16.7 16.8 16.9 17.0

.46943 .48680 .50457 .52274 .54132

.29048 .30731 .32496 .34347 .36286

.17974 .19400 .20929 .22567 .24323

.11122 .12247 .13479 .14828 .16304

.06882 .07731 .08681 .09743 .10929

.04259 .04881 .05591 .06401 .07326

(table continues)

118

Chapter 4


Table 4-18. Erlang C Delay Loss Probability (N = 19 Servers Continued) Delay Loss Probability P (>t) for T,IT2 =


P(>O)

13.1 13.2 13.3 13.4 13.5

.08997 .09553 .10135 .10742 .11375

.00008 .00009 .00011 .00013 .00015

.00002 .00003 .00003 .00004 .00005

.00001 .00001 .00001 .00001 .00002

.00001

13.6 13.7 13.8 13.9 14.0

.12035 .12723 .13439 .14184 .14957

.00018 .00022 .00026 .00031 .00037

.00006 .00008 .00009 .00011 .00014

.00002 .00003 .00003 .00004 .00005

.00001 .00001 .00001 .00001 .00002

.00001 .00001

14.1 14.2 14.3 14.4 14.5

.15761 .16595 .17460 .18356 .19284

.00044 .00052 .00062 .00074 .00087

.00017 .00020 .00024 .00029 .00035

.00006 .00008 .00009 .00012 .00014

.00002 .00003 .00004 .00005 .00006

.00001 .00001 .00001 .00002 .00002

14.6 14.7 14.8 14.9 15.0

.20245 .21239 .22266 .23327 .24422

.00103 .00122 .00144 .00170 .00201

.00043 .00052 .00062 .00075 .00090

.00018 .00022 .00027 .00033 .00041

.00007 .00009 .00012 .00015 .00018

.00003 .00004 .00005 .00006 .00008

15.1 15.2 15.3 15.4 15.5

.25552 .26718 .27919 .29156 .30429

.00237 .00280 .00329 .00388 .00456

.00109 .00131 .00157 .00189 .00227

.00050 .00061 .00075 .00092 .00113

.00023 .00029 .00036 .00045 .00056

.00010 .00013 .00017 .00022 .00028

15.6 15.7 15.8 15.9 16.0

.31740 .33088 .34473 .35896 .37357

.00537 .00631 .00741 .00870 .01021

.00272 .00326 .00391 .00468 .00560

.00138 .00168 .00206 .00252 .00307

.00070 .00087 .00109 .00135 .00169

.00035 .00045 .00057 .00073 .00093

16.1 16.2 16.3 16.4 16.5

.38857 .40396 .41973 .43590 .45247

.01197 .01403 .01644 .01925 .02253

.00670 .00802 .00958 .01144 .01366

.00375 .00458 .00558 .00680 .00829

.00210 .00262 .00325 .00404 .00503

.00118 .00149 .00190 .00240 .00305

16.6 16.7 16.8 16.9 17.0

.46943 .48680 .50457 .52274 .54132

.02635 .03081 .03601 .04206 .04911

.01631 .01945 .02319 .02764 .03292

.01009 .01228 .01494 .01816 .02207

.00624 .00775 .00962 .01193 .01479

.00386 .00489 .00619 .00784 .00992

1.2

1.4

1.6

1.8

2.0

Chapter 4

119




P(>O) 0.2

0.4

0.6

0.8

1.0

14.1 14.2 14.3 14.4 14.5

.09910 .10488 .11091 .11719 .12372

.03045 .03288 .03547 .03824 .04118

.00936 .01031 .01134 .01248 .01371

.00288 .00323 .00363 .00407 .00456

.00088 .00101 .00116 .00133 .00152

.00027 .00032 .00037 .00043 .00051

14.6 14.7 14.8 14.9 15.0

.13052 .13758 .14492 .15253 .16043

.04432 .04767 .05122 .05500 .05902

.01505 .01651 .01810 .01983 .02171

.00511 .00572 .00640 .00715 .00799

.00174 .00198 .00226 .00258 .00294

.00059 .00069 .00080 .00093 .00108

15.1 15.2 15.3 15.4 15.5

.16862 .17710 .18588 .19496 .20435

.06328 .06781 .07261 .07769 .08308

.02375 .02596 .02836 .03096 .03378

.00891 .00994 .01108 .01234 .01373

.00335 .00381 .00433 .00492 .00558

.00126 .00146 .00169 .00196 .00227

15.6 15.7 15.8 15.9 16.0

.21405 .22407 .23441 .24508 .25608

.08878 .09482 .10120 .10794 .11506

.03683 .04012 .04369 .04754 .05170

.01528 .01698 .01886 .02094 .02323

.00634 .00718 .00814 .00922 .01044

.00263 .00304 .00352 .00406 .00469

16.1 16.2 16.3 16.4 16.5

.26741 .27908 .29109 .30345 .31616

.12258 .13052 .13889 .14771 .15700

.05619 .06104 .06626 .07190 .07796

.02576 .02855 .03162 .03500 .03872

.01181 .01335 .01508 .01703 .01923

.00541 .00624 .00720 .00829 .00955

16.6 16.7 16.8 16.9 17.0

.32922 .34264 .35642 .37055 .38506

.16679 .17709 .18794 .19934 .21133

.08450 .09153 .09910 .10723 .11598

.04281 .04731 .05225 .05769 .06365

.02169 .02445 .02755 .03103 .03493

.01099 .01264 .01453 .01669 .01917

17.1 17.2 17.3 17.4 17.5

.39993 .41518 .43079 .44679 .46316

.22392 .23715 .25104 .26562 .28092

.12537 .13546 .14630 .15792 .17039

.07020 .07738 .08525 .09389 .10335

.03930 .04420 .04968 .05582 .06268

.02201 .02525 .02895 .03318 .03802

17.6 17.7 17.8 17.9 18.0

.47991 .49705 .51457 .53248 .55077

.29696 .31378 .33140 .34986 .36920

.18376 .19808 .21344 .22988 .24748

.11371 .12505 .13746 .15104 .16589

.07036 .17894 .08853 .09924 .11120

.04354 .04983 .05702 .06521 .07454

(table continues)

120

Chapter 4



=20 Servers


P(>O)

1.2

1.4

1.6

1.8

Continued)

= 2.0

14.1 14.2 14.3 14.4 14.5

.09910 .10488 .11091 .11719 .12372

.00008 .00010 .00012 .00014 .00017

.00003 .00003 .00004 .00005 .00006

.00001 .00001 .00001 .00002 .00002

.00001

14.6 14.7 14.8 14.9 15.0

.13052 .13758 .14492 .15253 .16043

.00020 .00024 .00028 .00034 .00040

.00007 .00008 .00010 .00012 .00015

.00002 .00003 .00004 .00004 .00005

.00001 .00001 .00001 .00002 .00002

.00001 .00001

15.1 15.2 15.3 15.4 15.5

.16862 .17710 .18588 .19496 .20435

.00047 .00056 .00066 .00078 .00092

.00018 .00021 .00026 .00031 .00038

.00007 .00008 .00010 .00012 .00015

.00002 .00003 .00004 .00005 .00006

.00001 .00001 .00002 .00002 .00003

15.6 15.7 15.8 15.9 16.0

.21405 .22407 .23441 .24508 .25608

.00109 .00129 .00152 .00179 .00211

.00045 .00054 .00066 .00079 .00095

.00019 .00023 .00028 .00035 .00043

.00008 .00010 .00012 .00015 .00019

.00003 .00004 .00005 .00007 .00009

16.1 16.2 16.3 16.4 16.5

.26741 .27908 .29109 .30345 .31616

.00248 .00292 .00343 .00404 .00474

.00114 .00137 .00164 .00196 .00235

.00052 .00064 .00078 .00096 .00117

.00024 .00030 .00037 .00047 .00058

.00011 .00014 .00018 .00023 .00029

16.6 16.7 16.8 16.9 17.0

.32922 .34264 .35642 .37055 .38506

.00557 .00653 .00766 .00898 .01052

.00282 .00338 .00404 .00483 .00577

.00143 .00174 .00213 .00260 .00317

.00072 .00090 .00112 .00140 .00174

.00037 .00047 .00059 .00075 .00095

17.1 17.2 17.3 17.4 17.5

.39993 .41518 .43079 .44679 .46316

.01232 .01442 .01687 .01973 .02306

.00690 .00824 .00983 .01173 .01399

.00386 .00471 .00573 .00697 .00848

.00216 .00269 .00334 .00415 .00515

.00121 .00154 .00195 .00246 .00312

17.6 17.7 17.8 17.9 18.0

.47991 .49705 .51457 .53248 .55077

.02694 .03146 .03672 .04284 .04997

.01667 .01986 .02365 .02815 .03349

.01032 .01254 .01523 .01850 .02245

.00638 .00791 .00981 .01215 .01505

.00395 .00500 .00632 .00799 .01009

Chapter 4


121


P(>O)

15.1 15.2 15.3 15.4 15.5


=

0.2

0.4

0.6

0.8

1.0

.10806 .11403 .12025 .12671 .13342

.03320 .03575 .03846 .04134 .04441

.01020 .01121 .01230 .01349 .01478

.00314 .00351 .00393 .00440 .00492

.00096 .00110 .00126 .00144 .00164

.00030 .00035 .00040 .00047 .00055

15.6 15.7 15.8 15.9 16.0

.14038 .14761 .15510 .16286 .17089

.04767 .05114 .05482 .05873 .06287

.01619 .01772 .01938 .02118 .02313

.00550 .00614 .00685 .00764 .00851

.00187 .00213 .00242 .00275 .00313

.00063 .00074 .00086 .00099 .00115

16.1 16.2 16.3 16.4 16.5

.17921 .18781 .19669 .20587 .21535

.06726 .07191 .07683 .08204 .08756

.02524 .02753 .03001 .03270 .03560

.00947 .01054 .01172 .01303 .01447

.00356 .00404 .00458 .00519 .00588

.00133 .00155 .00179 .00207 .00239

16.6 16.7 16.8 16.9 17.0

.22513 .23522 .24561 .25632 .26735

.09338 .09954 .10603 .11289 .12013

.03873 .04212 .04578 .04972 .05398

.01607 .01782 .01976 .02190 .02425

.00666 .00754 .00853 .00965 .01090

.00276 .00319 .00368 .00425 .00490

17.1 17.2 17.3 17.4 17.5

.27870 .29037 .30238 .31471 .32738

.12776 .13580 .14427 .15319 .16257

.05857 .06351 .06883 .07456 .08073

.02685 .02970 .03284 .03629 .04009

.01231 .01389 .01567 .01767 .01991

.00564 .00650 .00748 .00860 .00989

17.6 17.7 17.8 17.9 18.0

.34039 .35374 .36744 .38148 .39587

.17245 .18283 .19375 .20522 .21726

.08737 .09450 .10216 .11039 .11924

.04426 .04884 .05387 .05939 .06544

.02242 .02524 .02840 .03195 .03591

.01136 .01305 .01498 .01719 .01971

18.1 18.2 18.3 18.4 18.5

.41062 .42572 .44118 .45700 .47319

.22991 .24318 .25710 .27170 .28700

.12872 .13891 .14983 .16153 .17408

.07207 .07934 .08731 .09603 .10558

.04035 .04532 .05088 .05709 .06404

.02259 .02589 .02965 .03394 .03884

18.6 18.7 18.8 18.9 19.0

.48974 .50665 .52394 .54159 .55962

.30304 .31984 .33744 .35585 .37512

.18752 .20191 .21732 .23381 .25145

.11603 .12746 .13996 .15363 .16856

.07180 .08047 .09014 .10094 .11299

.04443 .05080 .05805 .06632 .07574

(table continues)

122

Chapter 4




P(>O) 1.2

1.4

1.6

1.8

2.0

15.1 15.2 15.3 15.4 15.5

.10806 .11403 .12025 .12671 .13342

.00009 .00011 .00013 .00015 .00018

.00003 .00003 .00004 .00005 .00006

.00001 .00001 .00001 .00002 .00002

.00001 .00001

15.6 15.7 15.8 15.9 16.0

.14038 .14761 .15510 .16286 .17089

.00022 .00026 .00030 .00036 .00042

.00007 .00009 .00011 .00013 .00016

.00002 .00003 .00004 .00005 .00006

.00001 .00001 .00001 .00002 .00002

.00001 .00001

16.1 16.2 16.3 16.4 16.5

.17921 .18781 .19669 .20587 .21535

.00050 .00059 .00070 .00082 .00097

.00019 .00023 .00027 .00033 .00040

.00007 .00009 .00011 .00013 .00016

.00003 .00003 .00004 .00005 .00007

.00001 .00001 .00002 .00002 .00003

16.6 16.7 16.8 16.9 17.0

.22513 .23522 .24561 .25632 .26735

.00115 .00135 .00159 .00187 .00220

.00048 .00057 .00069 .00082 .00099

.00020 .00024 .00033 .00036 .00044

.00008 .00010 .00013 .00016 .00020

.00003 .00004 .00006 .00007 .00009

17.1 17.2 17.3 17.4 17.5

.27870 .29037 .30238 .31471 .32738

.00259 .00304 .00357 .00419 .00491

.00119 .00142 .00170 .00204 .00244

.00054 .00066 .00081 .00099 .00121

.00025 .00031 .00039 .00048 .00060

.00011 .00015 .00018 .00023 .00030

17.6 17.7 17.8 17.9 18.0

.34039 .35374 .36744 .38148 .39587

.00576 .00674 .00790 .00924 .01082

.00292 .00349 .00416 .00497 .00594

.00148 .00180 .00220 .00268 .00326

.00075 .00093 .00116 .00144 .00179

.00038 .00048 .00061 .00077 .00098

18.1 18.2 18.3 18.4 18.5

.41062 .42572 .44118 .45700 .47319

.01265 .01479 .01728 .02018 .02356

.00708 .00845 .01007 .01200 .01429

.00397 .00483 .00587 .00713 .00867

.00222 .00276 .00342 .00424 .00526

.00124 .00157 .00199 .00252 .00319

18.6 18.7 18.8 18.9 19.0

.48974 .50665 .52394 .54159 .55962

.02749 .03207 .03739 .04358 .05077

.01701 .02024 .02408 .02863 .03403

.01053 .01278 .01551 .01881 .02281

.00651 .00807 .00999 .01236 .01529

.00403 .00509 .00643 .00812 .01025

Chapter 4


123


P(>O)

16.1 16.2 16.3 16.4 16.5


0.4

0.6

0.8

1.0

.11683 .12297 .12935 .13597 .14283

.03590 .03855 .04137 .04436 .04755

.01103 .01208 .01323 .01448 .01583

.00339 .00379 .00423 .00472 .00527

.00104 .00119 .00135 .00154 .00175

.00032 .00037 .00043 .00050 .00058

16.6 16.7 16.8 16.9 17.0

.14995 .15731 .16494 .17282 .18097

.05092 .05450 .05830 .06232 .06658

.01729 .01888 .02061 .02247 .02449

.00587 .00654 .00728 .00810 .00901

.00199 .00227 .00257 .00292 .00331

.00068 .00079 .00091 .00105 .00122

17.1 17.2 17.3 17.4 17.5

.18940 .19810 .20707 .21634 .22589

.07108 .07585 .08089 .08621 .09184

.02668 .02904 .03160 .03436 .03734

.01001 .01112 .01234 .01369 .01518

.00376 .00426 .00482 .00546 .00617

.00141 .00163 .00188 .00217 .00251

17.6 17.7 17.8 17.9 18.0

.23573 .24586 .25630 .26704 .27808

.09778 .10404 .11065 .11761 .12495

.04056 .04403 .04777 .05180 .05614

.01682 .01863 .02062 .02281 .02523

.00698 .00788 .00890 .01005 .01134

.00289 .00334 .00384 .00443 .00509

18.1 18.2 18.3 18.4 18.5

.28943 .30110 .31308 .32539 .33801

.13268 .14081 .14938 .15838 .16785

.06082 .06585 .07127 .07709 .08335

.02788 .03080 .03400 .03753 .04139

.01278 .01440 .01622 .01827 .02055

.00586 .00674 .00774 .00889 .01021

18.6 18.7 18.8 18.9 19.0

.35096 .36424 .37785 .39180 .40608

.17780 .18826 .19924 .21077 .22286

.09008 .09730 .10506 .11338 .12231

.04564 .05029 .05540 .06099 .06713

.02312 .02599 .02921 .03281 .03684

.01171 .01343 .01540 .01765 .02022

19.1 19.2 19.3 19.4 19.5

.42070 .43567 .45097 .46663 .48263

.23555 .24886 .26280 .27742 .29273

.13189 .14215 .15315 .16493 .17755

.07384 .08120 .08925 .09806 .10769

.04134 .04638 .05201 .05830 .06532

.02315 .02649 .03031 .03466 .03962

19.6 19.7 19.8 19.9 20.0

.49898 .51568 .53274 .55015 .56792

.30876 .32554 .34310 .36147 .38069

.19106 .20551 .22097 .23751 .25518

.11822 .12974 .14231 .15605 .17106

.07315 .08190 .09166 .10253 .11466

.04527 .05170 .05903 .06737 .07686

(table continues)

Chapter 4

124



P(>O)

16.1 16.2 16.3 16.4 16.5


1.2

1.4

1.6

1.8

.11683 .12297 .12935 .13597 .14283

.00010 .00012 .00014 .00016 .00019

.00003 .00004 .00004 .00005 .00006

.00001 .00001 .00001 .00002 .00002

.00001 .00001

16.6 16.7 16.8 16.9 17.0

.14995 .15731 .16494 .17282 .18097

.00023 .00027 .00032 .00038 .00045

.00008 .00009 .00011 .00014 .00017

.00003 .00003 .00004 .00005 .00006

.00001 .00001 .00001 .00002 .00002

.00001 .00001 .00001

17.1 17.2 17.3 17.4 17.5

.18940 .19810 .20707 .21634 .22589

.00053 .00062 .00074 .00087 .00102

.00020 .00024 .00029 .00035 .00041

.00007 .00009 .00011 .00014 .00017

.00003 .00004 .00004 .00005 .00007

.00001 .00001 .00002 .00002 .00003

17.6 17.7 17.8 17.9 18.0

.23573 .24586 .25630 .26704 .27808

.00120 .00141 .00166 .00195 .00229

.00050 .00060 .00072 .00086 .00103

.00021 .00025 .00031 .00038 .00046

.00009 .00011 .00013 .00017 .00021

.00004 .00005 .00006 .00007 .00009

18.1 18.2 18.3 18.4 18.5

.28943 .30110 .31308 .32539 .33801

.00269 .00315 .00369 .00433 .00507

.00123 .00147 .00176 .00211 .00252

.00056 .00069 .00084 .00103 .00125

.00026 .00032 .00040 .00050 .00062

.00012 .00015 .00019 .00024 .00031

18.6 18.7 18.8 18.9 19.0

.35096 .36424 .37785 .39180 .40608

.00593 .00694 .00812 .00950 .01110

.00301 .00359 .00428 .00511 .00609

.00152 .00185 .00226 .00275 .00334

.00077 .00096 .00119 .00148 .00183

.00039 .00050 .00063 .00080 .00101

19.1 19.2 19.3 19.4 19.5

.42070 .43567 .45097 .46663 .48263

.01296 .01513 .01766 .02061 .02403

.00726 .00864 .01029 .01225 .01457

.00406 .00494 .00600 .00728 .00884

.00227 .00282 .00350 .00433 .00536

.00127 .00161 .00204 .00257 .00325

19.6 19.7 19.8 19.9 20.0

.49898 .51568 .53274 .55015 .56792

.02801 .03264 .03802 .04427 .05152

.01733 .02060 .02448 .02908 .03454

.01073 .01301 .01577 .01911 .02315

.00664 .00821 .01016 .01256 .01552

.00411 .00518 .00654 .00825 .01040

2.0

Chapter 4


125


P(>O)

17.1 17.2 17.3 17.4 17.5

Delay Loss Probability P (>1) for T,IT 2 =

0.2

0.4

0.6

0.8

1.0

.12539 .13169 .13822 .14498 .15198

.03853 .04128 .04420 .04730 .05059

.01184 .01294 .01414 .01543 .01684

.00364 .00406 .00452 .00504 .00561

.00112 .00127 .00145 .00164 .00187

.00034 .00040 .00046 .00054 .00062

17.6 17.7 17.8 17.9 18.0

.15922 .16671 .17445 .18244 .19069

.05407 .05776 .06166 .06579 .07015

.01836 .02001 .02179 .02372 .02581

.00624 .00693 .00770 .00855 .00949

.00212 .00240 .00272 .00308 .00349

.00072 .00083 .00096 .00111 .00128

18.1 18.2 18.3 18.4 18.5

.19921 .20799 .21704 .22637 .23598

.07477 .07964 .08478 .09021 .09594

.02806 .03049 .03312 .03595 .03901

.01053 .01168 .01294 .01433 .01586

.00395 .00447 .00505 .00571 .00645

.00148 .00171 .00197 .00228 .00262

18.6 18.7 18.8 18.9 19.0

.24587 .25604 .26650 .27726 .28831

.10198 .10835 .11505 .12211 .12955

.04230 .04585 .04967 .05378 .05821

.01755 .01940 .02144 .02369 .02615

.00728 .00821 .00926 .01043 .01175

.00302 .00347 .00400 .00459 .00528

19.1 19.2 19.3 19.4 19.5

.29966 .31131 .32326 .33553 .34810

.13736 .14559 .15423 .16332 .17286

.06297 .06809 .07359 .07950 .08584

.02887 .03184 .03511 .03870 .04263

.01323 .01489 .01675 .01884 .02117

.00607 .00696 .00799 .00917 .01051

19.6 19.7 19.8 19.9 20.0

.36099 .37420 .38772 .40157 .41574

.18289 .19341 .20444 .21602 .22816

.09265 .09996 .10780 .11621 .12522

.04694 .05167 .05684 .06251 .06872

.02378 .02670 .02997 .03363 .03772

.01205 .01380 .01580 .01809 .02070

20.1 20.2 20.3 20.4 20.5

.43024 .44506 .46022 .47571 .49153

.24089 .25422 .26819 .28282 .29813

.13487 .14522 .15629 .16814 .18082

.07552 .08295 .09108 .09996 .10968

.04228 .04738 .05308 .05943 .06652

.02367 .02706 .03093 .03533 .04035

20.6 20.7 20.8 20.9 21.0

.50769 .52419 .54103 .55821 .57573

.31415 .33091 .34844 .36677 .38593

.19439 .20890 .22441 .24099 .25869

.12029 .13188 .14453 .15834 .17341

.07443 .08325 .09308 .10404 .11624

.04606 .05256 .05995 .06836 .07792

(table continues)

Chapter 4

126



P(>O)

17.1 17.2 17.3 17.4 17.5


=

1.2

1.4

1.6

1.8

.12539 .13169 .13822 .14498 .15198

.00011 .00012 .00015 .00017 .00021

.00003 .00004 .00005 .00006 .00007

.00001 .00001 .00002 .00002 .00002

.00001 .00001

17.6 17.7 17.8 17.9 18.0

.15922 .16671 .17445 .18244 .19069

.00024 .00029 .00034 .00040 .00047

.00008 .00010 .00012 .00014 .00017

.00003 .00003 .00004 .00005 .00006

.00001 .00001 .00002 .00002 .00002

.00001 .00001 .00001

18.1 18.2 18.3 18.4 18.5

.19921 .20799 .21704 .22637 .23598

.00056 .00066 .00077 .00091 .00107

.00021 .00025 .00030 .00036 .00043

.00008 .00010 .00012 .00014 .00018

.00003 .00004 .00005 .00006 .00007

.00001 .00001 .00002 .00002 .00003

18.6 18.7 18.8 18.9 19.0

.24587 .25604 .26650 .27726 .28831

.00125 .00147 .00173 .00202 .00237

.00052 .00062 .00074 .00089 .00107

.00022 .00026 .00032 .00039 .00048

.00009 .00011 .00014 .00017 .00022

.00004 .00005 .00006 .00008 .00010

19.1 19.2 19.3 19.4 19.5

.29966 .31131 .32326 .33553 .34810

.00278 .00326 .00381 .00446 .00522

.00127 .00152 .00182 .00217 .00259

.00058 .00071 .00087 .00106 .00129

.00027 .00033 .00041 .00051 .00064

.00012 .00016 .00020 .00025 .00032

19.6 19.7 19.8 19.9 20.0

.36099 .37420 .38772 .40157 .41574

.00610 .00713 .00833 .00973 .01136

.00309 .00369 .00439 .00524 .00623

.00157 .00191 .00232 .00282 .00342

.00079 .00098 .00122 .00151 .00188

.00040 .00051 .00064 .00081 .00103

20.1 20.2 20.3 20.4 20.5

.43024 .44506 .46022 .47571 .49153

.01325 .01546 .01802 .02101 .02447

.00742 .00883 .01050 .01249 .01484

.00416 .00504 .00612 .00742 .00900

.00233 .00288 .00357 .00441 .00546

.00130 .00165 .00208 .00262 .00331

20.6 20.7 20.8 20.9 21.0

.50769 .52419 .54103 .55821 .57573

.02850 .03318 .03861 .04491 .05223

.01764 .02094 .02487 .02951 .03501

.01091 .01322 .01601 .01939 .02347

.00675 .00835 .01031 .01274 .01573

.00418 .00527 .00664 .00837 .01055

2.0

Chapter 4


127

Table 4-23. Erlang C Delay Loss Probability (N = 24 Servers) Delay Loss Probability P (>t) for T,fT2 =


P(>O)

18.1 18.2 18.3 18.4 18.5

.13375 .14019 .14684 .15373 .16085

.04110 .04395 .04696 .05016 .05354

18.6 18.7 18.8 18.9 19.0

.16820 .17580 .18364 .19172 .20006

19.1 19.2 19.3 19.4 19.5

0.2

0.4

0.6

0.8

1.0

.01263 .01378 .01502 .01637 .01782

.00388 .00432 .00480 .00534 .00593

.00119 .00135 .0015-1 .00174 .00197

.00037 .00042 .00049 .00057 .00066

.05712 .06091 .06491 .06914 .07360

.01940 .02110 .02294 .02493 .02708

.00659 .00731 .00811 .00899 .00996

.00224 .00253 .00287 .00324 .00366

.00076 .00088 .00101 .00117 .00135

.20866 .21751 .22663 .23601 .24566

.07831 .08328 .08853 .09405 .09988

.02939 .03189 .03458 .03748 .04061

.01103 .01221 .01351 .01494 .01651

.00414 .00468 .00528 .00595 .00671

.00155 .00179 .00206 .00237 .00273

19.6 19.7 19.8 19.9 20.0

.25558 .26578 .27626 .28703 .29807

.10601 .11247 .11927 .12642 .13393

.04397 .04759 .05149 .05568 .06018

.01824 .02104 .02223 .02452 .02704

.00757 .00852 .00960 .01080 .01215

.00314 .00361 .00414 .00476 .00546

20.1 20.2 20.3 20.4 20.5

.30941 .32104 .33296 .34518 .35770

.14184 .15014 .15886 .16802 .17763

.06502 .07022 .07580 .08178 .08821

.02980 .03284 .03616 .03981 .04380

.01366 .01536 .01725 .01938 .02175

.00626 .00718 .00823 .00943 .01080

20.6 20.7 20.8 20.9 21.0

.37053 .38366 .39709 .41084 .42490

.18772 .19829 .20938 .22101 .23319

.09510 .10249 .11041 .11889 .12798

.04818 .05297 .05822 .06396 .07024

.02441 .02738 .03070 .03441 .03855

.01237 .01415 .01619 .01851 .02115

21.1 21.2 21.3 21.4 21.5

.43927 .45396 .46897 .48430 .49995

.24595 .25931 .27329 .28792 .30323

.13771 .14812 .15926 .17118 .18392

.07710 .08461 .09281 .10177 .11155

.04317 .04833 .05408 .06050 .06766

.02417 .02761 .03152 .03597 .04104

21.6 21.7 21.8 21.9 22.0

.51592 .53223 .54886 .56581 .58310

.31925 .33599 .35348 .37177 .39087

.19754 .21210 .22766 .24427 .26201

.12224 .13390 .14662 .16050 .17563

.07564 .08453 .09443 .15045 .11773

.04680 .05336 .06082 .06929 .07892

(table continues)

Chapter 4

128

Table 4-23. Erlang C Delay Loss Probability (N





P(>O)

18.1 18.2 18.3 18.4 18.5

.13375 .14019 .14684 .15373 .16085

.00011 .00013 .00016 .00019 .00022

18.6 18.7 18.8 18.9 19.0

.16820 .17580 .18364 .19172 .20006

19.1 19.2 19.3 19.4 19.5

1.2

1.4

=

1.6

1.8

.00003 .00004 .00005 .00006 .00007

.00001 .00001 .00002 .00002 .00002

.00001 .00001 .00001

.00026 .00030 .00036 .00042 .00050

.00009 .00011 .00013 .00015 .00018

.00003 .00004 .00004 .00005 .00007

.00001 .00001 .00002 .00002 .00002

.00001 .00001 .00001

.20866 .21751 .22663 .23601 .24566

.00058 .00069 .00081 .00095 .00111

.00022 .00026 .00031 .00038 .00045

.00008 .00010 .00012 .00015 .00018

.00003 .00004 .00005 .00006 .00007

.00001 .00001 .00002 .00002 .00003

19.6 19.7 19.8 19.9 20.0

.25558 .26578 .27626 .28703 .29807

.00130 .00153 .00179 .00210 .00245

.00054 .00065 .00077 .00092 .00110

.00022 .00027 .00033 .00041 .00050

.00009 .00012 .00014 .00018 .00022

.00004 .00005 .00006 .00008 .00010

20.1 20.2 20.3 20.4 20.5

.30941 .32104 .33296 .34518 .35770

.00287 .00336 .00393 .00459 .00536

.00132 .00157 .00187 .00223 .00266

.00060 .00073 .00089 .00109 .00132

.00028 .00034 .00043 .00053 .00066

.00013 .00016 .00020 .00026 .00033

20.6 20.7 20.8 20.9 21.0

.37053 .38366 .39709 .41084 .42490

.00626 .00731 .00854 .00996 .01161

.00317 .00378 .00450 .00536 .00637

.00161 .00195 .00237 .00288 .00350

.00081 .00101 .00125 .00155 .00192

.00041 .00052 .00066 .00083 .00105

21.1 21.2 21.3 21.4 21.5

.43927 .45396 .46897 .48430 .49995

.01351 .01577 .01837 .02139 .02489

.00758 .00901 .01070 .01271 .01510

.00424 .00515 .00624 .00756 .00916

.00238 .00294 .00363 .00449 .00555

.00133 .00168 .00212 .00267 .00337

21.6 21.7 21.8 21.9 22.0

.51592 .53223 .54886 .56581 .58310

.02896 .03369 .03917 .04553 .05290

.01792 .02127 .02523 .02991 .03456

.01109 .01342 .01625 .01965 .02377

.00686 .00847 .01046 .01291 .01593

.00425 .00535 .00674 .00849 .01068

2.0

Chapter 4


129


P(>O)

19.1 19.2 19.3 19.4 19.5


0.4

0.6

0.8

1.0

.14191 .14846 .15524 .16223 .16946

.04360 .04654 .04965 .05293 .05641

.01340 .01459 .01588 .01727 .01878

.00412 .00457 .00508 .00564 .00625

.00127 .00143 .00162 .00184 .00208

.00039 .00045 .00052 .00060 .00069

19.6 19.7 19.8 19.9 20.0

.17691 .18460 .19252 .20069 .20910

.06008 .06396 .06805 .07237 .07692

.02040 .02216 .02405 .02610 .02830

.00693 .00768 .00850 .00941 .01041

.00235 .00266 .00300 .00339 .00383

.00080 .00092 .00106 .00122 .00141

20.1 20.2 20.3 20.4 20.5

.21776 .22667 .23584 .24527 .25495

.08173 .08679 .09213 .09774 .10366

.03067 .03323 .03599 .03895 .04214

.01151 .01272 .01406 .01552 .01713

.00432 .00487 .00549 .00619 .00697

.00162 .00187 .00215 .00247 .00283

20.6 20.7 20.8 20.9 21.0

.26490 .27512 .28561 .29637 .30741

.10988 .11642 .12330 .13053 .13813

.04558 .04926 .05323 .05749 .06207

.01890 .02085 .02298 .02532 .02789

.00784 .00882 .00992 .01115 .01253

.00325 .00373 .00428 .00491 .00563

21.1 21.2 21.3 21.4 21.5

.31873 .33033 .34221 .35439 .36685

.14611 .15448 .16328 .17250 .18217

.06698 .07225 .07790 .08396 .09046

.03070 .03379 .03717 .04087 .04492

.01407 .01580 .01773 .01989 .02231

.00645 .00739 .00846 .00968 .01108

21.6 21.7 21.8 21.9 22.0

.37960 .39265 .40600 .41965 .43359

.19231 .20294 .21408 .22575 .23796

.09743 .10489 .11288 .12144 .13060

.04936 .05421 .05952 .06533 .07167

.02501 .02802 .03139 .03514 .03934

.01267 .01448 .01655 .01891 .02159

22.1 22.2 22.3 22.4 22.5

.44781 .46240 .47727 .49244 .50793

.25075 .26413 .27813 .29277 .30807

.14039 .15087 .16208 .17406 .18686

.07861 .08618 .09445 .10348 .11333

.04401 .04923 .05504 .06152 .06874

.02464 .02812 .03208 .03658 .04169

22.6 22.7 22.8 22.9 23.0

.52372 .53984 .55626 .57301 .59008

.32407 .34079 .35826 .37650 .39554

.20053 .21514 .23073 .24738 .26514

.12409 .13581 .14860 .16254 .17773

.07678 .08574 .09570 .10680 .11914

.04751 .05412 .06164 .07017 .07986

(table continues)

Chapter 4

130


P(>O)

19.1 19.2 19.3 19.4 19.5




1.2

1.4

1.6

1.8

.14191 .14846 .15524 .16223 .16946

.00012 .00014 .00017 .00020 .00023

.00004 .00004 .00005 .00006 .00008

.00001 .00001 .00002 .00002 .00003

.00001 .00001 .00001

19.6 19.7 19.8 19.9 20.0

.17691 .18460 .19252 .20069 .20910

.00027 .00032 .00038 .00044 .00052

.00009 .00011 .00013 .00016 .00019

.00003 .00004 .00005 .00006 .00007

.00001 .00001 .00002 .00002 .00003

.00001 .00001 .00001

20.1 20.2 20.3 20.4 20.5

.21776 .22667 .23584 .24527 .25495

.00061 .00071 .00084 .00098 .00115

.00023 .00027 .00033 .00039 .00047

.00009 .00010 .00013 .00016 .00019

.00003 .00004 .00005 .00006 .00008

.00001 .00002 .00002 .00002 .00003

20.6 20.7 20.8 20.9 21.0

.26490 .27512 .28561 .29637 .30741

.00135 .00158 .00185 .00216 .00253

.00056 .00067 .00080 .00095 .00114

.00023 .00028 .00034 .00042 .00051

.00010 .00012 .00015 .00018 .00023

.00004 .00005 .00006 .00008 .00010

21.1 21.2 21.3 21.4 21.5

.31873 .33033 .34221 .35439 .36685

.00296 .00346 .00404 .00471 .00550

.00136 .00162 .00193 .00229 .00273

.00062 .00076 .00092 .00112 .00136

.00028 .00035 .00044 .00054 .00067

.00013 .00017 .00021 .00026 .00033

21.6 21.7 21.8 21.9 22.0

.37960 .39265 .40600 .41965 .43359

.00642 .00749 .00873 .01017 .01185

.00325 .00387 .00460 .00547 .00650

.00165 .00200 .00243 .00294 .00357

.00083 .00103 .00128 .00158 .00196

.00042 .00053 .00067 .00085 .00107

22.1 22.2 22.3 22.4 22.5 22.6 22.7 22.8 22.9 23.0

.44781 .46240 .47727 .49244 .50793 .52372 .53984 .55626 .57301 .59008

.01380 .01606 .01869 .02175 .02529 .02940 .03417 .03970 .04611 .05353

.00773 .00917 .01089 .01293 .01534 .01819 .02157 .02557 .03029 .03588

.00433 .00524 .00635 .00769 .00930 .01126 .01362 .01647 .01990 .02405

.00242 .00299 .00370 .00457 .00564 .00697 .00860 .01060 .01308 .01612

.00136 .00171 .00216 .00272 .00342 .00431 .00543 .00683 .00859 .01081

2.0

5 Binomial Distribution

The Binomial distribution is used in lieu of the Poisson distribution for applications such as final trunk groups between a small PBX or remote switching unit and a central office (CO). It is based on the following assumptions: • • • • 5.1

Calls are served in random order. There are a finite number of sources. Blocked calls are held. Holding time is exponential or constant. BINOMIAL FORMULA

The Binomial formula is given in Equation 5.1. 5-1

p=

L

i=N

(S-I)! a i (I_a)(S-I-I) i! (S-l-i)!

(5.1)

where P = Binomial loss probability S = Number of sources 131

132

Chapter 5


N = Number of trunks in full-availability group a = Traffic offered to group in Erlangs per source 5.2

BINOMIAL COMPUTER PROGRAM

The following computer program can be used to calculate Equation 5.1 to determine Binomial loss probabilities. Required inputs are the number of sources, the number of servers in the group, and the total traffic offered to the groupexpressedin Erlangs(ErlangsareconvertedtoErlangspersource inStep 140). 100 REM BINOMIAL LOSS PROBABILITY CALCULATION 110 120 130 140 150 160 170 180 190 200 210 220 230

5.3

INPUT "ENTER NUMBER OF SOURCES (S) ";S INPUT "ENTER NUMBER OF SERVERS (N) ";N INPUT "ENTER OFFERED TRAFFIC IN ERLANGS (A)";A LET A=A/S FOR I=N TO (8-1) LET X=l FOR J=l TO I LET X=X*(S-J)/J NEXT J LET P=P+X*AAI*(l-A)A(S-r-l) NEXT I PRINT USING lip = #.#####";P END

BINOMIAL LOSS PROBABILITY TABLES

Binomialloss probabilitytables(Tables5-1 through5-6) are usedto determine the lossprobability(gradeof service)whentrafficisofferedtoa groupof trunks from a finite group of sources. Blanks (no data) in the tables indicate that the loss probability is less than 0.000005-not necessarily zero, but essentially nonblocking for practical applications. The following examples illustrate typical table usage. Example 5-1 Determine the loss probability for a 60-line digital remote switching unit connected to the local CO by a 24-channel trunk group, given that nonlocal busyhour traffic is 15 Erlangs.

Chapter 5

Binomial Distribution a

133

Erl) = (15 =0.25 Erl per line (source) 60 lines

UsingTable 5-6 (60 sources) (pages 144-145), select the a row for 0.25 Erlangs, the P column for 24 servers, and read .00587 at the intersection.

Example 5-2 For the remote switching unit of Example 5-1, determine the offered traffic in Erlangs and CCS to achieve a grade of service of 0.01 or better. Using Table 5-6 (60 sources), select the N column for 24 servers and read down until .00979 is found. Read across the a row to determine that 0.26 Erlangs per source (15.6 Erlangs) of traffic can be offered. (15.6 Erl)(36 CCS/Erl) =561.6 CCS

Example 5-3 Determine the number of CO trunks required for a 30-line PBX in a small office if nonlocal busy-hour traffic is 0.25 Erlangs per subscriber and a grade of service of 0.05 or better is required. Using Table 5-3 (30 sources) (pages 137-138), select the a row for 0.25 Erlangs per source and read across until .03903 is found. Read up the N column to determine that at least 12 trunks are required.

134

Chapter 5


Table 5-1. Binomial Loss Probability (8 = 10 Sources) Loss Probability (P) for N=

Traffic per Source (ain Erl)

3

4

0.01 0.02 0.03 0.04 0.05

.00008 .00061 .00198 .00448 .00836

.00002 .00009 .00027 .00064

.00001 .00003

0.06 0.07 0.08 0.09 0.10

.01380 .02091 .02979 .04048 .05297

.00128 .00227 .00372 .00570 .00833

.00008 .00017 .00031 .00055 .00089

.00001 .00002 .00004 .00006

0.11 0.12 0.13 0.14 0.15

.06725 .08326 .10093 .12016 .14085

.01169 .01585 .02094 .02691 .03393

.00138 .00206 .00296 .00414 .00563

.00011 .00018 .00028 .00043 .00063

.00001 .00001 .00002 .00003 .00005

0.16 0.17 0.18 0.19 0.20

.16289 .18614 .21046 .23573 .26180

.04202 .05121 .06153 .07301 .08564

.00748 .00976 .01250 .01575 .01958

.00091 .00127 .00173 .00232 .00307

.00007 .00011 .00016 .00022 .00031

.00001 .00001 .00001 .00002

0.21 0.22 0.23 0.24 0.25

.28853 .31579 .34342 .37131 .39932

.09943 .11436 .13041 .14754 .16573

.02403 .02914 .03497 .04155 .04893

.00398 .00511 .00646 .00808 .00999

.00043 .00059 .00078 .00103 .00134

.00003 .00004 .00006 .00008 .00011

0.26 0.27 0.28 0.29 0.30

.42733 .45523 .48290 .51025 .53717

.18490 .20502 .22601 .24781 .27034

.05713 .06620 .07616 .08702 .09881

.01224 .01486 .01788 .02135 .02529

.00173 .00220 .00277 .00346 .00429

.00014 .00019 .00026 .00033 .00043

0.31 0.32 0.33 0.34 0.35

.56358 .58941 .61459 .63904 .66273

.29352 .31728 .34152 .36916 .39111

.11153 .12519 .13978 .15529 .17172

.02976 .03479 .04041 .04666 .05359

.00527 .00643 .00778 .00936 .01118

.00056 .00071 .00089 .00112 .00140

0.36 0.37 0.38 0.39 0.40

.68559 .70760 .72872 .74893 .76821

.41627 .44157 .46692 .49222 .51739

.18904 .20722 .22622 .24602 .26657

.06121 .06958 .07870 .08862 .09935

.01328 .01569 .01842 .02153 .02503

.00173 .00212 .00259 .00315 .00380

5

6

7

8

Chapter 5

135


Table 5-2. Binomial Loss Probability (5 = 20 Sources) Loss Probability (P) for N

=

Traffic per Source (a in Erl)

3

4

5

6

0.01 0.02 0.03 0.04 0.05

.00086 .00610 .01826 .03840 .06655

.00003 .00049 .00219 .00612 .01324

.00003 .00020 .00074 .00201

.00001 .00007 .00024

.00001 .00002

0.06 0.07 0.08 0.09 0.10

.10207 .14392 .19084 .24148 .29456

.02430 .03985 .06016 .08527 .11500

.00444 .00851 .01471 .02347 .03519

.00064 .00144 .00285 .00514 .00859

.00007 .00020 .00045 .00091 .00170

.00001 .00002 .00006 .00013 .00027

0.11 0.12 0.13 0.14 0.15

.34883 .40324 .45683 .50885 .55868

.14897 .18667 .22750 .27079 .31585

.05016 .06854 .09043 .11578 .14444

.01352 .02022 .02899 .04007 .05370

.00295 .00484 .00756 .01132 .01633

.00053 .00095 .00162 .00262 .00408

0.16 0.17 0.18 0.19 0.20

.60585 .65004 .69104 .72873 .76311

.36199 .40854 .45490 .50052 .54491

.17618 .21068 .24756 .28639 .32671

.07001 .08911 .11102 .13571 .16306

.02282 .03101 .04108 .05323 .06760

.00613 .00891 .01257 .01731 .02328

0.21 0.22 0.23 0.24 0.25

.79422 .82217 .84710 .86920 .88866

.58767 .62848 .66706 .70325 .73691

.36806 .40996 .45196 .49363 .53458

.19292 .22505 .25920 .29504 .33224

.08429 .10337 .12485 .14871 .17488

.03066 .03962 .05032 .06290 .07746

0.26 0.27 0.28 0.29 0.30

.90569 .92051 .93332 .94435 .95378

.76798 .79646 .82238 .84580 .86683

.57445 .61294 .64979 .68479 .71778

.37046 .40931 .44844 .48750 .52614

.20322 .23357 .26573 .29947 .33450

.09409 .11286 .13378 .15683 .18197

0.31 0.32 0.33 0.34 0.35

.96180 .96858 .97429 .97907 .98304

.88559 .90222 .91687 .92970 .94086

.74864 .77730 .80374 .82797 .85000

.56405 .60095 .63659 .67074 .70324

.37055 .40731 .44448 .48176 .51883

.20909 .23808 .26876 .30095 .33443

0.36 0.37 0.38 0.39 0.40

.98634 .98905 .99127 .99307 .99454

.95052 .95882 .96592 .97179 .97704

.86992 .88781 .90377 .91792 .93039

.73392 .76271 .78951 .81430 .83708

.55543 .59126 .62610 .65972 .69193

.36896 .40427 .44010 .47618 .51222

7

8

(table continues)

Chapter 5

136

Table 5-2. Binomial Loss Probability (S Traffic per Source (a in Erl)


=20 Sources Continued)

Loss Probability (P) for N =

9

10

11

12

13

14

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

.00001 .00002 .00004

0.11 0.12 0.13 0.14 0.15

.00008 .00015 .00028 .00050 .00084

.00001 .00002 .00004 .00008 .00014

.00001 .00002

0.16 0.17 0.18 0.19 0.20

.00136 .00212 .00319 .00467 .00666

.00025 .00042 .00067 .00104 .00158

.00004 .00007 .00012 .00019 .00031

.00001 .00002 .00003 .00005

0.21 0.22 0.23 0.24 0.25

.00928 .01266 .01693 .02225 .02875

.00233 .00336 .00474 .00656 .00890

.00048 .00074 .00110 .00160 .00229

.00008 .00013 .00021 .00032 .00048

.00001 .00002 .00003 .00005 .00008

.00001 .00001

0.26 0.27 0.28 0.29 0.30

.03658 .04589 .05681 .06945 .08392

.01189 .01562 .02022 .02582 .03255

.00321 .00442 .00599 .00800 .01054

.00071 .00103 .00147 .00205 .00282

.00013 .00020 .00029 .00043 .00062

.00002 .00003 .00005 .00007 .00011

0.31 0.32 0.33 0.34 0.35

.10029 .11862 .13894 .16124 .18549

.04054 .04991 .06079 .07328 .08747

.01370 .01758 .02229 .02796 .03469

.00383 .00513 .00678 .00886 .01144

.00088 .00122 .00169 .00230 .00309

.00016 .00023 .00034 .00048 .00067

0.36 0.37 0.38 0.39 0.40

.21163 .23954 .26910 .30016 .33252

.10346 .12129 .14101 .16261 .18609

.04262 .05184 .06249 .07467 .08847

.01461 .01847 .02312 .02867 .03523

.00411 .00541 .00704 .00906 .01156

.00093 .00128 .00173 .00231 .00307

Chapter 5

137


Table 5-3. Binomial Loss Probability (8

=30 Sources)



9

10

11

12

13

14

0.11 0.12 0.13 0.14 0.15

.00301 .00542 .00916 .01465 .02234

.00072 .00141 .00260 .00451 .00742

.00015 .00032 .00065 .00121 .00215

.00003 .00006 .00014 .00029 .00055

.00001 .00003 .00006 .00012

.00001 .00002

0.16 0.17 0.18 0.19 0.20

.03270 .04616 .06309 .08377 .10838

.01165 .01757 .02556 .03600 .04926

.00363 .00586 .00908 .01359 .01970

.00099 .00171 .00284 .00451 .00694

.00024 .00044 .00078 .00132 .00215

.00005 .00010 .00019 .00034 .00059

0.21 0.22 0.23 0.24 0.25

.13697 .16946 .20561 .24510 .28746

.06566 .08544 .10878 .13575 .16631

.02775 .03805 .05094 .06669 .08554

.01034 .01496 .02109 .02901 .03903

.00340 .00519 .00772 .01117 .01578

.00098 .00159 .00249 .00380 .00565

0.26 0.27 0.28 0.29 0.30

.33214 .37852 .42595 .47376 .52130

.20030 .23747 .27745 .31980 .36400

.10765 .13312 .16195 .19403 .22918

.05143 .06647 .08437 .10530 .12938

.02180 .02984 .03911 .05094 .06522

.00818 .01160 .01610 .02191 .02927

0.31 0.32 0.33 0.34 0.35

.56794 .61312 .65636 .69724 .73545

.40949 .45566 .50191 .54767 .59239

.26711 .30746 .34977 .39357 .43832

.15663 .18699 .22034 .25645 .29502

.08217 .10196 .12472 .15050 .17928

.03843 .04963 .06313 .07912 .09779

0.36 0.37 0.38 0.39 0.40

.77076 .80302 .83218 .85825 .88131

.63556 .67675 .71562 .75187 .78532

.48346 .52844 .57274 .61586 .65733

.33568 .37799 .42149 .46565 .50996

.21099 .24546 .28243 .32160 .36258

.11928 .14366 .17098 .20116 .23410

0.41 0.42 0.43 0.44 0.45

.90149 .91897 .93396 .94667 .95733

.81584 .84339 .86799 .88972 .90871

.69679 .73391 .76845 .80023 .82915

.55390 .59699 .63875 .67878 .71672

.40496 .44826 .49198 .53563 .57873

.26961 .30742 .34719 .38856 .43109

0.46 0.47 0.48 0.49 0.50

.96619 .97347 .97939 .98415 .98794

.92514 .93919 .95109 .96104 .96929

.85520 .87839 .89883 .91663 .93198

.75229 .78526 .81551 .84293 .86753

.62079 .66139 .70014 .73672 .77087

.47430 .51773 .56088 .60327 .64446

(table continues)

138

Chapter 5

Table 5-3. Binomial Loss Probability (5 Traffic per Source (a in Erl)

=30 Sources

Loss Probability (P) for N

17


15

16

0.16 0.17 0.18 0.19 0.20

.00001 .00002 .00004 .00008 .00014

.00001 .00002 .00003

.00001

0.21 0.22 0.23 0.24 0.25

.00025 .00043 .00071 .00114 .00178

.00006 .00010 .00018 .00030 .00050

.00001 .00002 .00004 .00007 .00012

.00001 .00001 .00003

0.26 0.27 0.28 0.29 0.30

.00272 .00404 .00587 .00835 .01165

.00080 .00124 .00189 .00281 .00411

.00020 .00034 .00054 .00084 .00128

0.31 0.32 0.33 0.34 0.35

.01596 .02149 .02846 .03709 .04764

.00587 .00825 .01138 .01545 .02063

0.36 0.37 0.38 0.39 0.40

.06031 .07534 .09291 .11316 .13621

0.41 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0.50

18

Continued)

= 19

20

.00005 .00008 .00013 .00022 .00035

.00001 .00002 .00003 .00005 .00008

.00001 .00001 .00002

.00191 .00279 .00402 .00569 .00791

.00054 .00083 .00125 .00184 .00267

.00014 .00022 .00034 .00052 .00079

.00003 .00005 .00008 .00013 .00020

.02715 .03522 .04506 .05690 .07095

.01083 .01460 .01940 .02542 .03288

.00380 .00534 .00737 .01004 .01348

.00117 .00171 .00246 .00348 .00485

.00031 .00048 .00071 .00105 .00152

.16211 .19086 .22237 .25650 .29303

.08743 .10645 .12820 .15273 .18008

.04119 .05297 .06604 .08139 .09922

.01786 .02336 .03018 .03853 .04862

.00667 .00906 .01215 .01608 .02103

.00218 .00307 .00427 .00586 .00795

.33169 .37214 .41396 .45674 .50000

.21021 .24303 .27835 .31596 .35554

.11966 .14284 .16882 .19760 .22913

.06068 .07491 .09150 .11064 .13247

.02718 .03474 .04392 .04495 .06802

.01066 .01412 .01850 .02396 .03071

0.11 0.12 0.13 0.14 0.15

Chapter 5


139

Table 5-4. Binomial Loss Probability (5 = 40 Sources) Loss Probability (P) for N =


13

14

15

16

17

0.11 0.12 0.13 0.14 0.15

.00017 .00041 .00090 .00180 .00337

.00004 .00010 .00024 .00052 .00106

.00001 .00002 .00006 .00014 .00030

.00001 .00003 .00008

.00001 .00002

0.16 0.17 0.18 0.19 0.20

.00593 .00991 .01591 .02413 .03550

.00200 .00358 .00608 .00987 .01539

.00061 .00117 .00212 .00366 .00605

.00004 .00035 .00067 .00123 .00216

.00017 .00009 .00019 .00038 .00070

.00001 .00002 .00005 .00010 .00021

0.21 0.22 0.23 0.24 0.25

.05045 .06950 .09305 .12134 .15447

.02311 .03356 .04724 .06464 .08615

.00961 .01473 .02183 .03139 .04387

.00363 .00588 .00918 .01388 .02037

.00125 .00213 .00351 .00559 .00862

.00039 .00070 .00122 .00205 .00333

0.26 0.27 0.28 0.29 0.30

.19231 .23456 .28071 .33008 .38185

.11208 .14256 .17758 .21695 .26028

.05972 .07936 .10310 .13113 .16352

.02907 .04040 .05482 .07271 .09441

.01291 .01879 .02666 .03692 .04998

.00523 .00798 .01184 .01714 .02422

0.31 0.32 0.33 0.34 0.35

.43512 .48895 .54238 .59453 .64459

.30705 .35655 .40801 .46055 .51327

.20018 .24085 .28510 .33236 .38196

.12017 .15010 .18420 .22230 .26411

.06624 .08604 .10965 .13726 .16892

.03346 .04525 .05998 .07799 .09958

0.36 0.37 0.38 0.39 0.40

.69187 .73584 .77609 .81239 .84463

.56531 .61584 .66413 .70956 .75165

.43310 .48496 .53669 .58747 .63653

.30916 .35685 .40652 .45737 .50862

.20457 .24400 .28684 .33261 .38070

.12499 .15432 .18760 .22469 .26533

0.41 0.42 0.43 0.44 0.45

.87285 .89720 .91790 .93524 .94956

.79005 .82458 .85516 .88185 .90480

.68320 .72692 .76727 .80394 .83679

.55943 .60905 .65676 .70192 .74405

.43043 .48102 .53170 .58169 .63023

.30914 .35559 .40407 .45387 .50424

0.46 0.47 0.48 0.49 0.50

.96122 .94050 .97797 .98373 .98815

.92426 .97957 .95387 .96472 .97338

.86578 .89098 .91257 .93079 .94594

.78276 .81779 .84903 .87646 .90021

.67667 .72043 .76104 .79817 .83161

.55441 .60362 .65116 .69640 .73880

18

(table continues)

Chapter 5

140



=40 Sources

Continued)

Loss Probability (P) for N = 24

21

22

23

.00003 .00006 .00011 .00021 .00037

.00001 .00001 .00003 .00006 .00011

.00001 .00001 .00003

.00001

.00193 .00309 .00480 .00727 .01073

.00065 .00109 .00177 .00281 .00434

.00020 .00035 .00060 .00099 .00160

.00005 .00010 .00018 .00032 .00054

.00001 .00003 .00005 .00009 .00016

.00001 .00001 .00002 .00004

0.31 0.32 0.33 0.34 0.35

.01547 .02180 .03007 .04067 .05395

.00653 .00960 .01380 .01942 .02679

.00251 .00385 .00578 .00847 .01216

.00088 .00141 .00220 .00337 .00504

.00028 .00047 .00076 .00122 .00190

.00008 .00014 .00024 .00040 .00065

0.36 0.37 0.38 0.39 0.40

.07027 .08995 .11326 .14036 .17133

.03625 .04817 .06290 .08076 .10206

.01712 .02363 .03203 .04266 .05588

.00738 .01059 .01491 .02062 .02803

.00289 .00432 .00633 .00910 .01283

.00103 .00160 .00244 .00365 .00535

0.41 0.42 0.43 0.44 0.45

.20613 .24458 .28639 .33113 .37825

.12700 .15573 .18827 .22454 .26431

.07202 .09139 .11425 .14079 .17110

.03745 .04923 .06373 .08125 .10209

.01779 .02426 .03254 .04297 .05589

.00770 .01089 .01515 .02075 .02796

0.46 0.47 0.48 0.49 0.50

.42711 .47701 .52719 .57689 .62537

.30726 .35292 .40071 .44998 .50000

.20517 .24288 .28396 .32805 .37463

.12648 .15458 .18646 .22205 .26120

.07164 .09052 .11282 .13873 .16839

.03711 .04853 .06257 .07956 .09980

19

20

0.16 0.17 0.18 0.19 0.20

.00001 .00003 .00006

.00001 .00001

0.21 0.22 0.23 0.24 0.25

.00011 .00021 .00039 .00069 .00117

0.26 0.27 0.28 0.29 0.30

0.11 0.12 0.13 0.14 0.15

Chapter 5


141

Table 5-5. Binomial Loss Probability (8 = 50 Sources) Loss Probability (P) for N =

Traffic per Source

(a in Ert)

15

16

17

18

19

20

0.11 0.12 0.13 0.14 0.15

.00017 .00044 .00102 .00218 .00429

.00004 .00012 .00031 .00072 .00154

.00001 .00004 .00009 .00022 .00051

.00001 .00002 .00006 .00015

.00001 .00002 .00004

.00001

0.16 0.17 0.18 0.19 0.20

.00787 .01360 .02226 .03469 .05175

.00303 .00561 .00978 .01619 .02557

.00107 .00213 .00396 .00696 .01166

.00035 .00074 .00148 .00276 .00491

.00011 .00024 .00051 .00101 .00191

.00003 .00007 .00016 .00034 .00069

0.21 0.22 0.23 0.24 0.25

.07419 .10260 .13730 .17830 .22527

.03870 .05636 .07922 .10779 .14236

.01866 .02865 .04235 .06047 .08362

.00831 .01347 .02097 .03146 .04562

.00343 .00586 .00962 .01518 .02310

.00131 .00236 .00409 .00679 .01086

0.26 0.27 0.28 0.29 0.30

.27752 .33405 .39366 .45494 .51646

.18291 .22913 .28039 .33576 .39412

.11227 .14666 .18678 .23232 .28269

.06410 .08746 .11612 .15030 .19000

.03401 .04853 .06726 .09075 .11937

.01676 .02503 .03627 .05106 .06999

0.31 0.32 0.33 0.34 0.35

.57684 .63478 .68922 .73933 .78452

.45417 .51456 .57396 .63116 .68509

.33705 .39433 .45332 .51274 .57132

.23492 .28451 .33798 .39432 .45241

.15334 .19263 .23698 .28588 .33857

.09354 .12208 .15582 .19472 .23856

0.36 0.37 0.38 0.39 0.40

.82449 .85917 .88869 .91335 .93358

.73494 .78009 .82021 .85519 .88511

.62789 .68140 .73104 .77617 .81644

.51100 .56889 .62493 .67811 .72758

.39412 .45142 .50933 .56664 .62226

.28684 .33886 .39372 .45037 .50770

0.41 0.42 0.43 0.44 0.45

.94988 .96277 .97278 .98042 .98615

.91023 .93094 .94771 .96103 .97142

.85169 .88198 .90751 .92865 .94583

.77272 .81314 .84865 .87927 .90518

.67517 .72453 .76971 .81029 .84605

.56455 .61983 .67255 .72186 .76711

0.46 0.47 0.48 0.49 0.50

.99036 .99340 .99557 .99707 .99810

.97939 .98538 .98981 .99302 .99530

.95953 .97027 .97852 .98474 .98935

.92670 .94424 .95828 .96931 .97781

.87697 .90322 .92508 .94295 .95728

.80786 .84386 .87507 .90163 .92380

(table continues)

Chapter 5

142


Table 5-5. Binomial Loss Probability (S = 50 Sources Continued) Traffic per Source (a in Erl)


21

22

23

0.16 . 0.17 0.18 0.19 0.20

.00001 .00002 .00005 .00011 .00023

.00001 .00003 .00007

.00001 .00002

.00001

0.21 0.22 0.23 0.24 0.25

.00046 .00080 .00161 .00282 .00473

.00015 .00030 .00059 .00108 .00191

.00005 .00010 .00020 .00039 .00072

.00001 .00003 .00006 .00013 .00025

.00001 .00002 .00004 .00008

.00001 .00002

0.26 0.27 0.28 0.29 0.30

.00767 .01200 .01818 .02675 .03824

.00325 .00534 .00847 .01303 .01945

.00125 .00221 .00367 .00590 .00920

.00047 .00085 .00147 .00248 .00405

.00016 .00030 .00055 .00097 .00165

.00005 .00010 .00019 .00035 .00063

0.31 0.32 0.33 0.34 0.35

.05324 .07229 .09585 .12429 .15778

.02824 .03993 .05507 .07418 .09772

.01394 .02054 .02950 .04133 .05655

.00640 .00984 .01472 .02146 .03053

.00273 .00438 .00683 .01037 .01536

.00108 .00181 .00294 .00466 .00718

0.36 0.37 0.38 0.39 0.40

.19633 .23970 .28743 .33887 .39314

.12602 .15927 .19747 .24042 .28767

.07568 .09916 .12730 .16031 .19819

.04245 .05771 .07681 .10018 .12815

.02219 .03134 .04329 .05854 .07758

.01080 .01585 .02274 .03192 .04386

0.41 0.42 0.43 0.44 0.45

.44925 .50611 .56260 .61764 .67024

.33861 .39239 .44806 .50456 .56078

.24075 .28759 .33809 .39148 .44680

.16092 .19849 .24070 .28718 .33733

.10082 .12860 .16111 .19840 .24030

.05905 .07798 .10107 .12864 .16090

0.46 0.47 0.48 0.49 0.50

.71954 .76490 .80583 .84208 .87356

.61566 .66820 .71757 .76306 .80420

.50303 .55907 .61387 .66644 .71591

.39040 .44546 .50151 .55747 .61227

.28645 .33632 .38915 .44404 .50000

.19791 .23953 .28542 .33506 .38772

24

24

26

0.11 0.12 0.13 0.14 0.15

(table continues)

Chapter 5

143


Table 5-5. Binomial Loss Probability (5 = 50 Sources Continued) Traffic per Source (a in Erl)


28

29

30

31

32

0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25

.00001

0.26 0.27 0.28 0.29 0.30

.00001 .00003 .00006 .00012 .00022

.00001 .00002 .00004 .00007

.00001 .00002

.00001

0.31 0.32 0.33 0.34 0.35

.00040 .00069 .00118 .00194 .00312

.00013 .00025 .00044 .00075 .00126

.00004 .00008 .00015 .00027 .00047

.00001 .00002 .00005 .00009 .00016

.00001 .00001 .00003 .00005

.00001 .00002

0.36 0.37 0.38 0.39 0.40

.00488 .00746 .01112 .01620 .02311

.00205 .00326 .00505 .00764 .01132

.00080 .00132 .00213 .00335 .00515

.00029 .00049 .00083 .00136 .00217

.00009 .00017 .00030 .00051 .00084

.00003 .00005 .00010 .00018 .00030

0.41 0.42 0.43 0.44 0.45

.03226 .04415 .05926 .07805 .10094

.01641 .02328 .03239 .04418 .05915

.00775 .01141 .01646 .02328 .03229

.00339 .00518 .00776 .01139 .01637

.00137 .00218 .00339 .00516 .00769

.00051 .00084 .00136 .00216 .00334

0.46 0.47 0.48 0.49 0.50

.12829 .16030 .19705 .23841 .28409

.07777 .10045 .12755 .15931 .19580

.04395 .05875 .07715 .09959 .12643

.02309 .03197 .04346 .05804 .07620

.01125 .01614 .02273 .03144 .04272

.00506 .00754 .01100 .01577 .02219

Chapter 5

144


Table 5-6. Binomial Loss Probability (8 = 60 Sources) Traffic per Source (a in Erl)

Loss Probability (P) for N=

16

18

20

22

0.11 0.12 0.13 0.14 0.15

.00048 .00125 .00287 .00601 .01154

.00004 .00013 .00035 .00086 .00193

.00001 .00003 .00009 .00024

.00001 .00002

0.16 0.17 0.18 0.19 0.20

.02060 .03443 .05431 .08137 .11639

.00396 .00757 .01354 .02283 .03651

.00057 .00125 .00225 .00485 .00872

.00006 .00016 .00036 .00079 .00159

.00001 .00002 .00004 .00010 ' .00022

.00001 .00002

0.21 0.22 0.23 0.24 0.25

.15967 .21096 .26940 .33356 .40165

.05567 .08126 .11401 .15426 .20189

.01485 .02409 .03736 .05561 .07970

.00304 .00549 .00945 .01555 .02453

.00048 .00096 .00185 .00337 .00587

.00006 .00013 .00028 .00057 .00109

0.26 0.27 0.28 0.29 0.30

.47157 .54122 .60858 .67195 .72996

.25631 .31644 .38079 .44761 .51502

.11032 .14784 .19227 .24320 .29982

.03721 .05445 .07705 .10564 .14066

.00979 .01571 .02428 .03627 .05244

.00201 .00354 .00599 .00978 .01541

0.31 0.32 0.33 0.34 0.35

.78173 .82678 .86505 .89679 .92253

.58117 .64437 .70321 .75664 .80398

.36092 .42499 .49038 .55536 .61830

.18221 .23004 .28355 .34175 .40335

.07353 .10020 .13288 .17179 .21681

.02349 .03469 .04974 .06936 .09416

0.36 0.37 0.38 0.39 0.40

.94292 .95873 .97072 .97962 .98608

.84494 .87957 .90817 .93128 .94953

.67775 .73255 .78188 .82525 .86250

.46688 .53075 .59337 .65330 .86250

.26749 .32305 .38240 .44422 .50703

.12464 .16108 .20348 .25155 .30467

0.41 0.42 0.43 0.44 0.45

.99068 .99388 .99606 .99751 .98846

.96364 .97430 .98219 .98790 .99194

.89379 .91948 .94010 .95629 .96873

.76039 .80595 .84565 .87947 .90753

.56932 .62964 .68669 .73941 .78702

.36191 .42212 .48392 .54588 .60655

0.46 0.47 0.48 0.49 0.50

.99907 .99945 .99968 .99982 .99990

.99474 .99664 .99789 .99871 .99923

.97806 .98492 .98985 .99331 .99568

.93055 .94880 .96300 .97380 .98183

.82903 .86526 .89581 .92099 .94126

.66458 .71883 .76838 .81262 .85120

24

26

(table continues)

Chapter 5

145



= 60 Sources Continued)


30

32

34

36

38

0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25

.00001 .00001 .00003 .00007 .00016

.00001 .00002

0.26 0.27 0.28 0.29 0.30

.00032 .00062 .00115 .00206 .00355

.00004 .00008 .00017 .00034 .00064

.00001 .00002 .00004 .00009

.00001

0.31 0.32 0.33 0.34 0.35

.00590 .00947 .01473 .02224 .03261

.00116 .00203 .00343 .00562 .00892

.00018 .00034 .00062 .00111 .00191

.00002 .00004 .00009 .00017 .00032

.00001 .00002 .00004

0.36 0.37 0.38 0.39 0.40

.04652 .06466 .08765 .11601 .15010

.01376 .02063 .03013 .04288 .05955

.00320 .00518 .00818 .01255 .01877

.00058 .00102 .00173 .00288 .00464

.00008 .00015 .00028 .00051 .00089

.00001 .00002 .00004 .00007 .00013

0.41 0.42 0.43 0.44 0.45

.19003 .23563 .28644 .34170 .40037

.08077 .10708 .13890 .17644 .21967

.02736 .03894 .05414 .07360 .09789

.00730 .01118 .01671 .02439 .03479

.00151 .00250 .00403 .00633 .00971

.00024 .00043 .00075 .00127 .00210

0.46 0.47 0.48 0.49 0.50

.46121 .52284 .58383 .64281 .69854

.26826 .32159 .37878 .43869 .50000

.12749 .16270 .20360 .25001 .30146

.04853 .06623 .08852 .11591 .14880

.01455 .02130 .03052 .04278 .05874

.00338 .00532 .00820 .01234 .01817

6 Engset Distribution

The Engset distribution, also known as the Erlang-Engset distribution, is used in lieu of the Erlang B distribution for dimensioning nonqueued (immediate service) common equipment pools that have a limited number of sources, such as small rural switching units and line concentrators. It is based on the following assumptions: • • • • 6.1

Calls are served in random order. There are a finite number of sources. Blocked calls are cleared. Holding time is constant or exponential. ENGSET FORMULA

The Engset formula, also known as the truncated Binomial formula, is given in Equation 6.1. Note that P is present on both sides of the equation, requiring an iterative (i.e., successive approximation) calculation process. When P is very small, such that the value of I - P approaches unity, the approximate Engset formula of Equation 6.2 can be used. 147

Chapter 6

148

P=

(S-I)! [ A ] N! (S-I-N)! S -A (I-P)

£

(5-1)! [ A ;=0 i! (S -I-i)! S -A (I-P)

p-::::;

(S -I)! N! (S-l-N)! N

L

Engset Distribution

N

]i

(6.1 )

(6.2)

(5-1)! ;=0 i! (S -I-i)!

where P =Engset loss probability S = Number of sources N = Number of servers in full-availability group A = Traffic offered to group in Erlangs

6.2 ENGSET COMPUTER PROGRAM The following iterative computer program can be used to calculate Equation 6.1 to determine Engset loss probabilities with an accuracy of ± 0.0000 I as set in Step 220. More accurate results can be obtained by reducing this parameter; this will result in more iterationsand longer run times, however. A noniterative, faster program can be written to calculate Equation 6.2 but the results, even when P is very small, will be less accurate. Required inputs for this program are the number of sources, the number of servers in the group, and the traffic offered to the group expressed in Erlangs. 100 110 120 130 140 150 160 170 180

REM ENGSET LOSS PROBABILITY CALCULATION INPUT "ENTER NUMBER OF SOURCES (S)";S INPUT "ENTER NUMBER OF SERVERS (N) ";N INPUT "ENTER OFFERED TRAFFIC IN ERLANGS (A) ";A LET PO=Pl LET X=l LET Y=l FOR 1=1 TO N

LET X=X*A/((S-A*(l-PO) )*(S-I)/1

Chapter 6

Engset Distribution

190 200 210 220 230 240

6.3

149

LET Y=X+Y NEXT I LE~ P1=X/Y

IF ABS(PO-P1».00001 GOTO 140 PRINT USING lip END

=

#.#####";P1

ENGSET LOSS PROBABILITY TABLES

Engset loss probability tables (Tables 6-1 through 6-6) are used to determine the loss probability (grade of service) when traffic is offered to a group of servers from a finite group of sources. Blanks (no data) in the tables indicate that the loss probability is less than O.000005-not necessarily zero, but essentially nonblocking for practical applications. The following examples illustrate typical table usage:

Example 6-1 Determine the loss probability for a line concentrator with 60 subscribers, 18 channels, and offered traffic of 0.2 Erlangs per subscriber (source). A = (60 sub)(0.2 Erl/sub) = 12 Erlangs Using Table 6-6 (60 sources) (pages 161-164), select the A row for 12 Erlangs and the P column for 18 servers, and read .01787 at the intersection.

Example 6-2 For the line concentrator of Example 6-1, determine the offered traffic in Erlangs and CCS to achieve a grade of service of 0.005 or better. Using Table 6-6 (60 sources), select the N column for 18 servers and read down until .00436 is found. Read across the A row to determine that 10.2 Erlangs of traffic can be offered. (10.2 Erl)(36 CCS/Erl) = 367.2 CCS

Example 6-3 Determine the number of trunks required for a 40-line remote switching unit if nonlocal busy-hour traffic is 0.1 Erlangs per subscriber and a grade of service of 0.01 or better is required. A

=(40 lines )(0.1 Erl/line) =4 Erlangs

Using Table 6-4 (40 sources) (pages 155-157), select the A row for 4 Erlangs and read over until .00897 is found. Read up the N column to determine that at least 9 trunks are required.

Chapter 6

150

Engset Distrihution

Table 6-1. Engset Loss Probability (S = 10 Sources) Loss Probability (P) for N =


2

0.1 0.2 0.3 0.4 0.5

.00336 .01250 .02619 .04335 .06305

0.6 0.7 0.8 0.0 1.0

4

5

.00008 .00060 .00189 .00421 .00772

.00002 .00009 .00026 .00061

.00001 .00003

.08453 .10717 .13046 .15401 .17749

.01251 .01862 .02604 .03471 .04455

.00120 .00210 .00340 .00515 .00743

.00008 .00016 .00030 .00051 .00083

.00001 .00002 .00003 .00006

1.1 1.2 1.3 1.4 1.5

.20070 .22344 .24561 .26711 .28789

.05544 .06727 .07991 .09323 .10709

.01028 .01374 .01785 .02263 .02809

.00127 .00188 .00268 .00370 .00498

.00010 .00017 .00027 .00040 .00059

.00001 .00001 .00002 .00003 .00004

1.6 1.7 1.8 1.9 2.0

.30792 .32720 .34571 .36347 .38050

.12139 .13599 .15080 .16573 .18069

.03421 .04099 .04840 .05640 .06495

.00655 .00845 .01070 .01332 .01635

.00083 .00116 .00157 .00210 .00275

.00007 .00010 .00015 .00021 .00029

2.1 2.2 2.3 2.4 2.5

.39681 .41244 .42740 .44174 .45547

.19562 .21044 .22512 .23961 .25387

.07401 .08353 .09344 .10371 .11427

.01980 .02367 .02799 .03276 .03797

.00354 .00450 .00564 .00699 .00857

.00040 .00055 .00073 .00095 .00123

2.6 2.7 2.8 2.9 3.0

.46862 .48123 .49332 .50491 .51603

.26788 .28161 .29505 .30819 .32103

.12507 .13607 .14721 .15846 .16976

.04361 .04967 .05614 .06300 .07021

.01039 .01248 .01485 .01751 .02049

.00158 .00200 .00251 .00311 .00383

3.1 3.2 3.3 3.4 3.5

.52671 .53696 .54682 .55629 .56540

.33351 .34572 .35760 .36916 .38041

.18109 .19241 .20369 .21491 .22604

.07776 .08562 .09375 .10213 .11072

.02379 .02741 .03137 .03566 .04029

.00468 .00566 .00680 .00811 .00961

3.6 3.7 3.8 3.9 4.0

.57417 .58261 .59075 .59859 .60615

.39135 .40199 .41234 .42239 .43216

.23707 .24797 .25874 .26935 .27980

.11949 .12842 .13747 .14662 .15584

.04524 .05051 .05609 .06196 .06812

.01130 .01321 .01535 .01772 .02035

3

6

7

Chapter 6

151

Engset Distribution

Table 6-2. Engset Loss Probability (5 = 20 Sources) Offered Traffic (A in Erl)


3

4

5

6

7

0.2 0.4 0.6 0.8 1.0

.01442 .04874 .09301 .14103 .18920

.00083 .00561 .01609 .03237 .05371

.00003 .00046 .00199 .00538 .01123

.00003 .00018 .00067 .00177

.00001 .00007 .00022

.00001 .00002

1.2 1.4 1.6 1.8 2.0

.23553 .27908 .31947 .35665 .39076

.07894 .10685 .13634 .16652 .19669

.01987 .03140 .04567 .06236 .08108

.00380 .00707 .01185 .01832 .02658

.00057 .00124 .00240 .00423 .00689

.00007 .00017 .00039 .00078 .00142

2.2 2.4 2.6 2.8 3.0

.42200 .45061 .47684 .50092 .52307

.22635 .25515 .28288 .30939 .33464

.10136 .12276 .14488 .16736 .18991

.03664 .04842 .06176 .07646 .09230

.01054 .01533 .02134 .02863 .03719

.00242 .00389 .00593 .00866 .01219

3.2 3.4 3.6 3.8 4.0

.54348 .56233 .57977 .59596 .61100

.35860 .38130 .40276 .42305 .44222

.21229 .23432 .25586 .27681 .29710

.10904 .12646 .14433 .16247 .18071

.04699 .05793 .06992 .08282 .09648

.01658 .02192 .02823 .03552 .04377

4.2 4.4 4.6 4.8 5.0

.62501 .63808 .65031 .66177 .67252

.46033 .47745 .49364 .50896 .52346

.31670 .33558 .35374 .37118 .38790

.19891 .21695 .23475 .25222 .26931

.11076 .12552 .14062 .15595 .17138

.05293 .06294 .07371 .08516 .09718

5.2 5.4 5.6 5.8 6.0

.68264 .69216 .70114 .70962 .71765

.53720 .55024 .56261 .57437 .58554

.40394 .41931 .43403 .44813 .46165

.28599 .30222 .31798 .33327 .34808

.18684 .20223 .21749 .23256 .24740

.10967 .12253 .13576 .14901 .16247

6.2 6.4 6.6 6.8 7.0

.72526 .73247 .73932 .74583 .75204

.59618 .60632 .61598 .62520 .63401

.47460 .48701 .49891 .51032 .52127

.36241 .37627 .38967 .40261 .41511

.26198 .27626 .29022 .30385 .31715

.17597 .18946 .20288 .21619 .22935

7.2 7.4 7.6 7.8 8.0

.75795 .76359 .76898 .77413 .77906

.64243 .65049 .65821 .66560 .67269

.53179 .54189 .55159 .56092 .56990

.42719 .43884 .45010 .46098 .47148

.33010 .34270 .35496 .36688 .37845

.24234 .25513 .26769 .28001 .29209

(table continues)

Chapter 6

152

Engset Distribution

Table 6-2. Engset Loss Probability (S = 20 Sources Continued) Offered Traffic (A in Erl)


8

9

10

11

1.2 1.4 1.6 1.8 2.0

.00001 .00002 .00005 .00012 .00024

.00001 .00001 .00003

2.2 2.4 2.6 2.8 3.0

.00045 .00080 .00133 .00212 .00323

.00007 .00013 .00024 .00042 .00070

.00001 .00002 .00004 .00007 .00012

.00001 .00002

3.2 3.4 3.6 3.8 4.0

.00475 .00676 .00933 .01254 .01644

.00111 .00170 .00251 .00361 .00505

.00021 .00035 .00055 .00085 .00127

4.2 4.4 4.6 4.8 5.0

.02110 .02653 .03276 .03976 .04753

.00690 .00921 .01205 .01545 .01947

5.2 5.4 5.6 5.8 6.0

.05602 .06517 .07493 .08522 .09596

6.2 6.4 6.6 6.8 7.0 7.2 7.4 7.6 7.8 8.0

12

13

.00003 .00006 .00010 .00016 .00026

.00001 .00001 .00003 .00004

.00001

.00184 .00261 .00362 .00492 .00655

.00040 .00060 .00089 .00128 .00180

.00007 .00011 .00018 .00027 .00040

.00001 .00002 .00003 .00005 .00007

.02414 .02946 .03544 .04208 .04935

.00857 .01102 .01395 .01739 .02138

.00248 .00337 .00449 .00589 .00761

.00058 .00083 .00117 .00162 .00219

.00011 .00017 .00025 .00036 .00051

.10709 .11853 .13021 .14207 .15404

.05720 .06561 .07451 .08386 .09359

.02592 .03104 .03673 .04297 .04975

.00969 .01216 .01507 .01845 .02231

.00293 .00387 .00503 .00645 .00818

.00071 .00099 .00135 .00181 .00240

.16607 .17813 .19015 .20211 .21398

.10365 .11398 .12452 .13523 .14605

.05704 .06480 .07299 .08156 .09046

.02667 .03153 .03690 .04276 .04909

.01023 .01266 .01548 .01872 .02240

.00315 .00408 .00522 .00661 .00827

0.2 0.4 0.6 0.8 1.0

Chapter 6

153

Engset Distribution

Table 6-3. Engset Loss Probability (8 = 30 Sources) Loss Probability (P) for N =

Offered Traffic (Ain Erl)

4

5

0.3 0.6 0.9 1.2 1.5

.00018 .00229 .00899 .02198 .04151

.00001 .00023 .00139 .00457 .01084

1.8 2.1 2.4 2.7 3.0

.06672 .09619 .12837 .16189 .19567

3.3 3.6 3.9 4.2 4.5

7

8

9

.00002 .00017 .00076 .00228

.00002 .00010 .00039

.00001 .00006

.00001

.02096 .03518 .05327 .07466 .09861

.00534 .01053 .01831 .02893 .04239

.00112 .00260 .00523 .00937 .01536

.00020 .00054 .00125 .00255 .00469

.00003 .00009 .00025 .00059 .00122

.22893 .26115 .29201 .32133 .34903

.12434 .15111 .17830 .20543 .23213

.05847 .07682 .09697 .11846 .14082

.02342 .03365 .04599 .06028 .07628

.00795 .01257 .01875 .02661 .03619

.00229 .00400 .00654 .01010 .01485

4.8 5.1 5.4 5.7 6.0

.37513 .39964 .42265 .44424 .46448

.25813 .28327 .30742 .33053 .35259

.16366 .18663 .20947 .23196 .25395

.09367 .11214 .13138 .15110 .17105

.04744 .06023 .07438 .08967 .10588

.02093 .02841 .03730 .04756 .05909

6.3 6.6 6.9 7.2 7.5

.48348 .50132 .51809 .53386 .54871

.37559 .39356 .41252 .43054 .44764

.27532 .29601 .31597 .33518 .35362

.19102 .21084 .23039 .24954 .26825

.12277 .14013 .15778 .17553 .19326

.07176 .08540 .09984 .11490 .13042

7.8 8.1 8.4 8.7 9.0

.56270 .57592 .58840 .60021 .61139

.46387 .47929 .49395 .50788 .52113

.37131 .38826 .40449 .42002 .43489

.28643 .30408 .32115 .33765 .35357

.21085 .22820 .24525 .26194 .27823

.14624 .16223 .17826 .19425 .21011

9.3 9.6 9.9 10.2 10.5

.62200 .63206 .64163 .65072 .65939

.53374 .54576 .55721 .56814 .57857

.44911 .46273 .47577 .48825 .50021

.36891 .38369 .39793 .41163 .42482

.29409 .30950 .32446 .33896 .35300

.22576 .24117 .25628 .27107 .28551

10.8 11.1 11.4 11.7 12.0

.66764 .67552 .68305 .69024 .68712

.58854 .59807 .60719 .61592 .62429

.51167 .52266 .53321 .54333 .55305

.43751 .44972 .46148 .47280 .48370

.36659 .37974 .39245 .40473 .41661

.29960 .31331 .32665 .33961 .35220

6

(table continues)

Chapter 6

154

Engset Distribution



10

11

12

13

14

1.8 2.1 2.4 2.7 3.0

.00001 .00004 .00012 .00027

.00001 .00002 .00005

.00001

3.3 3.6 3.9 4.2 4.5

.00057 .00109 .00196 .00329 .00525

.00012 .00026 .00051 .00093 .00160

.00002 .00005 .00011 .00023 .00042

.00001 .00002 .00005 .00010

.00001 .00002

4.8 5.1 5.4 5.7 6.0

.00798 .01164 .01634 .02218 .02922

.00263 .00413 .00622 .00901 .01264

.00075 .00127 .00205 .00318 .00476

.00019 .00034 .00059 .00098 .00156

.00004 .00008 .00015 .00026 .00045

.00001 .00002 .00003 .00006 .00011

6.3 6.6 6.9 7.2 7.5

.03745 .04684 .05731 .06876 .08106

.01719 .02274 .02932 .03694 .04556

.00689 .00967 .01319 .01754 .02277

.00240 .00358 .00519 .00729 .01000

.00073 .00116 .00178 .00265 .00383

.00019 .00033 .00053 .00084 .00128

7.8 8.1 8.4 8.7 9.0

.09407 .10766 .12168 .13603 .15057

.05513 .06555 .07674 .08858 .10095

.02891 .03596 .04391 .05269 .06225

.01338 .01749 .02240 .02812 .03466

.00541 .00746 .01005 .01326 .01713

.00191 .00278 .00394 .00547 .00743

9.3 9.6 9.9 10.2 10.5

.16521 .17987 .19446 .20893 .22323

.11375 .12686 .14020 .15367 .16719

.07249 .08334 .09469 .10645 .11853

.04199 .05009 .05889 .06833 .07833

.02171 .02704 .03310 .03989 .04738

.00988 .01289 .01652 .02079 .02573

10.8 11.1 11.4 11.7 12.0

.23732 .25117 .26474 .27803 .29102

.18071 .19417 .20751 .22070 .23370

.13084 .14331 .15587 .16846 .18103

.08880 .09967 .11086 .12230 .13390

.05552 .06425 .07351 .08322 .09333

.03136 .03765 .04459 .05214 .06025

15

0.3 0.6 0.9 1.2 1.5

Chapter 6

Engset Distribution

155



4

5

6

7

8

9

0.4 0.8 1.2 1.6 2.0

.00058 .00649 .02303 .05115 .08837

.00004 .00093 .00497 .01475 .03168

.00011 .00087 .00348 .00939

.00001 .00013 .00068 .00233

.00002 .00011 .00049

.00002 .00009

2.4 2.8 3.2 3.6 4.0

.13016 .17592 .22062 .26366 .30425

.05562 .08524 .11872 .15428 .19048

.01982 .03531 .05565 .08007 .10752

.00595 .01243 .02245 .03628 .05376

.00152 .00374 .00778 .01424 .02353

.00033 .00097 .00233 .00485 .00897

4.4 4.8 5.2 5.6 6.0

.34205 .37699 .40915 .43869 .46582

.22625 .26088 .29396 .32525 .35468

.13688 .16720 .19767 .22771 .25692

.07442 .09757 .12248 .14846 .17489

.03582 .05103 .06882 .08875 .11029

.01514 .02363 .03459 .04794 .06349

6.4 6.8 7.2 7.6 8.0

.49075 .51369 .53483 .55435 .57240

.38225 .40802 .43209 .45455 .47552

.28505 .31192 .33748 .36171 .38461

.20131 .22735 .25275 .27734 .30100

.13292 .15618 .17966 .20306 .22613

.08089 .09978 .11975 .14044 .16150

8.4 8.8 9.2 9.6 10.0

.58913 .60467 .61913 .63262 .64522

.49511 .51343 .53057 .54664 .56173

.40624 .42665 .44591 .46408 .48122

.32367 .34533 .36599 .38565 .40435

.24869 .27061 .29181 .31224 .33187

.18267 .20373 .22450 .24484 .26467

10.4 10.8 11.2 11.6 12.0

.65702 .66809 .67848 .68826 .69748

.57590 .58923 .60179 .61365 .62484

.49742 .51272 .52719 .54090 .55388

.42213 .43903 .45510 .47038 .48491

.35071 .36875 .38602 .40253 .41832

.28392 .30256 .32055 .33788 .35457

12.4 12.8 13.2 13.6 14.0

.70618 .71440 .72219 .72957 .73658

.63544 .64547 .65498 .66402 .67260

.56620 .57789 .58900 .59957 .60963

.49874 .51191 .52446 .53643 .54785

.43341 .44784 .46163 .47483 .48746

.37061 .38603 .40083 .41505 .42871

14.4 14.8 15.2 15.6 16.0

.74324 .74958 .75561 .76137 .76686

.68077 .68855 .68597 .70306 .70983

.61922 .62837 .63711 .64545 .65344

.55875 .56917 .57914 .58867 .59781

.49954 .51112 .52221 .53284 .54304

.44182 .45441 .46650 .47812 .48930

(table continues)

Chapter 6

156

Engset Distribution

Table 6-4. Engset Loss Probability (S = 40 Sources Continued) Loss Probability (P) for N


10

0.4 0.8 1.2 1.6 2.0

.00001

2.4 2.8 3.2 3.6 4.0

=

11

12

13

14

.00006 .00022 .00061 .00144 .00299

.00001 .00004 .00014 .00038 .00088

.00001 .00003 .00009 .00023

.00001 .00002 .00005

.00001

4.4 4.8 5.2 5.6 6.0

.00561 .00964 .01541 .02313 .03292

.00183 .00347 .00607 .00991 .01525

.00053 .00110 .00212 .00377 .00628

.00014 .00031 .00066 .00128 .00231

.00003 .00008 .00018 .00039 .00076

.00001 .00002 .00005 .00010 .00022

6.4 6.8 7.2 7.6 8.0

.04475 .05847 .07385 .09059 .10839

.02225 .03103 .04158 .05375 .06748

.00989 .01479 .02114 .02904 .03848

.00392 .00630 .00965 .01493 .01989

.00139 .00240 .00394 .00617 .00926

.00044 .00082 .00145 .00242 .00387

8.4 8.8 9.2 9.6 10.0

.12694 .14598 .16525 .18456 .20375

.08245 .09844 .11521 .13251 .15015

.04942 .06170 .07517 .08963 .10486

.02701 .03551 .04534 .05642 .06861

.01335 .01856 .02499 .03266 .04154

.00594 .00876 .01248 .01720 .02301

10.4 10.8 11.2 11.6 12.0

.22267 .24124 .25939 .27706 .29421

.16793 .18571 .20335 .22077 .23789

.12068 .13688 .15332 .16984 .18634

.08174 .09565 .11016 .12511 .14035

.05057 .06264 .07462 .08737 .10073

.02994 .03798 .04709 .05718 .06815

12.4 12.8 13.2 13.6 14.0

.31083 .32691 .34244 .35743 .37190

.25464 .27099 .28691 .30237 .31738

.20270 .21886 .23476 .25034 .26557

.15575 .17120 .18660 .20188 .21697

.11457 .12874 .14313 .15763 .17215

.07987 .09221 .10505 .11826 .13173

14.4 14.8 15.2 15.6 16.0

.38584 .39928 .41222 .42470 .43673

.33192 .34600 .35962 .37280 .38554

.28044 .29491 .30900 .32268 .33596

.23183 .24641 .26069 .27465 .28827

.18661 .20095 .21512 .22908 .24280

.14536 .15907 .17278 .18642 .19995

15

(table continues)

Chapter 6

Engset Distribution

157

Table 6-4. Engset Loss Probability (5 = 40 Sources Continued) Offered Traffic (A in Erl)


17

18

19

20

4.4 4.8 5.2 5.6 6.0

.00001 .00003 .00006

.00001 .00001

6.4 6.8 7.2 7.6 8.0

.00013 .00026 .00048 .00085 .00146

.00003 .00007 .00014 .00027 .00049

.00001 .00002 .00004 .00008 .00015

.00001 .00002 .00004

8.4 8.8 9.2 9.6 10.0

.00001

.00238 .00372 .00562 .00819 .01156

.00086 .00143 .00228 .00352 .00525

.00028 .00049 .00084 .00136 .00215

.00008 .00015 .00028 .00048 .00079

.00002 .00004 .00008 .00015 .00026

.00001 .00001 .00002 .00004 .00008

10.4 10.8 11.2 11.6 12.0

.01583 .02108 .02735 .03465 .04292

.00758 .01063 .01448 .01922 .02490

.00327 .00484 .00694 .00968 .01316

.00128 .00199 .00300 .00441 .00629

.00045 .00074 .00117 .00181 .00272

.00014 .00025 .00041 .00067 .00106

12.4 12.8 13.2 13.6 14.0

.05216 .06222 .07303 .08445 .09639

.03153 .03908 .04752 .05677 .06674

.01744 .02257 .02859 .03548 .04321

.00875 .01187 .01573 .02037 .04321

.00397 .00565 .00784 .01064 .02583

.00162 .00242 .00353 .00502 .00697

14.4 14.8 15.2 15.6 16.0

.10873 .12137 .13420 .14716 .16017

.07734 .08847 .10001 .11188 .12399

.05172 .06095 .07079 .08117 .09198

.03211 .03920 .04703 .05557 .06472

.01828 .02323 .02896 .03544 .04266

.00945 .01255 .01631 .02079 .02599

21

0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0

Chapter 6

158

Engset Distribution



8

9

10

11

12

13

2.5 3.0 3.5 4.0 4.5

.00215 .00600 .01329 .02493 .04121

.00052 .00174 .00455 .00982 .01835

.00011 .00044 .00137 .00341 .00724

.00002 .00010 .00037 .00105 .00254

.00002 .00009 .00029 .00079

.00002 .00007 .00022

5.0 5.5 6.0 6.5 7.0

.06183 .08602 .11279 .14117 .17029

.03059 .04659 .06599 .08817 .11238

.01350 .02270 .03507 .05054 .06875

.00531 .00991 .01681 .02633 .03858

.00187 .00388 .00725 .01241 .01969

.00059 .00137 .00282 .00528 .00911

7.5 8.0 8.5 9.0 9.5

.19947 .22819 .25612 .28302 .30877

.13790 .16407 .19038 .21641 .24187

.08921 .11133 .13455 .15837 .18235

.05341 .07054 .08952 .10991 .13124

.02932 .04129 .05549 .07162 .08934

.01465 .02215 .03174 .04339 .05695

10.0 10.5 11.0 11.5 12.0

.33329 .35657 .37864 .39952 .41927

.26657 .29037 .31321 .33504 .35587

.20618 .22960 .25245 .27459 .29595

.15312 .17519 .19718 .21889 .24015

.10826 .12801 .14826 .16872 .18917

.07219 .08881 .10647 .12488 .14375

12.5 13.0 13.5 14.0 14.5

.43795 .45561 .47231 .48812 .50309

.37571 .39459 .41254 .42961 .44585

.31649 .33620 .35507 .37312 .39036

.26085 .28092 .30031 .31899 .33697

.20940 .22929 .24874 .26766 .28602

.16284 .18195 .20092 .21962 .23796

15.0 15.5 16.0 16.5 17.0

.51728 .53074 .54352 .55566 .56721

.46128 .47597 .48995 .50326 .51595

.40684 .42257 .43760 .45196 .46457

.35423 .37080 .38669 .40193 .41653

.30378 .32093 .33747 .35339 .36872

.30378 .27329 .29021 .30661 .32246

17.5 18.0 18.5 19.0 19.5

.57821 .58868 .59867 .60821 .61732

.52805 .39765 .55063 .56118 .57126

.47878 .53960 .50331 .51479 .52579

.43053 .44395 .45682 .46916 .48100

.38347 .49132 .41128 .42439 .43699

.33779 .35258 .36685 .38061 .39387

20.0 20.5 21.0 21.5 22.0

.62602 .63436 .64234 .64999 .65733

.58091 .59016 .59902 .60752 .61568

.53634 .54644 .55615 .56546 .57441

.49237 .50329 .51378 .52387 .53357

.44912 .46078 .47200 .48281 .49321

.40666 .41899 .43088 .44234 .45340

(table continues)

Chapter 6

Engset Distribution

159

Table 6-5. Engset Loss Probability (S = 50 Sources Continued) Loss Probability (P) for N =


14

15

2.5 3.0 3.5 4.0 4.5

.00002 .00007

.00002

5.0 5.5 6.0 6.5 7.0

.00017 .00043 .00099 .00203 .00382

.00004 .00013 .00031 .00071 .00145

7.5 8.0 8.5 9.0 9.5

.00665 .01083 .01663 .02422 .03368

10.0 10.5 11.0 11.5 12.0

17

18

19

.00001 .00003 .00009 .00022 .00050

.00001 .00002 .00007 .00016

.00001 .00002 .00005

.00001

.00274 .00482 .00795 .01239 .01834

.00103 .00196 .00347 .00579 .00915

.00035 .00072 .00138 .00247 .00418

.00011 .00025 .00050 .00097 .00175

.00003 .00008 .00017 .00035 .00067

.04497 .05795 .07238 .08803 .10461

.02594 .03523 .04615 .05857 .07229

.01377 .01981 .02736 .03645 .04700

.00670 .01024 .01497 .02104 .02851

.00299 .00486 .00754 .01120 .01601

.00122 .00211 .00348 .00548 .00829

12.5 13.0 13.5 14.0 14.5

.12186 .13955 .15745 .17541 .19326

.08708 .10271 .11896 .13561 .15248

.05890 .07197 .08600 .10080 .11616

.03738 .04758 .05900 .07147 .08483

.02207 .02942 .03807 .04793 .05890

.01205 .01689 .02291 .03013 .03855

15.0 15.5 16.0 16.5 17.0

.21091 .22827 .24527 .26186 .27802

.16942 .18630 .20302 .21950 .23568

.13189 .14786 .16390 .17992 .19581

.09888 .11346 .12839 .14354 .15879

.07084 .08359 .09697 .11085 .12506

.04809 .05865 .07010 .08229 .09508

17.5 18.0 18.5 19.0 19.5

.29371 .30894 .32369 .33797 .35179

.25152 .26697 .28203 .29667 .31089

.21150 .22695 .24209 .25691 .27137

.17403 .18918 .20417 .21894 .23346

.13950 .15403 .16857 .18305 .19739

.10833 .12190 .13568 .14957 .16349

20.0 20.5 21.0 21.5 22.0

.36514 .37805 .39052 .40257 .41421

.32469 .33807 .35103 .36359 .37574

.28547 .29920 .31254 .32550 .33809

.24769 .26160 .27520 .28845 .30137

.21156 .22550 .23919 .25260 .26573

.17736 .19112 .20472 .21813 .23132

16

(table continues)

160

Chapter 6

Engset Distribution



20

21

7.5 8.0 8.5 9.0 9.5

.00001 .00002 .00005 .00011 .00024

.00001 .00001 .00003 .00008

.00001 .00002

.00001

10.0 10.5 11.0 11.5 12.0

.00046 .00084 .00148 .00247 .00394

.00016 .00031 .00058 .00102 .00173

.00005 .00010 .00021 .00039 .00070

.00001 .00003 .00007 .00014 .00026

.00001 .00002 .00004 .00009

.00001 .00001 .00003

12.5 13.0 13.5 14.0 14.5

.00606 .00895 .01277 .01762 .02358

.00280 .00437 .00657 .00953 .01338

.00119 .00196 .00311 .00475 .00701

.00047 .00081 .00136 .00218 .00339

.00017 .00031 .00054 .00092 .00151

.00006 .00011 .00020 .00036 .00062

15.0 15.5 16.0 16.5 17.0

.03067 .03886 .04809 .05827 .06927

.01822 .02410 .03104 .03901 .04796

.01002 .01388 .01869 .02448 .03127

.00508 .00739 .01043 .01429 .01904

.00238 .00363 .00537 .00771 .01075

.00102 .00164 .00254 .00383 .00560

17.5 18.0 18.5 19.0 19.5

.08096 .09321 .10589 .11887 .13207

.05778 .06837 .07960 .09135 .10351

.03903 .04770 .05719 .06740 .07821

.02474 .03138 .03893 .04734 .05652

.01459 .01929 .02488 .03137 .03873

.00796 .01100 .01480 .01944 .02492

20.0 20.5 21.0 21.5 22.0

.14538 .15872 .17203 .18526 .19836

.11597 .12864 .14142 .15424 .16705

.08951 .10121 .11319 .12536 .13766

.06638 .07680 .08769 .09896 .11050

.04689 .05578 .06531 .07538 .08589

.03126 .03843 .04635 .05497 .06420

22

23

24

25

2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0

Chapter 6

161

Engset Distribution

Table 6-6. Engset Loss Probability (8 = 60 Sources) Offered Traffic (A in Erl)


9

10

11

12

13

14

3.6 4.2 4.8 5.4 6.0

.00574 .01350 .02630 .04449 .06761

.00183 .00507 .01136 .02167 .03648

.00052 .00170 .00439 .00950 .01785

.00013 .00051 .00153 .00376 .00791

.00003 .00014 .00048 .00134 .00318

.00001 .00003 .00014 .00044 .00116

6.6 7.2 7.8 8.4 9.0

.09460 .12421 .15523 .18669 .21784

.05564 .07852 .10420 .13169 .16012

.02997 .04591 .06534 .08760 .11191

.01470 .02464 .03794 .05444 .07372

.00656 .01209 .02026 .03136 .04538

.00266 .00541 .00992 .01665 .02591

9.6 10.2 10.8 11.4 12.0

.24817 .27737 .30523 .33168 .35670

.18877 .21710 .24470 .27144 .29705

.13752 .16377 .19010 .21613 .24156

.09518 .11819 .14215 .16653 .19093

.06206 .08096 .10158 .12337 .14586

.03780 .05221 .06883 .08724 .10701

12.6 13.2 13.8 14.4 15.0

.38031 .40255 .42349 .44320 .46177

.32149 .34473 .36679 .38770 .40749

.26619 .28991 .31265 .33437 .35508

.21503 .23861 .26150 .28361 .30488

.16864 .19139 .21386 .23586 .25727

.12768 .14888 .17028 .19160 .21266

15.6 16.2 16.8 17.4 18.0

.47926 .49575 .51131 .52601 .53991

.42622 .44395 .46074 .47665 .49172

.37480 .39356 .41140 .42835 .44448

.32529 .34482 .36349 .38131 .39833

.27800 .29800 .31724 .33573 .35345

.23330 .25342 .27295 .29183 .31006

18.6 19.2 19.8 20.4 21.0

.55306 .56552 .57734 .58856 .59922

.50602 .51959 .53248 .54475 .55641

.45981 .47440 .48829 .50152 .51413

.41457 .43006 .44485 .45896 .47245

.37044 .38671 .40228 .41718 .43145

.32761 .34449 .36071 .37629 .39124

21.6 22.2 22.8 23.4 24.0

.60937 .61903 .62825 .63704 .64543

.56753 .57812 .58824 .59789 .60712

.52616 .53764 .54861 .55910 .56913

.48533 .49764 .50942 .52070 .53150

.44511 .45819 .47073 .48274 .49427

.40560 .41837 .43260 .44530 .45749

24.6 25.2 25.8 26.4 27.0

.65346 .66114 .66849 .67554 .68230

.61579 .62441 .63251 .64027 .64773

.57874 .58794 .59676 .60523 .61336

.54184 .55177 .56129 .57044 .57922

.50532 .51593 .52612 .53592 .54534

.46921 .48047 .49130 .50171 .51174

(table continues)

162

Chapter 6

Engset Distribution

Table 6-6. Engset Loss Probability (S = 60 Sources Continued) Loss Probability (P) for N ==


15

16

17

18

3.6 4.2 4.8 5.4 6.0

.00001 .00004 .00013 .00039

.00001 .00004 .00012

.00001 .00003

.00001

6.6 7.2 7.8 8.4 9.0

.00099 .00222 .00445 .00812 .01365

.00034 .00083 .00183 .00364 .00663

.00011 .00029 .00069 .00150 .00296

.00003 .00009 .00024 .00057 .00122

.00001 .00003 .00008 .00020 .00047

.00001 .00002 .00007 .00016

9.6 10.2 10.8 11.4 12.0

.02138 .03146 .04387 .05844 .07484

.01117 .01760 .02613 .03680 .04953

.00539 .00912 .01446 .02164 .03079

.00240 .00436 .00741 .01183 .01787

.00099 .00193 .00352 .00600 .00965

.00038 .00079 .00155 .00282 .00484

12.6 13.2 13.8 14.4 15.0

.09271 .11165 .13131 .15136 .17154

.06409 .08019 .09749 .11566 .13439

.04188 .05476 .06922 .08496 .10170

.02569 .03531 .04666 .05960 .07387

.01471 .02135 .02966 .03963 .05115

.00784 .01205 .01768 .02483 .03354

15.6 16.2 16.8 17.4 18.0

.19162 .21146 .23091 .24991 .26837

.15341 .17251 .19151 .21026 .22868

.11914 .13702 .15512 .17326 .19129

.08923 .10541 .12217 .13927 .15655

.06405 .07809 .09306 .10871 .12483

.04375 .05535 .06815 .08193 .09650

18.6 19.2 19.8 20.4 21.0

.28627 .30358 .32030 .33641 .35194

.24668 .26420 .28122 .29771 .31367

.20909 .22658 .24369 .26038 .27661

.17383 .19099 .20795 .22461 .24092

.14123 .15775 .17425 .19064 .20683

.11164 .12717 .14292 .15875 .17456

21.6 22.2 22.8 23.4 24.0

.36688 .38127 .39511 .40842 .42124

.32909 .34398 .35835 .37220 .38556

.,29236 .30764 .32243 .33673 .35056

.25685 .27237 .28746 .30211 .31632

.22274 .23834 .25359 .26846 .28294

.19024 .20574 .22098 .23593 .25055

24.6 25.2 25.8 26.4 27.0

.43356 .44543 .45685 .46785 .47845

.39845 .41086 .42284 .43438 .44552

.36393 .37684 .38931 .40135 .41299

.33008 .34342 .35632 .36881 .38090

.29702 .31069 .32396 .33684 .34932

.26483 .27875 .29231 .30549 .31830

19

20

(table continues)

Chapter 6

163

Engset Distribution

Table 6-6. Engset Loss Probability (S =60 Sources Continued) Offered Traffic (A in Erl)


25

26

.00001 .00004 .00008 .00019 .00039

.00001 .00003 .00007 .00015

.00001 .00002 .00005

.00001 .00002

.00179 .00310 .00509 .00798 .01194

.00077 .00141 .00246 .00407 .00644

.00031 .00060 .00110 .00193 .00323

.00011 .00024 .00046 .00086 .00151

.00004 .00009 .00018 .00035 .00066

.02826 .03727 .04766 .05926 .07192

.01715 .02369 .03161 .04086 .05134

.00974 .01414 .01977 .02668 .03487

.00516 .00789 .01159 .01639 .02239

.00255 .00411 .00635 .00944 .01351

.00118 .00200 .00325 .00508 .00763

18.6 19.2 19.8 20.4 21.0

.08543 .09961 .11426 .12924 .14439

.06291 .07540 .08862 .10241 .11661

.04429 .05481 .06630 .07860 .09154

.02960 .03802 .04755 .05807 .06946

.01867 .02499 .03246 .04104 .05064

.01105 .01547 .02096 .02755 .03523

21.6 22.2 22.8 23.4 24.0

.15960 .17477 .18981 .20468 .21930

.13108 .14568 .16031 .17489 .18936

.10496 .11873 .13271 .14680 .16091

.08155 .09421 .10727 .12063 .13417

.06114 .07240 .08428 .09665 .10938

.04394 .05357 .06401 .07514 .08681

24.6 25.2 25.8 26.4 27.0

.23366 .24772 .26146 .27487 .28794

.20364 .21771 .23153 .24506 .25831

.17496 .18888 .20264 .21619 .22951

.14779 .16140 .17496 .18839 .20167

.12235 .13547 .14865 .16181 .17491

.09890 .11130 .12391 .13663 .14940

21

22

6.6 7.2 7.8 8.4 9.0

.00001 .00002 .00005

.00001 .00002

9.6 10.2 10.8 11.4 12.0

.00013 .00030 .00063 .00123 .00225

.00004 .00011 .00024 .00050 .00097

12.6 13.2 13.8 14.4 15.0

.00388 .00634 .00983 .01457 .02068

15.6 16.2 16.8 17.4 18.0

23

3.6 4.2 4.8 5.4 6.0

(table continues)

164

Chapter 6

Table 6-6. Engset Loss Probability (S Offered Traffic (A in Erl)

Engset Distribution

=60 Sources Continued)

Loss Probability (P ) for N =

27

28

29

30

31

32

3.6 4.2 4.8 5.4 6.0 6.6 7.2 7.8 8.4 9.0 9.6 10.2 10.8 11.4 12.0

.00001

12.6 13.2 13.8 14.4 15.0

.00001 .00003 .00007 .00014 .00027

.00001 .00002 .00005 .00010

.00001 .00002 .00004

.00001 .00001

15.6 16.2 16.8 17.4 18.0

.00051 .00091 .00155 .00255 .00403

.00020 .00038 .00069 .00119 .00198

.00008 .00015 .00029 .00052 .00091

.00003 .00006 .00011 .00021 .00039

.00001 .00002 .00004 .00008 .00016

.00001 .00001 .00003 .00006

18.6 19.2 19.8 20.4 21.0

.00612 .00898 .01273 .01746 .02323

.00317 .00487 .00723 .01039 .01443

.00153 .00247 .00384 .00578 .00840

.00069 .00117 .00190 .00300 .00457

.00029 .00052 .00088 .00145 .00232

.00011 .00021 .00038 .00066 .00110

21.6 22.2 22.8 23.4 24.0

.03006 .03791 .04671 .05635 .06671

.01945 .02548 .03252 .04050 .04935

.01183 .01616 .02144 .02770 .03489

.00674 .00962 .01331 .01790 .02342

.00358 .00535 .00774 .01087 .01481

.00178 .00278 .00420 .00617 .00878

24.6 25.2 25.8 26.4 27.0

.07768 .08914 .10096 .11305 .12531

.05897 .06924 .08006 .09130 .10286

.04298 .05186 .06144 .07162 .08227

.02986 .03720 .04536 .05425 .06378

.01963 .02536 .03197 .03942 .04763

.01214 .01632 .02136 .02726 .03402

Definitions

Address signaling

See register signaling.

Alternate routing

Deterministic routing arrangement in a switching center whereby traffic overflows to another trunk group when all the circuits in the primary trunk group are busy.

Blocked calls cleared

Calls failing to find idle traffic resources are cleared from the system and do not reappear.

Blocked calls delayed

Calls failing to find idle traffic resources are held in a waiting queue untila trafficresourcebecomesavailable.

Blocked calls held

Calls failing to find idle traffic resources are held for an interval equal to one call-holding time. If a traffic resource becomes available during 165

166

Definitions

that interval, it will be used to service the call. Blocked traffic

Portion of traffic that cannot be handled by a traffic resource (offered traffic minus carried traffic).

Busy hour

Continuous sixty-minute period of the day when the highest usage occurs.

Call

Seriesofdialing attemptsto thesame subscriber, where the last attempt results in a completed call or isabandoned.

Call attempt

Any effort, even an error, on the part of a traffic source to obtain service. Ratio of the number of call attempts to completed calls.

Call-attempt factor Call-holding time

Length of time during which traffic resources are held for a call.

Carried traffic

Traffic intensity handled by a traffic resource.

Centum-call-seconds

See unit call.

Common equipment

Common equipment, or servers, are those resources that are used on a shared basis. Trunk groups, signaling registers, and operator positions areexamplesof commonequipment.

Completed call

Callattemptthat issuccessfullycompleted (called subscriber answered).

Congestion

See grade of service.

Cut through

Circuit is completed between a calling and a called subscriber.

Definitions

167

Degree of congestion

Frequency with which a call will encounter some measure of inconvenience in completing a call (i.e., the probability that a call will be blocked or delayed).

Delay system

System in which call attempts are held in a waiting queue until suitable resources are available to handle calls.

DP signaling

Dial-pulse signaling (or rotary-dial signaling) employs a sequence of on-hook, off-hook pulses, at a nominal rate of 10 pulses per second, to represent dialed digits.

DTMF signaling

Dual-tone multifrequency signaling (or tone dialing) uses unique pairs of low- and high-band tones to represent digits.

Erlang

Unit of traffic that represents a circuit occupied for one hour (e.g., one call held for an hour, sixty calls held for one minute each, or any combination totaling one call-hour).

Essentially nonblocking

Grade of service (congestion) that is considered negligible for a specific application. Direct trunk group between two switches where alternate routing is not employed.

Final trunk group

Finite sources

The ratio of sources to servers is 5111al1.

Full availability

Every source has equal access to every server.

Definitions

168

Grade of service

Measure of the probability that a percentage of the offered traffic will be blocked or delayed (used synonymously to define level of traffic congestion or service objective).

High-usage trunk group

Trunk group between two switches engineered to carry high traffic loads and offer significant overflow traffic to an alternate route (another highusage trunk group or a final trunk group).

Hundred-call-seconds

See unit call.

Immediate connection

The time required to set up a call is negligible when compared with the associated call-holding time.

Independent sources

Source of calls (subscribers) are independent as far as originating calls are concerned.

Infinite sources

The ratio ofsources to servers is very large.

Line signaling

Signaling arrangement, also known as supervision, that controls the setup, holding, charging, and releasing of call connections.

Loss system

System in which a call attempt is rejected when an idle resource is unavailable to handle the call.

MF signaling

Multifrequency signaling, also referred to as MF 2/6 signaling, uses unique pairs of six tones to represent line signals and address digits on interoffice trunks.

MFC signaling

Multifrequency-compelled signaling is similar to MF signaling except that for each pair of tones there is a

169

Definitions

complementary pair that is sent back to the calling switch to verify that the correct signal was received at the called switch. Network

Arrangement and interconnection of a number of nodes in a telecommunication system, such as the public switched telephone network, corporate private networks, and local area networks.

Nonblocking

Grade of service (congestion) level of zero at all traffic intensities up to and including one Erlang per channel.

Offered traffic

Traffic intensity submitted to a traffic resource.

Overflow traffic

Traffic intensity that cannot be handled by a traffic resource and is offered to (overflows to) an alternate resource.

Peakedness

Abnormal variations in traffic intensity caused by alternate routing overflow traffic or other system variances such as special rates, holidays, floods, and disasters.

Probability

Likelihood of the occurrence of a specific event.

Probability distribution

Probabilities associated with every possible event of a set of events under consideration.

Queuing system

System operating on a delay basis such that when no idle traffic resource is available, a new call arrival waits in queue until a resource becomes available.

Definitions

170

Random calls

Calls initiated by subscribers are independent of calls initiated by other subscribers.

Register-holding time

See server-holding time.

Register signaling

Signaling arrangement, also known as address signaling, that transmits the called termination's address digits (directory number) through the network to the terminating switch.

Server

See common equipment.

Server-holding time

Length oftime during which a traffic resource is held to process a call attempt.

Service objective

See grade of service.

Traffic density

Number of simultaneous calls at a given moment.

Traffic intensity

Average traffic density (occupancy) during a one-hour period.

Traffic flow

Product of the number of calls during a period of time and their average duration (call-holding time).

Traffic resource

Any resource provided to handle traffic, such as trunks, signaling registers, and switching matrices.

Unit call

Unit oftraffic , also known as centumcall-seconds (CCS) or hundred-callseconds, representing a circuit occupied for one hundred seconds.

Glossary

BASIC

Beginner's All-purpose Symbolic Instruction Code (programming language)

CCITT

International Telephone and Telegraph Consultative Committee

ccs

centum-call-seconds (or hundredcall-seconds

CO

central office

DP

dial pulse

DTMF

dual-tone multifrequency

Erl

Erlang

FIFO FORTRAN

first-in, first-out

GOS MF

grade of service

FORmula TRANslation (programming language) multifrequency 171

172

MFC

Glossary

PSTN

multifrequency compelled private branch exchange public switched telephone network

UC

unit call

PBX


D. Bear, PrinciplesofTelecommunication-Traffic Engineering. London: Institution of Electrical Engineers, 1980. P. Beckmann, Elementary Queuing Theory and Telephone Traffic. Geneva, IL: abc TeleTraining, 1977. V.E.Benes, "OnTrunkswithNegativeExponential Holding TimesServinga Renewal Process," Bell System Technical Journal, vol. 38, no. 1, 1959. G.S.Berkeley, "TrafficandDelay Formulae," J. PostOfficeElec.Engineers(London), vol. 29, 1936. J. G. van Bosse, "Delay Distribution in Communication Systems with Partially Ordered Queues," IEEE Trans. Commun. Sys., vol. 9, no. 3, 1965. J. R. Boucher, Voice Teletraffic Systems Engineering. Norwood, MA: Artech House, 1988. E. Brockmeyer, H. Halstrom, and A. Jensen, "The Life and Works of A.K. Erlang," Trans. Danish Acad. Sci., no. 2, 1948. G.A. Campbell, "Probability Curves Showing Poisson's Exponential Summations," Bell System Tech. J., vol. 2, no. 1, 1925. J. W. Cohen, "The Generalized Engset Formula," Philips Telecommun. Rev. (Amsterdam), vol. 18, no. 4, 1957. 173

174


C. D. Crommelin, "Delay Probability Formula When the Holding Times Are Constant," J. Post Office Elec. Engineers (London), vol. 25, 1932. C. D. Crommelin, "Delay Probability Formulae," J. Post Office Elec. Engineers (London), vol. 26, 1933. A. Elldin and G. Lind, Elementary Traffic Theory, ed. 2., Stockholm: L. M. Ericcson, 1964. A. K. Erlang, "Solutionof Some Problems in theTheoryof Probabilityof Significance in Automatic Telephone Exchanges," J. Post Office Elec. Engineers (London), vol. 10, issue 4, 1917. T. C. Fry, Probability and Its Engineering Uses, ed. 2., New York: D. Van Nostrand, 1965. R. Holm, "The Validity of Erlang's Trunk Congestion Formula," J. Post Office Elec. Engineers (London), vol. 17, 1925. M. M. Jung, "Loss ProbabilityCharts Calculated with the Formula of Engset," Philips Telecommun. Rev. (Amsterdam), vol. 23, no. 4, 1962. G. Lind, "Accuracy of Traffic Measurements in a Poisson Traffic Process," Proc. lEE (London), vol. 23, no. 4, 1967.

E. C. Molina, "Application of the Theory of Probability to Telephone Trunking Problems," Bell System Tech. J., vol. 6, no. 3,1927. E. C. Molina, Poisson's Exponential Binomial Limit, New York: D. Van Nostrand, 1942. I. Molnar, Delay Probability Chartsfor Telephone Traffic Where the Holding Times

Are Constant, Chicago: Automatic Electric, 1952. 1. Riordan, "Delay Curves for Calls Served at Random," Bell System Tech. .I., vol, 40, no. 3, 1953.

J. Riordan, "Delay for the Last Come First Served Service and the Busy Period," Bell System Tech. J., vol. 38, no. 4, 1951. R. Syski, "Determination of Waiting Time in the Simple Delay System," Automatic Electric Tech. J., vol. 13, 1957. R. Syski, "The Theory of Congestion in Lost-CallSystems," Automatic Electric Tech. J., vol. 9, 1953. Telephone Traffic Theory Tables and Charts, Munich: Siemens AG, 1970. Teletraffic Engineering Manual. Stuttgart: ITT Standard Electrik Lorenz AG, 1966.

R. I. Wilkinson, "Working Curves for Delayed Exponential Calls Served in Random Order," Bell System Tech. J., vol. 32, no. 2, 1953.

Index

A Alternate routing, 2, 7,33,48,73,165,167 Assumptions, 10, 11, 14, 19 Availability full, 10,14,20,34,80,132,148 limited, 14

B Binomial distribution, 14, 131 computer program, 132 formula, 14-16, 131, 147 tables, 132, 134-145 Blocked calls cleared, 11,14,15,19,33,147,165 delayed, 11, 14, 15,79, 165 disposition, 11 held, 11, 14, 15, 19, 131, 165 Blocked traffic, definition, 166

175

176

Busy hour, 2, 7, 8,10,73,81,166

c Call, definition, 166 attempt, 10, 11, 166 attempt factor, 8-10, 166 completed, 9, 10, 166 disposition, 9, 10, 14, 15 holding time, see Holding time, call Carried traffic, definition, 2, 166 Centum-call-seconds, 2, 166 See also Unit call Common equipment, 2, 33,79,81, 166 Congestion, 11, 13, 166 degree of, 167 See also Grade of service Crommelin-Pollaczak, 12, 15

D Delay, 79-82 dial tone, 13 formula, 15 loss, 80, 81 operator answer, 13 system, 167 E

Engset distribution, 14, 147 computer program, 148 formula, 14-16, 148 tables, 144-164 Erlang, A. K., 1,173, 174 Erlang, definition, 1, 167 Erlang B distribution, 14, 16,17,19,33,48,73 computer program, 34 formula, 14-16, 33

Index

Index

tables, 34, 36-47 Erlang C distribution, 14, 79 computer program, 80 formulas, 14-16, 79 tables, 81, 83-130 F

Final trunk group, see Trunk group, final Finite sources, see Sources, finite Full availability, see Availability, full G

Grade of service, 12, 13,20,21,34,35,49,73,82, 132, 133, 149, 168

H High-usage trunk group, see Trunk group, high usage Holding time call, 7,8,10 distribution, 11, 15 constant, 11, 19,33,131,147 exponential, 11, 19,33,79,131,147 server, 8,9 variations, 12 Hundred-call-seconds, 2, 168 See also Unit call

Immediate service, 33, 147 Independent sources, see Sources, independent Infinite sources, see Sources, infinite Interpolation, 16 L

Loss system, 168

177

178

Lost calls, 11 Limited availability, see Availability, limited

N Neal-Wilkinson, 73 Network, 2, 169 Nonblocking, 12, 169 absolutely, 12 essentially, 81, 132, 149, 167

o Occupancy, 1 Offered traffic, definition, 2, 169 Overflow traffic, 48, 73, 169 p Peakedness, 73, 169 factor, 73, 74 Peaked traffic, 73 Poisson distribution, 14, 16, 17, 19, 131 computer program, 20 formula, 14-16, 19 tables, 20, 22-31, 73 Primary route, 7 Probability distribution, 13, 19, 169

Q Queue, 8, 11,33,79 Queuing system, 169

R Random calls, 170 Random traffic, 73 Register signaling, 2, 170

Index

Index

s Secondary route, 7 Server, definition, 170 Server-holding time, see Holding time, server Service objective, 1, 12, 170 See also Grade of service Smooth traffic, 73 Sources number of, 11, 14,15,19,33,79,131,147 fini te, 11, 14, 131, 147, 167 independent, 168 infinite, 11, 14, 19, 33, 79, 168

T Traffic density, 1, 10,170 flow, 170 intensity, 1, 2, 170 units, 1,2 volume, 13, 17 Trunk group, 2, 8 alternate route, 7 final, 7, 19,21,33,73,131,167 high usage, 7, 33, 35,48, 168 primary route, 7 secondary route, 7

u Unit call, 2, 170

w Wilkinson, 73

179

Traffic System Design Handbook

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