TYPES OF FLOW •
•
UNINTERRUPTED Flow occurring at long sections of road where ehicles are not re!uired to sto" #$ an$ cause e%ternal to the traffic strea&' INTERRUPTED Flow occurring at intersections or driewa$s where ehicles are re!uired to sto" #$ an$ cause outside the traffic strea& ( such as traffic signs signs )STOP or YIELD*+ traffic signal lights+etc'
,-.OR TR-FFI/ 0-RI-1LES 0-RI-1LES 2Uninterru"ted flow can #e descri#ed using an$ of the following traffic aria#les3 4' Flow Flow rate rate or olu olu&e &e 5' S"eed 6' Dens Densit$ it$ or or conce concentr ntrati ation on •
Flow Rate or 0olu&e 7is defined as the nu&#er of ehicle "assing a "oint during a s"ecified "eriod of ti&e' It is often referred to as olu&e when &easured oer an hour'
N q= T
-gain if the o#seration o#seration "eriod T is set to one hour+ hour+ q is called olu&e and will hae a unit of ehicles "er hour' In general+ flow rate hae units li8e ehicles "er &inute or ehicles "er da$'
•
S"eed
7is defined as the rate of &otion in distance distance "er unit ti&e' When descri#ing traffic strea+ two t$"es of s"eed are used3 ti&e &ean s"eed and s"ace &ean s"eed' a* Ti&e Ti&e &ea &ean n s"e s"eed ed -lso called s"ot s"eed+ ti&e &ean s"eed is si&"l$ the arith&etic arith&etic &ean of the s"eeds of ehicles "assing a "oint within a gien interal of ti&e' Strictl$ s"ea8ing+ distance or length of road &ust #e 8nown in order to &easure s"eed' 9oweer+ with the use of s"eed radar+ s"ot s"eed can #e &easured at a certain "oint on the road' -lso+ s"ot s"eed can #e reasona#l$ &easured if a "oint is a""ro%i&ated #$ a short distance sa$ 4:7:; & of road' This distance is nor&all$ called trap length '
Δx ui= t i
Where3
ui
< s"eed of ehicle I, in 8ilo&etre "er hour
t i
< ti&e it ta8es for ehicle to traerse the tra" length
Δx
< tra" length+ in &eters
=nowing the indiidual s"eeds of n ehicles o#sered within the ti&e T+ the &ean s"eed or s"ot s"eed of the traffic strea& is gien #$ n
ut =
u ∑ = i
i
1
n
#* S"ace ,ean S"eed 7is used to descri#e the rate of &oe&ent of a traffic strea& within a gien section of road' It is the s"eed #ased on the aerage trael ti&e of ehicles in the strea& within the section' It is also called the har&onic &ean s"eed' If n ehicles are o#sered at an instant of ti&e t, the s"ace &ean s"eed is co&"uted as follows3 n us = n 1 ∑ = u i 1
•
i
Densit$ or /oncentration
7is defined as the nu&#er of ehicles in a gien length of road at an instant "oint in ti&e' If n ehicles are found within the section L+ densit$ k is co&"uted as3 n k = L
OT9ER T-FFI/ 0-RI-1LES There are other aria#les used to descri#e traffic flow' These aria#les+ howeer+ are si&"l$ ariants of the three aria#les descri#ed "reiousl$' 4' Ti&e 9eadwa$ 5' S"acing 6' Ti&e Occu"anc$ •
Ti&e 9eadwa$ 7is defined as the ti&e interal #etween "assage of consecutie ehicles at a s"ecified "oint on the road with a unit of ti&e "er ehicle' ht =
•
q
S"acing 7is the distance #etween two ehicles &easured fro& front #u&"er of ehicle to that of another' Si&ilar to the esti&ation of ti&e headwa$+ if there are n ehicles within a gien road section+ the su& of )n74* s"acing s, will #e al&ost e!ual to L' -erage s"acing+ therefore+ &a$ #e co&"uted as the inerse of densit$' s=
•
1
1
k
Ti&e Occu"anc$ 7it can onl$ #e &easured+ howeer+ if a detector is installed at a s"ecific "oint on the carriagewa$' It is defined as the total ti&e a detector is occu"ied diided #$ the total ti&e of o#seration'
2-ssu&ing that n ehicles were o#sered during the total ti&e of o#seration T, the ti&e occu"anc$ Ot ,is gien #$ n
t ∑ =
i
O t =
i
1
T
x 100
Where t i is the detection ti&e of the i th ehicle' Relationshi" of flow+ s"eed+ and densit$ - relationshi" e%ists a&ong the three &ost i&"ortant traffic aria#les3 flow rate+ s"ace &ean s"eed+ and densit$' - di&ensional anal$sis of the units will show that flow rate )eh>hr* is si&"l$ the "roduct of densit$ )eh>8&* and s"ace &ean s"eed )8&>hr* + or q = k % u +
-s &entioned earlier + densit$ is the &ost difficult aria#le to &easure' It can #e o#tained indirectl$ using this relation'
6'6'4 O#sered Relations
It is oftenti&es useful to deter&ine the relation #etween an$ two aria#les' Sure$s at the South Lu?on E%"resswa$ were conducted' Scattered "lots of the data are shown in figure 6':'
0olu&e7S"eed7Densit$ relations for the inner lane of South Lu?on E%"resswa$
6'6'5 E&"irical Relations S"eed7densit$ relation Figure 6':a shows that as densit$ increases+ s"eed decreases' Loo8ing at the scatter "lot+ it is eas$ to isuali?e that a linear relations &a$ #e assu&ed #etween the two aria#les')Note that othr highwa$s &a$ e%hi#it a trend other than linear'* This linear relation was first inestigated #$ @reenshield )@erlough and 9u#er 4AB:*' To descri#e this line+ the densit$ corres"onding to ?ero s"eed will #e called Ca& densit$ )8 C* and the s"eed corres"onding to ?ero densit$ will #e called free flow s"eed )u f *' Theoreticall$ + densit$ is not ?ero since at least one ehicle &ust #e "resent'
us
k ui=( 1− ) k
k
Typeequation here .
The e!uation of the line that gies the relation #etween s"eed and densit$ can #e easil$ deter&ined #$ ratio and "ro"ortion' ui < uf )47k/k j *
Deter&ine the relation #etween densit$ and s"eed'
Solution3 - co&&on wa$ of anal$?ing relation of two aria#les is through linear regression' The so7called #est fit line re"resents the data "oints with the least error' - scatter diagra& of the data "oints would show that a linear e!uation &a$ #e well suited for the anal$sis'
The regression line ta8es the for& u < a #8
where u ( s"eed 8 ( densit$ a+ # ( constants to #e deter&ined
The constants a and # are deter&ined using the following for&ulas' ) The reader is adised to refer to an$ statistics #oo8s for the deriation of these for&ulas' See -ng and Tang 4AB:'*
´ k u −n ku ∑ b= ∑ k −n ´k i
i
2
2
i
a =´u−b ´k
The correlation coefficient r is gien #$3 r= b
sk su
Where
2
s u=
1
1 ( u −´u ) ∧s = (k − ´k ) ∑ ∑ n −1 n −1 2
i
2
2
k
i
-re the ariances of u and 8+ res"ectiel$' The two aria#les will hae a er$ good correlation if the a#solute alue of r is close to 4';' To "erfor& the regression anal$sis+ it is conenient to "re"are the ta#le as shown3
Point
8
u
=u
=5
)876* 5
)u75':* 5
4 5 6 Su& ,ean
B: 4 45 4;; 665 6
: : 4; 6; 4B; 5':
66B: 45B: 45; 6;;; A;B;
:G5: 55: 5;4G 4;;;; 6G;4
G G5G 64 5A :
G'5: 4;G'5: 4;:G'5: 4:G'5: 6;5:
´ 9070 −4 ( 83 ) ( 42.5 ) k u −n ku ∑ = =−0.5959 b= 36014− 4 ( 83 ) ∑ k −n ´k i
i
2
2
2
i
a =´u−b ´k = 42.5−(−0.5959 ) ( 83 )= 91.96 1
2
( ui−´u ) = 4 −1 3205 =1008.33 ¿
2
s u=
1
n− 1
∑ ¿
s u=31.75
( k −k ´ ) = 13 8458 =2819.33 2
i
¿ 2
sk =
1
n −1
∑¿
s k =53.10
r= b
sk su
=−0.5959
53.10 31.75
=−0.9964
This is al&ost close to 74';+ which &eans that the correlation #etween the two aria#les is er$ high' There negatie sign confir&s that as densit$ increases+ s"eed decreases'
E%a&"le 6'B
Using the results of the "reious e%a&"le+ deter&ine the free flow s"eed and Ca& densit$'
Solution3 The densit$7s"eed relation o#tained fro& the "reious e%a&"le is
u < A4'AG7;':A:A 8
Free flow s"eed occurs when densit$ 8 < ;' uf 7;':A:A);* < A6'AG 8"h
.a& densit$ occurs when s"eed u < ;' ; < A4'AG ( ;':A:A 8 C = C < A4'AG
or
÷ ;':A:A < 4:'65 eh>8&
0olu&e7densit$ relation Su#stituting e!uation 6'A to the general relation )e!uation 6'*3 ! < 8u < 8 u C)478>8 C* < u C)8785>8 C*
)6'4;*
This e!uation e%"resses the relation of ! and 8 as "ara#olic'This can #e drawn as shown3
Due to the s$&&etr$ of the figure+ it can #e said that the &a%i&u& flow ! &a% occurs when the densit$ has a alue 8 & e!ual to half of Ca& densit$ 8 C ' 9oweer + when the relation cannot #e easil$ identified+ it is useful to differentiate the function and e!uate to ?ero to get the alue of 8 & corres"onding to &a%i&u& flow+ as follows3
(
)
2 k dq =u f 1− m =0 dk k j
k m =k j / 2
0olu&e7s"eed relation Fro& e!uation 6'A+ it can also shown that k =k j
( ) 1−
ui
uf
)6'44*
Su#stituting this in e!uation 6' gies a "ara#olic relation #etween ! and u i3
! < 8u4 < 8 C)u47u6 5>uf *
)6'45*
This relation is illustrated #elow3
-gain+ it can #e shown that &a%i&u& flow ! &a% occurs at s"eed u & e!ual to half of the free flow s"eed u f' Therefore+ the alue of the &a%i&u& flow+ also called ca"acit$ is q max= k m x um =
k j 2
x
u f
k j u f
2
4
=
@oing #ac8 to the s"eed ( densit$ relation+ ! &a% is shown to #e the shaded area of the rectangle'
E%a&"le 6' In the "reious e%a&"le +deter&ine the ca"acit$ of the rural highwa$ in one direction'
Solution3 -s alread$ shown+ the densit$7s"eed relation can #e &oldeled #$ a straight line' The for&ula for ! &a% can #e used to co&"ute for the ca"acit$' q max=
k j 2
x
u f 2
=
154.32 91.96 2
x
2
=3,547.82 vehicle / hr
/-P-/ITY -ND LE0EL OF SER0I/E /a"acit$ is defined as the &a%i&u& hourl$ rate at which "ersons or ehicles can reasona#l$ #e e%"ected to traerse a "oint or unifor& section of a lane or roadwa$ during a gien "eriod under "reailing roadwa$+ traffic and control conditions' On the other hand+ leel of serice )LOS* is a !ualitatie descri"tion of how a certain facilit$ is "erfor&ing'Traffic engineers rel$ on ca"acit$ and leel of serice anal$sis to deter&ine the width and nu&#er of lanes when "lanning for new facilities or when e%"anding e%isting facilities that are alread$ e%"eriencing congestion problems ' The Philippine Highwa Planning !anual )P9P,* deelo"ed #$ the Planning serice of the DPW9 "roides a ðodolog$ to carr$ out the "rocess of such anal$sis'The LOS conce"t uses !ualitatie &easures that characteri?e o"erational conditions within a traffic strea& and "erce"tion of these conditions #$ &otorists and "assengers'
Si% leels of serice are defined for each t$"e of facilit$ and are gien letter designation fro& - to F+ with - re"resenting the #est o"erating conditions and F the worst' Each leel re"resents a range of o"erating conditions defined #$ !uantitatie factors 8nown as &easures if effectieness' In the P9P, ðod+ LOS is defined #ased on the co&"uted olu&e and ca"acit$ ratio and the s"ace &ean s"eed of the traffic flow' The olu&e referred to is the hourl$ de&and olu&e' This ðod was si&ilar to the Highwa "apacit !anual #H"!$ ðod of 4AG:' The latest 9/, now considers densit$ as the &ain aria#le in deter&ining LOS' LEVEL OF
DESCRIPTION
SERVICE
-
Free flow+ with low olu&es and high s"eeds' Driers are irtuall$ unaffected #$ the "resence of others' Little or no restriction in
&aneuera#ilit$ and s"eed' 1
/
D
E
F
LE0EL OF SER0I/E 1 / D E F
The leel of co&fort and conenience "roided is so&ewhat less than at LOS -' Hone of sta#le flow with o"erating s"eeds #eginning to #e restricted so&ewhat #$ traffic conditions' Driers will hae reasona#le freedo& to select their s"eed #ut there is a decline in freedo& to &aneuer within the traffic strea& fro& LOS -' Still in ?one of sta#le flow+ #ut s"eed and &aneuera#ilit$ are &ost closel$ controlled #$ higher olu&es' ,ost of the driers are restricted in the freedo& to select their own s"eed' The leel of co&fort and conenience declines noticea#l$ at this leel -""roaches unsta#le flow' S"eed and freedo& to &aneuer are seerel$ restricted+ and drier e%"eriences a generall$ "oor leel of co&fort and conenience' S&all increases in traffic flow will generall$ cause o"erational "ro#le&s' Flow is unsta#le+ and there &a$ #e sto""ages of &o&entar$ condition' Re"resents o"erating conditions at or near ca"acit$ leel' -ll s"eeds are reduced to allow #ut relatiel$ unifor& alue' Freedo& to &aneuer within the traffic strea& is e%tre&el$ restricted + and it is generall$ acco&"lished #$ forcing a ehicle to gie wa$J to acco&&odate such &aneuer' Forced or #rea8down flow' The a&ount of traffic a""roaching a "oint e%ceeds the a&ount that can traerse the "oints' Kueues fro& #ehind such locations' O"eration within the !ueue is characteri?ed #$ sto"7and7go waes+ and is e%tre&el$ unsta#le' It is the "oint at which arrial flow causes the !ueue to for&'
0OLU,E7/-P-/ITY R-TIO Less than ;'5; ;'54 ( ;':; ;':4 ( ;'B; ;'B4 ( ;': ;'G ( 4';; @reater than 4'; ,easure of Effectieness for Different T$"es
T$"e of Facilit$
,easure of Effectieness
1asic e%"resswa$ seg&ents Weaing areas Ra&" Cunctions ,ultilane highwa$s Two7lane highwa$s Signali?ed intersections
Densit$ )"assenger car>8&>lane* -erage Trael S"eed )8&>hr* Flow Rates )"assenger car>hr* Densit$ )"assenger car>8&>lane* Percent Ti&e Dela$ ) * -erage Indiidual Sto""ed Dela$ )sec>eh* Resere /a"acit$ )"assenger car>hr* -erage Trael S"eed )8&>hr*
Unsignali?ed intersections -rterials
6': 9YDRODYN-,I/ -ND =INE,-TI/ ,ODELS OF TR-FFI/ Using fluid flow analog$+ &odels that are used to descri#e traffic flow will #e deelo"ed in this section' /onsider two "oints on a one7wa$ road assu&ed to #e ho&ogeneous3
Let Ni ( nu&#er of cars "assing station i during ti&e interal !i ( flow )olu&e* "assing station I during
∆ t
∆ t
∆ x ( distance #etween stations ∆ t ( duration of si&ultaneous counting at stations 4 and 5'
1$ definition+ qi =
N i
∆ t
)6'46*
Su""ose N4 M N5 )&eans traffic is #uilding u"*' Let
∆ N =( N 2− N 1)
With
Let
∆ q=
∆ N ∆ N = ∆ q ∆ t ∆ t
∆ k : increase in densit$ #etween stations 4 and 5 during "eriod
∆ t '
Then+
∆ k =
−( N 2− N 1 ) −∆ N ( period for buildin! ) = ∆x
∆x
−∆ N = ∆ k ∆ x
)6'4:*
Or e!uating e!uation 6'4 and e!uation 6'4:3 ∆k ∆ x −∆ q ∆ t = ∆ q ∆ k + = 0 (3.16 ) ∆ x ∆ t
If the &ediu& is considered continuous and finite ele&ents are allowed to #eco&e infinitesi&al3 " q " k + = 0 " x " t
This is well 8nown as the continuit$ e!uation'
With ! < u8
" ( uk ) " k + = 0 "x " t
)6'4B*
Recall that u = f#k$
E%"anding 3
" k " k " u + u +k =0 " t "x "x
)6'4*
-""l$ing chain rule3 du " u " x "u " k =u# = =u# dk "x"k "x "x
)6'4A*
Su#stituting e!uation 6'4A in e!uation 6'43
" k
# " k " k + u + k u " x = 0 " t "x
" k " k + ( u + k u# ) =0 " t "x
)6'5;*
-nalogous to fluid flow+ the e!uation of &otion e%"ressing the acceleration of traffic strea& at a gien "lace and ti&e is gien #$
2
du −c " k = dt k "x
)6'54*
Where c is a constant of "ro"ortionalit$' If
" k " x is "ositie+ then traffic flow has a tendenc$ to slow down'
On the other hand + if
" k " x is negatie+ the traffic flow tends to faster'
Let us generali?e the fluid7flow analog$ e!uation3 du " k =−c 2 k n dt "x
)6'55*
With s"eed u < f )%+ t* ' du " u " x " u " t du "u = + = u + dt " x " t "t "t dx " t
/o&#ining the a#oe with the general e!uation+ "u " u 2 n " k u + + c k =0 "x "t "x
#ut " u " u " k # " k = = u "x "k"t "x " k
# "u 2 n " k u + u " t + c k =0 "x "x
Using e!uation 6'4A3 " k " k 2 n " k =0 u # u + u # c k "x " t "x
Diiding the a#oe e!uation #$ u 3
(
2
n
)
" k c k " k + u + # =0 " t u "x
)6'56*
This e%actl$ has the sa&e for& as e!uation 6'5;' E!uating e!uations 6'5; and 6'56 3
(
n
2
)
" k c k " k " k " k + u + # = + ( u + k u# ) =0 " t "x u " x " t
(
2
n
)
c k # u + # =( u + k u ) u
( u# ) =c k n− u# = 2
2
1
du = c k ( n−1)/ 2 dk
/onsidering that u and 8 alwa$s hae an inerse relationshi"+ the negatie sign is added on the right side of the e!uation' du =−ck (n −1)/ 2 dk
)6'5*
We can now consider so&e s"ecific &odels+ the first of which is the @reenshields &odel )n<4*' du =−c k 0=−c d u=−cdk dk u=−ck + a
When 8<; u < u f ' Therefore a < u f % u < uf ( c8
also when u < ; + 8 < 8 C' Therefore
c=
uf k j
This gies the u78 relationshi" for @reenshields &odel 3 u=u f ( 1 −
k ) k j
Two &ore &odels can #e easil$ identified3
@reen#ergs &odel 3 n < 74
)6'5:*
Para#olic &odel 3 n < ;
Ta#le 6' su&&ari?es the different &acrosco"ic &odels de"ending on the alue of n3
Ta#le 6' ,acrosco"ic ,odels Ele&ent
n74
n<74
n<;
nM74
/onstant of "ro"ortionalit$
u f
um
uf
(n + 1 )u f
u78 relation
1/ 2
k j u=u f ( 1 −
k ) k j
u=u m ( ln
k j k
)
k j
k j
4
2
e
9
O"ti&u& s"eed+ U n
u j
c
6'G
2 k j
1 /2
k 1−( ) k j u =uf ¿
O"ti&u& densit$ + 8 n
2
(n +1)/ 2
2 k j
k j
u j 3
( )
k u=u f [ 1− k j
k j (
n +3
(
QUEUING THEORY
Kueuing at a gasoline station or at the toll gate+ falling un line to transact #usiness at the #an8 or Cust to get a &oie "ass+ !ueuing at a #us$ "ar8ing lot+ Cet
2
n+ 1 2
−2 /(n + 1)
)
n +1 )u n + 3 f
]
"lanes waiting #efore #eing gien the signal land or ta8eoff7these are eer$da$ occurrences that would surel$ test ones "atience' Kueuing anal$sis "roides was assessing the i&"acts of these actiities #$ 8nowing the &agnitude of ehicular dela$ and the e%tent of !ueue "ro"agated' The &odels that will discussed in this section are deried #ased on so&e assu&"tions related to arrial and de"arture "atterns and the "reailing !ueue disci"line' /onsider the s$ste& shown in figure 6'B'
Serice station In"ut
Out"ut
Figure 6'B Kueuing s$ste& The input is nor&all$ characteri?ed #$ so&e for& of arrial "attern usuall$ gien #$ its arrial distri#ution' The output generall$ de"ends on the !ueue disci"line and the serice &echanis& at the ser&ice station% The &ost co&&on t$"e of !ueue disci"line is the so7called FIFO or first7in first out+ i'e'+ the first one that arries at the serice station gets sered first and therefore the first to leae the s$ste& as well' )-nother t$"e of !ueue disci"line+ which has li&ited a""lication to traffic flow+ is the so7called LIFO or last7in first7out' T$"ical e%a&"les of this disci"line are the following3 the last rider of an eleator nor&all$ gets out first the last docu&ent "iled on to" gets signed first7not a reco&&ended "racticeQ* Serice &echanis& refers to the &anner custo&ers are sered at the station' For e%a&"le+ a toll #ooth that charges a single fee+ acce"ts onl$ a fi%ed a&ount+ and does not gie #ac8 an$ change will hae a fairl$ unifor& serice rate co&"ared to a #ooth that charges aria#le toll fees and gies #ac8 change u" to the last centao' =endalls notation is "o"ularl$ used to descri#e !ueuing s$ste&' It ta8es the for& $ % & % ' ( n )
Where - ( re"resents the in"ut or arrial "attern
1 ( re"resents the serice &echanis& /( re"resents nu&#er of serers n ( re"resents the li&it of the !ueue or users -rrials and de"artures &a$ either follow a rando& or deter&inistic "attern' ,ar8o ),* is used for rando& "rocesses while Deter&inistic )D* is used for "rocesses that are characteri?ed #$ regular or constant arrials or de"artures' T$"ical e%a&"les of these "rocesses are3
( % ( % 1 ( ) ) ( rando& arrial and de"arture )serice rate* one or single
serer infinite !ueue )no li&it* ( % ( % N ( ) ) 7 rando& arrial and de"arture N or &ulti"le serers infinite
!ueue * % * % 1 ( 100 ) 7 regular arrial regular serice rate or de"arture single serer
li&it of !ueue is 4;; - co&#ination of ,ar8o and deter&inistic "rocesses+ sa$ ,>D>4 &a$ also #e used' 6'G'4 D>D>4 Kueuing Due to the regularit$ of #oth arrials and de"artures+ it is &ore conenient to anal$se a D>D>4 !ueuing s$ste& gra"hicall$' -rrials and de"artures are easil$ re"resented #$ straight lines with the slo"es corres"onding to their rates'
6'G'5 !/'/( )ueuing
The ,>D>I !ueuing s$ste& assu&es that the arrial of ehicles follow a negatie e%"onential distri#utions+ a "ro#a#ilit$ distri#ution characteri?ed #$ rando&ness' De"arture is assu&ed to #e regular as in the D>D>4' The reader is adised to refer to other #oo8s on !ueuing theor$ for the deriation of for&ulas' Let ƛ7 arrial rate and 7 de"arture rate' Then += ƛ / , is the traffic densit$ or utili?ation factor' Note that if + < 1 then
ƛ < ,
+ which &eans that the s$ste& is sta#le'
Otherwise+ !ueue #eco&es longer and longer )unsta#le condition*' 1asic for&ulas for ,>D>4 a' -erage length of !ueue
´= m
2 +− +
2
2 ( 1− + )
#' -erage waiting ti&e ´=
+ 2 , ( 1− +)
c' -erage ti&e s"ent in the s$ste&
´t =
2− + 2 , (1 − +)
6'G'6 !/!/( )ueuing The ,>N>4 !ueuing s$ste& assu&es negatie e%"onential for #oth arrial and de"arture distri#utions' 1asic for&ulas for ,>,>43 -' -erage length of !ueue
´= m
ƛ
2
, ( , − ƛ )
1' -erage waiting ti&e
´= -
ƛ
, ( ,− ƛ )
/' -erage ti&e s"ent in the s$ste
´t =
1
( ,− ƛ )
6'G' !/!/* )ueuing When there is &ore than one serer+ such as in a toll gate shown in figure 6'A+ an arriing ehicle will #e a#le to "roceed to a acant gate+ if aaila#le'
Otherwise the drier &a$ hae to wait in !ueue if all gates are full' -gain the arrials are assu&ed with a rate of ƛ and the serice rate "er serer is , . + is still ƛ
efine as
,
' 9oweer+
+ N is defined as the utili?ation factor'
For ,>,>N + the alue of &a$ #e greater than 4 #ut
+ N &ust #e less than 4
for sta#le condition'
1asic for&ulas for ,>,>N3
a' -erage length of !ueue N + 1
+ m ´= o [ N / N
1
( ) + 1− N
2
]
)6'65*
Where .-= N −1
∑ = n
0
1 N
+
p N / ( 1 − ) N
Is the "ro#a#ilit$ of no units in the s$ste&'
)6'66*
#' -erage waiting ti&e ´=
+ + m ´ ƛ
−
1
,
)6'6*
