Nur Aini Masruroh

Queuing Theory

Outlines

Introduction

Birth-death process

Single server model

Multi server model

Introduction Involves the mathematical study of queues or waiting

line. The formulation of queues occur whenever the demand

for a service exceeds the capacity to provide that service. Decisions regarding the amount of capacity to provide

must be made frequently in industry and elsewhere. Queuing theory provides a means for decision makers to

study and analyze characteristics of the service facility for making better decisions.

Basic structure of queuing model Customers requiring service are generated over time by

an input source. These customers enter the queuing system and join a

queue. At certain times, a member of the queue is selected for

service by some rule know as the service disciple. The required service is then performed for the customer

by the service mechanism, after which the customer leaves the queuing system

The basic queuing process

Input source Queue Service

mechanismCustomers Served Customers

Queuing system

Characteristics of queuing models Input or arrival (interarrival) distribution Output or departure (service) distribution Service channels Service discipline Maximum number of customers allowed in the system Calling source

Kendall and Lee’s NotationKendall and Lee introduced a useful notation representing

the 6 basic characteristics of a queuing model.Notation: a/b/c/d/e/fwherea = arrival (or interarrival) distributionb = departure (or service time) distributionc = number of parallel service channels in the systemd = service disciplee = maximum number allowed in the system (service +

waiting)f = calling source

Conventional Symbols for a, bM = Poisson arrival or departure distribution (or equivalently

exponential distribution or service times distribution)D = Deterministic interarrival or service timesEk = Erlangian or gamma interarrival or service time

distribution with parameter kGI = General independent distribution of arrivals (or

interarrival times)G = General distribution of departures (or service times)

Conventional Symbols for d FCFS = First come, first served LCFS = Last come, first served SIRO = Service in random order GD = General service disciple

Transient and Steady StatesTransient state The system is in this state when its operating

characteristics vary with time. Occurs at the early stages of the system’s operation

where its behavior is dependent on the initial conditions.

Steady state The system is in this state when the behavior of the

system becomes independent of time. Most attention in queuing theory analysis has been

directed to the steady state results.

Queuing Model Symbolsn = Number of customers in the systems = Number of serverspn(t) = Transient state probabilities of exactly n customers in

the system at time tpn = Steady state probabilities of exactly n customers in the

systemλ = Mean arrival rate (number of customers arriving per unit

time)μ = Mean service rate per busy server (number of

customers served per unit time)

Queuing Model Symbols (Cont’d)ρ = λ/μ = Traffic intensityW = Expected waiting time per customer in the systemWq = Expected waiting time per customer in the queueL = Expected number of customers in the systemLq = Expected number of customers in the queue

Relationship Between L and WIf λn is a constant λ for all n, it can be shown that

L = λWLq = λ Wq

If λn are not constant then λ can be replaced in the above equations by λbar,the average arrival rate over the long run.

If μn is a constant μ for all n, thenW = Wq + 1/μ

Relationship Between L and W (cont’d)These relationships are important because: They enable all four of the fundamental quantities L, W,

Lq and Wq to be determined as long as one of them is found analytically.

The expected queue lengths are much easier to find than that of expected waiting times when solving a queuing model from basic principles.

Birth and Death ProcessMost elementary queuing models assume that the inputs

and outputs of the queuing system occur according to the birth and death process.

Birth :Refers to the arrival of a new customer into the queuing system.

Death: Refers to the departure of a served customer.Except for a few special cases, analysis of the birth and

death process is very difficult when the system is in transient condition.

However, it is relatively easy to derive the probability distribution of pn after the system has reached a steady state condition.

Rate Diagram for the Birth and Death Process Rate In = Rate Out Principle

For any state of the system n, the mean rate at which the entering incidents occurs must equal the mean rate at which the leaving incidents.

Balance equation The equations for the rate diagram can be formulated asfollows:State 0: μ1p1 = λ0 p0

State 1: λ0 p0 + μ2p2 = (λ1 + μ1)p1

State 2: λ1 p1 + μ3p3 = (λ2 + μ2)p2

….State n: λn-1 pn-1 + μn+1 pn+1 = (λn+ μn)pn

….

Balance equation (cont’d)

0

1

0

1

1 21

1100

1 11

021000

011

021

0123

0123

012

012

01

01

or1

1henceand

1or1obtainwe1Using

:State

:2State

:1State

:0State

pcpc

p

pppp

ppn

pp

pp

pp

nn

nn

n n

n

n nn

nnn n

nn

nnn

cn

Balance equation (cont’d) Expected number of customers in the system

Expected number of customers in the queue:

Furthermore where is the average arrival rate over the long runIt given by

0n

nnpL

sn

nq psnL )(

q

qLWLW ,

0n

nn p

Single server queuing models M/M/1/FCFS/∞/∞ Model

when the mean arrival rate λn and mean service μn are all constant we have

,...2,1for,1

Thus

11

1

11

where,...2,1for,

Therefore

,...2,1for,

11

0

1

0

0

np

p

npp

nc

nn

n

n

n

n

nn

nn

n

Single server queuing models (cont’d)

Consequently

11

11)1()1(

)1()1(

0

00

dd

dd

ddnL

n

n

n

n

n

n

Single server queuing models (cont’d)

qq

nnq

LW

LW

pLpnL

1istimewaitingexpectedThe

)(1)1(

Similarly2

01

Multi server queuing models M/M/s/FCFS/∞/∞ Model

When the mean arrival rate λn and mean service μn, are all constant, we have the following rate diagram

Multi server queuing models (cont’d)

,1,!

1!!

1where,,2,1!

Therefore

,1,!

,,2,1!

havewecasethisIn

0

00

ssnpss

ssn

psnpn

p

ssnss

snnc

sn

n

sn

n

n

sn

n

n

n

Multi server queuing models (cont’d)

1

)()!1(

thatfollowsIt

02

1

q

qq

q

s

q

WW

LW

LL

pss

L

Some General Comments Only very simple models allow analytic

determination of quantities of interests. That is, closed form solution can be obtained for

simple queuing models only. Transient versus steady state behavior

For some real world queuing systems, the transient behavior may be of interests to the decision makers.

For the more complex queuing systems, the quantities

of interests may be obtained through simulation.