Queueing Theory ALISH VIJI VARGHESE ASSISTANT PROFESSOR OF MATHEMATICS COLLEGE OF ENGINEERING THALASSERY November 24, 2011
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2
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Contents 1 Queueing Theory 5 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.1 Syllabus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.3 Application ion of Queueing ing Theory . . . . . . . . . . . . . . 5 1.1.4 Elements of Queueing System . . . . . . . . . . . . . . . . 6 1.1. 1.1.55 Oper Operat atin ingg Char Charac acte teri rist stic icss of Queu Queuei eing ng Syst System em . . . . . . 7 1.1.6 Distribution ions in queuing ing systems . . . . . . . . . . . . . . 7 1.1. 1.1.77 Clas Classi sific ficat atio ions ns of Queu Queuei eing ng Mo Mode dels ls . . . . . . . . . . . . . 8 1.2 Poisson Queueing Systems . . . . . . . . . . . . . . . . . . . . . . 9 1.2. 1.2.11 Mo Mode dell I- (M/M/1) M/M/1) : (∞/FIFO) /FIFO ) . . . . . . . . . . . . . . 9 1.2.2 Littles Formula . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2.3 Mode odel 2 (M/M/1) :( N/FIFO) . . . . . . . . . . . . . . . 10 1.2. 1.2.44 Model Model 3 (Ge (Gene nera raliz lized ed Model Model:: Birth Birth-D -Dea eath th Proce Process ss)) . . . . 10 1.2. 1.2.55 Model Model 4 (M/ (M/M/ M/C) C) :( ∞/FIFO) . . . . . . . . . . . . . . . 11 1.2.6 Mode odel 5 (M/M/C) :( N/ N/F FIFO) . . . . . . . . . . . . . . . 12 1.2.7 Mode odel 6 (M/M/C): (C/FIFO) . . . . . . . . . . . . . . . 12 1.2.8 Model 7 (M/M/R) (M/M/R) :( K/GD, K¿R)-M K¿R)-Mach achine-Re ine-Repairm pairman an model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.3 Non-Poisson Queueing Mode odels . . . . . . . . . . . . . . . . . . . . 13 1.3. 1.3.11 Model Model 1 (M/ (M/G/ G/1) 1):: ( ∞/GD) . . . . . . . . . . . . . . . . . 13 1.4 Problums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
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4
CONTENTS
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Chapter 1
Queueing Theory 1.1 1.1 1.1. 1.1.1 1
Intr In trodu oduct ctio ion n Syll Syllabu abuss
Basic structure of queueing models, exponential and Poisson distribution, the birth and death process , queueing models based on Poisson input and exponential services time, the basic model with constant arrival rate and service rate, finite queue, limited source Q models involving non exponential distributions, single service model with Poisson arrival and any services time distribution, Poisson arrival with constant service time, Poisson arrival with constant service time, Poisson arrival and Erlang service time priority disciplines.
1.1. 1.1.2 2
Defin Definit itio ion n
“The basic phenomenon of queueing arises whenever a shared facility needs to be accessed accessed for service service by a large large number number of jobs or customer customers. s. ” : Bose
1.1.3 1.1.3
Applic Applicati ation on of Queu Queuei eing ng Theor Theory y
Telecommunications, Traffic control, determining the sequence of computer operations, Predicting computer performance, Health services (eg. control of hospital bed assignments), airport traffic, airline ticket sales etc(At retail stores and
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6
CHAPTER CHAPTER 1. QUEUEING QUEUEING THEOR THEORY Y
Figure 1.1: Queueing System
1.1.4 1.1.4
Elemen Elements ts of Queuei Queueing ng Syste System m
Input or Arrival Process (a)Size of the queue No of potential customers waiting to be served. It can be finite or infinite .arrival may occur in batches of fixed or variable size or one by one. (b)Arrival distribution This gives the pattern by which the arrivals take place. place. It is usually described by the arrival arrival rate or inter-arriv inter-arrival al time. These are characterize characterized d by probability probability distributio distributions. ns. Commonly Commonly arrival arrival rate follows Poisson distribution or inter-arrival time follows exponential distribution. (c)Customers behavior if a customer decides not to enter the queue because of the queue size, then he is said to be balked . On the other hand if a customer decides to move from the queue after entering it because of impatience, he is said to have reneged . If a customer moves moves from one queue queue to another for personal personal gains, he is jockey for positions Inputt proces processs can be stationary or transient positions . Inpu (time dependent). Queue disciplines It is the rule by which customers are selected to service when a queue is formed. The most common queue discipline is first come first out (FCFO) or first in first out (FIFO). Last in first out (LIFO) and service in random order (SIRO) are the other queue disciplines. In priority discipline the service given according to some priority schemes. schemes. The service in priority priority disciplines disciplines are of two types:pre-emptive (customer with higher priority is served first.) and non-pre-emptive
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1.1. INTRODU INTRODUCTIO CTION N
7
Service Mechanism If there is infinite number of servers then all customers will be served on arrival and there will be no queue. If number number of customers customers is finite, then customers customers are served in some specific order. They may serve in batches of fixed or variable size called bulk service service system. The service system system is described described in terms terms of service service rate. Service facilities can be of the following types 1. Single queue queue one server, server, 2. Single queue several several servers, servers, 3. Several Several queues queues one server, server, 4. Several servers servers arranged in parallel channels (barber shop with more than one chair) or series channels. Capacity of the system. The source from which which costumers costumers are generated generated may may be b e finite or infinite. infinite. The size of the queue also can be finite or infinite.
1.1.5 1.1.5
Operating Operating Charact Characteris eristics tics of Queuei Queueing ng System System
The following are some important characteristics of a queueing system of general interest. 1. E (n) orL orL or Ls :- Expected number of customers in the system.(both in queue and service) 2. E (m) or Lq :- Expected number of customers in the queue.(excluding that in service) 3. E (v ) or W or W s :- Expected waiting time in the system.(service time and waiting time in queue) 4. E (w) or W q : - Expected waiting time in queue till the beginning of service 5. P : P : - Server Server utilization utilization factor or busy period or traffic intensit intensity y. It is the
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8
CHAPTER CHAPTER 1. QUEUEING QUEUEING THEOR THEORY Y
Distribution Distribution of arrivals arrivals (pure birth models): Let P n (t) denote the probability of n arrivals in a time interval of length t. If the the above above axioms axioms are satisfied, then P n(t) is given by P n(t) =
(λt) λt)n −λt e for n ≥ 0 n!
(1.1)
This is the Poisson distribution with mean λt Distribution of inter-arrival time Inter arrival time is the time between two successive successive arrivals. If arrival rate follows the exponential distribution then the inter arrival time follows the exponential distribution f (t) = λe−λt . The The mean inter arrival time is 1/λ 1/λ.. Distribution Distribution of departure departure As in the case of arrival rate, if P n (t) is the probability that n customers remaining after t times. If µ is the service rate or departure rate and N is the number of customers at time t=0 then (µt) µt)N −n −µt P n (t) = e : 1 ≤ n ≤ N (N − n)!
(1.2)
N
P 0 (t) = 1 −
P n (t)
(1.3)
n=1
This is known as truncated Poisson Law
1.1.7 1.1.7
Classi Classificat fication ionss of Queuei Queueing ng Models Models
Any queueing system can be completely specified by the symbolic form: ( a/b/c) a/b/c) : (c/e). c/e). First and second symbols (a&b) stands for type of distributio distributions ns of interinter-
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9
1.2. POISSON POISSON QUEUEIN QUEUEING G SYSTEMS SYSTEMS
1.2
Poisson oisson Queue Queueing ing System Systemss
1.2. 1.2.1 1
Mode Modell I- (M/M/1) M/M/1) : (∞/FIFO) /FIFO)
This This is single single chann channel, el, Poisso Poisson n input, input, exponen exponentia tiall servic servicee time time with with infinit infinitee system system capacit capacity y. This This type type of queueing queueing models can be solve solved d in three three steps. steps. The same procedure can be applied to other models also. Step I: Construction of differential-difference equations. Step II: Deriving Deriving the steady state difference difference equations. equations. Step III: Solution Solution of the steady-state steady-state difference difference equations. equations. The final step will gives the steady state solution as P n = Where ρ = length.
λ µ
n
λ µ
1−
λ µ
= ρ(1 − ρ)
(1.4)
< 1 and n > 0. This gives the probability probability distributio distribution n of queue
Characteristics of Model 1 • Probability of queue size being greater than n is given by ρn . • Average number of customers in the system is E (n) = Ls = • Average queue length is given by E (m) = Lq =
ρ2
1−ρ
=
• Average length of non empty queue E (m/m > 0) =
0) = P ( P (n > 1) = (fracλµ (fracλµ))2 V (n) = • The variance of queue length is given by V (
ρ
(1−ρ)2
ρ
1−ρ
=
λ µ−λ
λ2 µ(µ−λ) µ
(µ−λ) ,andP (m
=
>
λµ
(µ−λ)2
• Average waiting time of a customer in queue is given by E (w ) = W q = ρ µ(1−ρ)
=
λ µ(µ−λ)
• Average waiting time of a customer in the system including service time
E ( ) = W
1
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10
CHAPTER CHAPTER 1. QUEUEING QUEUEING THEOR THEORY Y
Note: The above mentioned characteristics are unchanged expect for service time distributions even if the queue discipline is SIRO or LIFO. So we write GD to represent these three disciplines.
1.2.3 1.2.3
Model Model 2 (M/ (M/M/1 M/1)) :( N/FIF N/FIFO) O)
In this model the maximum number of customers in the system is limited to N. as in the case of model 1, by solving the difference differential equation in steady state we have the following result (1 − ρ)ρn 1; 0 ≤ n ≤ N ρ= 1 − ρN +1 1 = ρ=1 N = 1
P n =
(1.8)
[These steady state probabilities exist even for ρ ≥ 0 ] Characteristics of model 2 • Average number of customers in the system
Ls =
ρ[1 − (N + 1)ρ 1)ρN + N ρn+1 ] (1 − ρ)(1 − ρN +1 )
• Average number of customers in the queue
Lq =
ρ2 [1 − N ρn−1 + (N (N − 1)ρ 1)ρn ] (1 − ρ)(1 − ρN +1 )
• The waiting time in the system and queue can be calculated using Littles and W q = W s − µ1 formula , i.e. W s = Lλ where λ = λeff = λ(1 − P N N ) s
1.2.4 1.2.4
Model 3 (Gene (Generaliz ralized ed Model: Birth-Death Birth-Death Process) Process)
This model deals with a queueing system having single service channel, Poisson
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11
1.2. POISSON POISSON QUEUEIN QUEUEING G SYSTEMS SYSTEMS
and solving them we have the steady state probability as P n =
λn−1 λn−2 · · · λ0 P 0 µn µn−1 · · · µ1
n≥1
and the value of P n can be obtained by using the condition Special Cases
∞
n=0
P n = 1
Case I When λn = λ for n ≥ 0; and µn = µ for n > 1 then the case become model 1. Case II When λn = λ for n ≥ 0 and µn = nµ for n > 1 P 0 = e−ρ and P n = n1! ρn e−ρ for n > 0 This model is known as Self-service ( M/M/∞) : (∞/FIFO).For /FIFO ).For this model Ls = ρ mode l and is represented by (M/M/ and Lq = 0; W q = 0.
1.2. 1.2.5 5
Model Model 4 (M/ (M/M/ M/C) C) :( ∞/FIFO)
Here we have C parallel service channels with service rate µ per service channel, arrival rate is λ. In effect the service rate of the service faculty faculty is nµ, nµ, if n < C and Cµ if n ≥ C . using generalized generalized model , the steady state state probabilities probabilities are given by 1 n ρ P 0 n! 1 = n−C ρn P 0 C C !
P n =
1 ≤ n ≤ C (1.9) n > C
Characteristics of model 4 • Average queue length is given by
L
λµ( λµ(λ/µ) λ/µ)C P 0
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12
CHAPTER CHAPTER 1. QUEUEING QUEUEING THEOR THEORY Y
1.2.6 1.2.6
Model Model 5 (M/ (M/M/C M/C)) :( N/FIF N/FIFO) O)
In this model the maximum number of customers in the system is limited to N (N ≥ C ). ). The steady state probabilities are given by 1 P n = n!
n
λ P 0 µ 1 λ = n−C C C ! µ
0 ≤ n < C (1.10)
n
P 0
C ≤ n ≤ N
Characteristics of model 5 • Average queue length is given by
Lq =
P 0 (Cρ) Cρ )C ρ +1 1 − ρN −C +1 )(N − C + C + 1)ρ 1)ρN −C − (1 − ρ)(N C !(1 !(1 − ρ)2
• Average number of customers in the system is given by C −1
Ls = Lq + C − P 0
n=0
(C − n)(ρC )(ρC )n n!
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13
1.3. NON-POIS NON-POISSON SON QUEUEING QUEUEING MODELS MODELS
Characteristics of model 7 • Average number of customers in the system is given by R−1
Ls = P 0
K n n n=0
λ µ
n
1 + R!
k
n=R
K n! n n −R n R R1
λ µ
n
• Average queue length R−1
Lq = Ls − R +
n=0
(R − n)
K n
λ µ
n
• Littles formula can be used to find the waiting time in system and queue with λ = λeff = λ[K − E (n)]
1.3
Non-P Non-Pois oisson son Qu Queu euein eing g Models Models
In such models arrivals or departures or both may not follows Poisson axioms. Analysis of such models is more complicated because the Poisson axioms do not hold. The following following technique techniquess are used to analysis a non-Poisson non-Poisson queue. queue.
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14
1.4 1.4
CHAPTER CHAPTER 1. QUEUEING QUEUEING THEOR THEORY Y
Prob Problu lums ms
1.1 A branch office of a large engineering firm has one on-line terminal that is connected to a central computer system during the normal eight-hour working day. Engineers, who work throughout the city, drive to the branch office to use the terminal terminal to make routine calculations. calculations. Statistics Statistics collected collected over a period of time indicate that the arrival pattern of people at the branch office to use the terminal has a Poisson (random) distribution, with a mean of 10 people coming to use the terminal each day. The distribution of time spent by an engineer at a terminal is exponential, with a mean of 30 minutes. minutes. The branch branch office receives receives complains from the staff about the termin terminal al servic service. e. It is reporte reported d that that individ individual ualss often often wait wait over over an hour hour to use the terminal and it rarely takes less than an hour and a half in the office to complete complete a few calculations. calculations. The manager manager is puzzled puzzled because the statistics show that the terminal is in use only 5 hours out of 8, on the average average.. This level of utilization utilization would would not seem to justify justify the acquisition acquisition of another another terminal. What insight can queueing queueing theory provide? provide? 1.2 A mechanic looks after 8 automatic machines; a machine breaks down, independent dependently ly of others, others, in accordanc accordancee with a Poisson process. The average average length of time for which a machine remains in working order is 12 hours. The duration of time a machine required for repair has an exponential distribution with mean 1 hour. Analysis the situation using queueing theory.