Chapter 1

Introduction to Queue Modelling

and Machine Repair Model

1.1 Introduction

Every moment we spent waiting for some type of service, we are part of a

queue. We wait in line in our cars in traffic jams or tool booths, we wait in line at

supermarkets to check out, we wait in line at barber shop or beauty parlor, we wait

in line at post offices etc. We, as customers, do not generally like these waits and

the managers of the establishments at which we wait also do not like us to wait,

since it may cost then business. Whey then is there waiting? The answer is, there

is facility for service available.

In the present chapter, we introduce queueing models and related some

definitions. We define machine interference problem which is finite source

queueing model.

Following are some queueing examples:

System Customers Server

Reception Desk People Receptionist

Hospital Patients Nurses

Airport Aeroplanes Runway

Road Network Cars etc. Traffic light

Grocery Shoppers Checkout

Station Computer Jobs CPU, Disk, CD

Table 1.1: Queueing examples.

2

1.2 Preliminaries of the Queueing System

Modern age is the information technology and e-governance age, require

innovations, that are based on modelling, analysing, designing, and finally

implementing new systems. The whole developing process as`sumes a well

organised team work of experts including engineers, computer scientists,

mathematicians, physicists etc. Modern info -communication networks are one

of the most complex systems where the reliability and efficiency of the

components play a very important role. For the better understanding of the

dynamic behaviour of the involved processes one has to deal with constructions

of mathematical models which describe the stochastic service of random

arriving requests. Queueing theory is one of the most commonly used

mathematical tool for the performance evaluation of such systems.

"Delay is the enemy of efficiency and waiting is the enemy of

utilization".

A flow of customers from infinite / finite population towards the service

facility forms a queue or waiting lines on accounts lack of capacity to serve

them all at a time. “The queue is the group of customer’s (people, tasks or

objects) waiting to be served”.

Though we are most aware of our own waiting time, queueing is for

most a problem for industry and government. The success of any organization

depends on maximizing the utilization of its resources. Every minute that an

employee spends waiting for another department and even minute that a job

spends waiting to be processed is money wasted. Thus, success of any

organization also depends on attracting and keeping customers. Every minute

that a customary spends waiting to be served translates into lost business and

lost revenue.

The basic components of queueing system are: queues, customers and

servers. The goal of the analysis of a queueing system is to finding analytical

expression for such performance measures as queue length, throughput and

3

utilization. Queueing is an aspect of modern life that we may encounter any

place and any time in our daily life. In the banks, people stand in a line in front

of the counters and wait for the service. Also, in the supermarkets, people wait

in the queue to pay for the goods. Although there are no people stand any line

in the barber shop, the customers will be served in the sequence they arrive.

The study of queueing theory or waiting lines is able to provide both a

theoretical background to the kind of system and the way in which the system

itself may be designed to provide some specified grade of service to its

customers. Generally, in queueing theory, a queueing model is used to imitate a

real queue situation. Also, the queueing behaviours can be analysed

mathematically by the performance measurement calculation. These measures

are very important since they are often relevant with the work efficiency or

economic losses of a real queueing system. In the queueing system, analysis is

more accurate and deterministic results are expected.

The first text book on the subject: Queues, Inventories and Maintenance

was written in [1958] by Morse,P. Satty, T.L., wrote his famous book:

Elements of Queueing Theory with Applications in [1961]. Kleinrock

completed his: Queueing Systems in [1975]. Queueing systems can be

modelled as network of queues, like waiting in line at a doctors clinic, railway

reservation centre, in a bank, a bus stop, etc. A queueing system can be

described as customers arriving for service, waiting for service, if the servers

are busy and leaving the system after own service being finished. Gross and

Harris [1985] studies Fundamentals of Queueing Theory. Little J.D.C. [1961]

and W.S. Jewell [1967] developed the basic understanding of a queueing

system and proof of the Little’s formula. Little's theorem states as: “The

average number of customers N can be determined by N = λt.

Where, λ = average customer arrival rate

And, t = average service time for a customer.

4

We explain some of the major characteristics of queueing systems which are

given below:

(i) Arrival pattern of customers.

(ii) Service pattern of servers.

(iii) Customers and customer behaviour.

1.2.1 Arrival Pattern of Customers

The arrival pattern or input to a queueing system is often measured in terms

of the average number of arrivals per some unit of time (mean arrival rate) or by

the average time between successive arrivals (mean inter-arrival time).

1.2.2 Service Pattern of Servers

Service pattern can be described by a rate (number of customers served

per unit of time).

Note:

1. If the system is empty, the service facility is idle. Service may also be

deterministic or probabilistic.

2. The service rate may depend on the number of customers waiting for

service. A server may work fasten if he/she sees that the queue is building

up or conversely, he / she may get flustered and become less efficient. The

situation in which service depends on the number of customers waiting is

known as state-dependent service.

1.2.3 Customers and Customer Behaviour

(a) Customers: By customers, we mean the arriving units that require

some service to be performed. The customer may be of persons,

vehicles, machines, etc., queues stands for a number of customers

waiting to be served.

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(b) Customer Behaviour

Some of the customer behaviours are: reneging, balking and jockeying.

Reneging is the act of leaving a queue before being served. Balking is

the act of not joining a queue upon arrival. Conceptually, the two concepts are

the same. The only difference is the timing of when the customer leaves (either

immediately in the case of balking or later in the case of reneging). Jockeying

is the act of switching from one queue to another.

Figure 1.1: Standard graphical notation for queues.

In above figure 1.1, we have shown a standard graphical notation for

queues. Until now, queueing systems have been well studied. For the shorthand

for describing queueing processes, Kendall’s [1953] notation was used. This

notation describes a single process as a series of symbols. We have presented

these notations in table 1.2.

1.3 Performance Measures in Queue

Let C be the number of identical servers and arriving unit requires

service from any of the server.

Define nP (t) Pr ob.= {exactly n units in the system at time t}. Then

(i) sL (t) = Expected number of units in the system at time t is given as

follows:

nn 1

nP (t)∞

=

∑

(1.1)

(ii) qL (t) = Expected Number of units in a queue at time t given by:

6

nn c 1

(n c)P (t)∞

= +

−∑

(1.2)

(iii) C(t) = Expected number of busy servers at time t is shown by:

1

1

( ) ( )∞ −

= =

+∑ ∑cn n

n c n

c P t nP t

(1.3)

(iv) ( )C t = Expected number of servers not busy at time

0

( ) ( ) ( ).∞

=

− = −∑ n

n

c c t c n P t

(1.4)

(v)0

( ) Prob.=

=∑c

n

n

P t {no arrival is waiting for service at time t i.e. no

queue}

( ) ( ).= +s qL L t c t

(1.5)

(vi) For steady state

( ) lim ( ).→∞

=n nt

P t P t (1.6)

sW = Expected waiting time in the system.

qW = Expected waiting time in the queue.

1.3.1 Little’s Formula

(i) ;s s q qL W L LWλ= =

(1.7)

Where, λ is an arrival rate of the system.

(ii)1

s qW Wµ

= +

(1.8)

Where, µ is service rate of the system.

(iii) Cλµ= . (1.9)

7

1.4 Queueing System

Figure 1.2: Queueing system.

Steady-state (behaviour is independent of time t)

( ) lim ( )n n nn

P t P t P→∞

= = (1.10)

Figure 1.3: Transient phase.

Generally, in many situations only first three symbols A / B / X are used,

because these are the three most important characteristics, the symbol Y is

omitted if there is no restriction and the queueing system is FIFO. With a single

queue, the most common performance measures are obtained as follows:

ρ : Is the server utilization which describes the fraction of time that the

server is busy or the mean fraction of active serves, in the case of

multiple servers.

λ : Is the throughput which describes the number of jobs where processing

is completed in a single unit of time.

Q : Is the queue length which is the number of jobs waiting in the queue at a

given time.

8

w : Is waiting time which is the time that the jobs spend in the queue

waiting to be served.

k : Is the number of jobs in the system at a given time.

Pn : Is the probability of a given number of jobs n in the system.

In the theory of queues almost every queueing system can be

represented by Kendall’s [1953] notation and analyzed as the above

performance measures.

1.4.1 Notations of Queueing Models

Due to wide range of applications, Statistical distributions

parameters and disciplines, Kendall, D.G [1953] represented Stochastic

Processes Occurring in the Theory of Queues and their Analysis by the

method of Imbedded Markov Chain and gave a short hand notation for

queueing systems. According to that notation, a queueing system is

described by the string A/B/X/Y/Z where:

A: Arrival distribution,

B: Service pattern,

X: Number of servers,

Y: System capacity, and

Z: Queueing discipline.

Standard symbols commonly used in queueing systems are presented

in Table 1.2.For example: M / D/ 5/ 50 / PRI: describes a queueing system

with exponential inter-arrival times, 5 servers each deterministic service

time, a system capacity of 50 places and a priority service discipline.

9

Note: 1. Exponential inter-arrival times directly relate to Poisson

Arrivals.

2. A system capacity of 50 places in a system with 5 servers

spacify a maximum queue size of 45.

3. All symbols are mandatory, as symbols Y and Z may be omitted

thus resulting in an abbreviated string A / B / X. In this case the

system capacity is unlimited and the queue discipline is FCFS. A

queuing system directed by M / M / 1 / ∞ / FCFS is generally

abbreviated by M / M / 1.

Characteristics Symbol Description

A – Inter-arrival distribution: D

CK

EK

G

GI

GEO

HK

M

ME

MAP

PH

Deterministic

Cox (K- phases)

Enlarge (K -phases)

General

General independent

Geometric (discrete)

Hyper-exponential

Exponential (Markov)

Matrix exponential

Markov arrival process

Phase type

B- Service time distribution: D

CK

EK

G

GI

Deterministic

Cox (K- phases)

Enlarge (K -phases)

General

General independent

10

KGEO

HK

M

ME

MAP

PH

SM

Geometric (discrete)

Hyper-exponential

Exponential (Markov)

Matrix exponential

Markov arrival process

Phase- type

Semi-Markov

X - Number of parallel servers: 1, 2, ..., ∞Y - System capacity: 1, 2, .., ∞Z –Queue- discipline: FCFS

LCFS

RSS

PRI

RR

PS

GD

First come first server

Last come first server

Random selection for service

Priority service

Random robin

Processor Sharing

General

Table 1.2: Notations of queueing models.

1.5 Queue Discipline

Queue discipline is a rule, according to which customers are selected for

service when a queue has been formed. Most common queue discipline is the

FCFS or “First Come First Served” or FIFO “First in First Out” rule

under which the customers are serviced in the strict order of their arrivals.

Other queue disciplines are LIFO “Last in First Out, SIRO “Selection for

service In Random Order” and a variety of priority schemes according to

which a customer’s service is done in preference over some other customer’s

service.

Under priority discipline, the service is of two types:

(i) preemptive

(ii) non-preemptive

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In preemptive discipline, the customers of high priority, are given

service over the low priority customers, while in non-preemptive discipline, a

customer of low priority is serviced before a customer of high priority. In case

of parallel channels “Fastest Server Rule” (FSR) is adopted, by parallel

channels, we mean a number of channels providing identical service facilities.

1.5.1 Cumulative Arrival and Departure Diagrams

The word cumulative means “resulting from accumulation”. A

cumulative diagram indicates how many customers have accumulated up-to

some point in time from an initial starting time. A cumulative arrival diagram

indicates that how many customers have arrived up-to a point in time and a

cumulative departure diagram indicates how many customers have departed

up-to a point in time.

Figure 1.4: Performance measures indicate whether system objectives and goals are achieved.

Cumulative diagrams are important because they provide many of the

measures of performance in one simple picture. Now, in the next section 1.6,

we explain some major terms and definitions which are used in the theory of

queues or waiting lines which are given below:

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1.6 Some Definitions

1.6.1 Poisson Distribution

The Poisson distribution is a probability distribution that arises from the

Poisson process. It is a discrete distribution i.e. its random variable is limited to

a set of distinct values. The Poisson distribution gets its name because it is the

probability distribution for the number of arrivals within any time period of a

Poisson process. This distribution expresses the probability of a given number

of events in a fixed interval of time and/or space, if these events occur with a

known average rate and independently of time, since the last event. Frank A.

Height [1967].

1.6.2 Characteristics of Poisson Distribution

(i) The Poisson distribution is discrete and defined over the set of non-

negative integers.

(ii) [ ( )]E N t tλ=

(iii) [ ( )]V N t tλ=

(iv) [ ( ) 0] tP N t e λ−= =

Where, E [ ] represents the expectation of the enclosed random variable, and

V [ ] represents the variance of the enclosed random variable. The Poisson

distribution has the distinguishing characteristic i.e. variance is equal to its

mean. The Poisson distribution can be used for the number of events in other

specified intervals such as distance, area or volume.

1.6.3 Exponential Distribution

The exponential distribution is contineous and defined over the set of

non-negative real numbers. For exponential distribution, the probability density

function ( )f x is given as follows:

13

( ) ; 0−= ≥xf x e xλλ (1.11)

Possesses only one parameters λ > 0 describing the average rate and the

probability distribution function

0( ) 1 ; 0− −= = − ≥∫ x

t tF x e dt e x

λ λλ . (1.12)

Mean 1

[ ]E xλ

= (1.13)

Variance 2

2

1[ ] [ ]V X E X

λ= = (1.14)

Figure 1.5: Explanation of exponential distribution for service facilities.

For service facilities, very often the average service time 1/s µ= is

specified with µ the average service rate. A similar description is available for

the arrivals and departure processes. In assuming an exponential distribution

for the arrival processes one addresses the distribution of the times between

subsequent arrivals t, called inter-arrival time t.

This is graphically illustrated in the figure 1.6.

14

Figure 1.6: Inter-arrival times in an arrival process.

Consider jt as the time between two arrivals 1andj jT T−

1j j jt T T−

= − (1.15)

Assuming jt for all j being exponentially distributed with parameter λ, i.e.

Pr{ } t

jt t e λ−≥ = (1.16)

Then no. of arrivals Nt with in [0, t] follows a Poisson distribution.

( )Pr{ } ( , )

!−

= = =

jt

t

tN J f j e

j

λλλ . (1.17)

Theorem: The exponential distribution is memory-less.

Proof: The proof is based on the definition of conditions probability. A random

variable T is said to be memory-less

0 0Pr{ | } Pr{ }T t t T t T t> + > = > . (1.18)

Now a random variable t is assumed to be exponentially distributed with

parameter λ, i.e.,

Pr{ } 1 −≤ = − tT t e λ . (1.19)

Hence,

00 0

0

Pr{ }Pr{ | }

Pr{ }

T t tT t t T t

T t

> +> + > =

>

0

0

( )

Pr{ }t t

t

t

ee T t

e

λλ

λ

− +−

−= = = > (1.20)

15

Hence, the theorem.

1.6.4 Memory-less Property

A random variable (r.v.) is said to be memory-less if the time until the

next event does not depand on how much time has already elapsed since the

last event. A random variable is memory-less if

{ | } { }P X s t X t X s> + > = > ∀ s, t ≥ 0. (1.21)

In the context of Poisson process, the memory-less property applies to

the inter-arrival time. From the law of conditional probability,

{ and }{ | }

{ }

P X s t X tP X s t X t

P X t

> + >> + > =

>

{ }

{ }

P X s t

P X t

> +=

> (1.22)

Substituting the exponential distribution function in above equation

We obtain

( )

{ | }s t

s

t

eP X s t X t e

e

λλ

λ

− +−

−> + > = = = {P X s> }. (2.23)

1.6.5 Gamma Probability Distribution Function

1

0

( )( ) 1 ; 1

n

t jn

Y

j

e tF t n

j

λ λ−−

=

= − ≥∑ (1.24)

Gamma Probability Density function is given by

1( ) ; 0n

nn t

Yf t t e tn

λλ− −= ≥ (1.25)

Where, n is the gamma function.

[Note: The gamma function is not a probability distribution function].

When n is an integer then the gamma function is defined as

16

1; 1n n n= − ≥ and integer.

The gamma function is continuous and defined over the domain 0t ≥ . The

mean and variance of the gamma distribution are:

[ ]n

E Tλ

=

2[ ]

nV T

λ=

1.6.6 Bernoulli Distribution

In probability theory and statistics, Bernoulli distribution, named after

Swiss scientist Jacob Bernoulli, is the probability distribution of a random

variable which takes value 1 with success probability p and value 0 with failure

probability q = 1 − p.

1.6.7 Bernoulli Random Variable

A random variable X is said to be a Bernoulli random variable if its

probability mass function (p.m.f.) is given as following:

( 0) 1P X p= = −

( 1)P X p= =

We assume X = 1 when the experimental outcome is successful and

X = 0 when it is failed.

[ ] 1. ( 1) 0. ( 0)E X p X p X p= = + = = (1.26)

2 2var[ ] [ ] [ ] (1 )X E X E X p p= − = − (1.27)

1.6.8 Binomial Distribution

Suppose we repeat a Bernoulli p experiment n times and count the

number X of successes, the distribution of X is called the Binomial B(n, p)

random variable

17

; ( ) k n kn

pmf p X k p qk

− = = (1.28)

Where, 1 ; 0,1,2,...,q p k n= − =

[ ]E X np=

var[ ] (1 )X np p= − .

Note:

1. Binomial distribution is a sum of independent and evenly distributed

Bernoulli trials.

2. Bernoulli trial is a random experiment with only two possible

outcomes.

3. Binomial experiment is a sequence of Bernoulli trials performed

independently.

1.6.9 Renewal Process

A renewal process is defined as a discrete time independent process

{ | 1,2,....}nX n = where X1, X2, .... are independent, identically distributed,

non-negative random variables.

Ex. Consider a system in which the repair (or replacement) after a failure is

performed, requiring negligible time. The times between successive

failure might well be independent, identically distributed random

variables {Xn | n = 1, 2, ....} of a renewal process.

1.6.10 Z-transform: An application in Queuing Systems. Let X is a

discrete random variable

( ) , 0,1,...= = =iP X i P i

0 1 2, , ...P P P

18

( )0

∞

=

=∑ i

i

i

g z Pz

1.6.11 Properties of Z-transform:

( )1 1,=g

( ) ( )0 10 , 0′= =P g P g

( )2

10

2′′=P g

( ) ( ) 1|=

= z

dE X g z

dz

( ) ( ) ( )22

1 12| |= =

= +z z

d dE X g z g z

dz dz

1.6.12 Matrix Geometric Method

In probability theory, the matrix geometric method is a method for the

analysis of a quasi-birth-death process. Continuous time Markov chain whose

transition rate matrices with a repetitive block structure. Harrison Peter G, Patel

Naresh, M.[1992].

1.6.13 Method Description

This method required a transition rate matrix with tri-diagonal block

structure which is given as follows:

00 01

10 1 2

0 1 2

0 1 2

B B

B A A

Q A A A

A A A

= ⋱ ⋱ ⋱

(1.29)

Where, B00, B01, A0, A1, A2 are matrices. To compute the stationary distribution

π, we have 0Qπ = and the balance equations are considered for sub-vectors πi

19

0 00 1 10 0B Bπ π+ =

0 01 1 1 2 0 0B A Aπ π π+ + =

⋮ ⋮ ⋮

1 2 1 1 0 0− ++ + =i i iA A Aπ π π

We observe that

11

i

i Rπ π −

= (1.31)

Above equation holds true where, R is the Neutes’ rate matrix. Amussen, S.R.

[2003] and Neutes, M [1981]. Numerically, we have

00 01

0 1

10 1 0

[ ] [0 0]B B

B A RAπ π

= + (1.32)

By using this relation, we can find 0 1,π π , and calculate iteratively all

the πi.

1.7 Simulation

Simulation is the technique that creates the data. According to Wesbters

dictionary, a simulation is the act of imitation. It is a word used in many

contexts. NASA uses chambers to simulate zero gravity for astronauts. United

airline uses computer programs to simulate Jet flight. The key characteristics of

Simulation are dynamic and time varying behaviour.

In the domain of Operations Research, Simulation is a powerful

technique for evaluating the behaviour. Simulation is a very powerful and

widely used management science technique for the analysis and study of

complex system. A simulation model usually takes the form of a set of

assumptions about the operation of the system, expressed as mathematical or

logical relations between the objects of interest in the system.hus the process of

designing a mathematical or logical model of a real-system and then

20

conducting computer based experiments with the model to describe, explain

and predict the behaviour of the real system.

Simulation fits in the following:

Figure 1.7: Simulation.

A system is a collection of entities that act and interact towards the

accomplishment of some logical end. Systems generally tend to be dynamic,

their status changes over time. To describe this status, we use the concept of the

state of a system.

Examples (Simulation Model)

1. Emergency room (beds, doctors, nurses)

2. Power integration-Semiconductor capacity, and random machine

downtime,

1.7.1 Value of Simulation

1. Empirical Method versus mathematical Model.

2. Allow we to calculate the extreme values not just the expected value.

i. Simulation is the actual running of the Model System to gain

insight into its performance.

ii. Simulation is used to better understand the expected performance

of the real system and to test the effectiveness of the system

design.

iii. Why use Simulation: Experimental system, new concepts, costly

experimentation, unsafe experimentation. Determines limits of

stress.

21

1.7.2 Simulation vs. Analytical Modelling

Advantage

i. Various performance measures

ii. Greater realism.

iii. Easier to understand.

iv. Model the steady state as well as the transit behaviour.

Disadvantage

i. May not provide, with the optimal solution.

ii. Time to construct model will be longer.

1.7.3 Simulation vs. Physical

Advantage

High speed, not descriptive, replication easy, control variations,

generally less costly.

Disadvantage

Realism

Validity

Representing System:

A collection of mutually interacting objects designed to accomplish a goal

(machine repair system).

1.7.4 Entities

Denotes an element / object with in boundary of the system such as

(machines, operators and repairmen) etc.

22

Entity: Work being performed an object.

Resource: Performing the work. Characteristics or property or an entity

(machine ID, Type of breakdown, time that machine went down).

Activity: Transform the state of an object usually over some time (repairman

service time, machine run time).

1.7.5 Queue: It is a set, used to model waiting

1.7.6 Types of Simulation Models

There are two types of simulation models which are given below:

1. Static and

2. Dynamic

1. A static simulation model is a representation of a system at a particular

point in time. Example: Monte Carlo Simulation.

2. Dynamic simulation is a representation of a system as it evolves over time.

With these two classifications, a simulation may be deterministic or

stochastic.

i. A deterministic simulation model is one that contains no random

variables.

ii. A stochastic simulation model contains one or more random

variables.

iii. Discrete event: State of system change only at discrete points in

time. Example: machine repair problem.

The simulation model provides a very flexible mechanism for evaluating

alternative policies. Simulation runs usually result in variation due to its

property and the random number used. Therefore, instead of depending on an

average output the confidence interval of output is usually used to illustrate the

numerical results. As mentioned by Banks [2004] “a confidence interval for the

23

performance measures being estimated by the simulation model is a basic

component of output analysis.

Simulation is an alternative approach to solve the machine interference

problem. Simulation is often used to solve complex problems that are too

complicated to be derived by mathematical modeling or through the analytical

approach. Simulation might not provide the exact solution, but it provided a

good estimate for existing problems. Guild and Harnett [1982] has worked on

the simulation study of the machine interference problem that involved N

machines and R operators with two type of stoppage. In continuation to the

application of the machine interference problem, Sztrik and Moller [2001]

discussed characteristics of stochastic simulation of the Markov modulated

finite source queueing systems also in [2002] Sztrik and Moller has studied on

simulation of machine interference on randomly changing environments.

1.8 Queueing Problems

Queueing problems can be classified into three types:

(i) Perpetual queue.

(ii) The predictable queue

(iii) The Stochastic queue

1.8.1 The Perpetual Queue

It is a worst type queue. A perpetual queue is a queue for which all

customers have to wait for service. All customers have to wait, because the

servers have insufficient capacity to handle the demand for the service.

1.8.2 The Predictable Queue

This type queue occurs when the arrival rate is known to exceed the

service capacity over finite intervals of time.

24

1.8.3 The Stochastic Queue

The Stochastic queues are not predictable. They occur by chance when

customers happen to arrive at a faster rather then they are served. Stochastic

queues can occur whether or not the service capacity exceeds the average

arrival rate.

1.9 Markov Chains

The stochastic process { , 0,1,2,...}nX n = is called a Markov chain, if

for j, k, 1 1,...., nj j−

(for any subset of I).

1 2 1 0 1{ | , ,..., }r n n n nP X k X j X j X j− − −

= = = =

1{ | }r n n jkP X k X j P−

= = = = (1.32)

The outcomes are called the states of the Markov chain. The transition

probabilities jkp are basic to the study of the structure of the Markov-chain.

1.9.1 Transition Matrix : The transition probabilities jkp satisfy

0; 1,≥ = ∀∑jk jk

k

p p j , where, (1.33)

11 12 13

21 22 23

p p p

P p p p

=

⋯

⋯

⋯ ⋯ ⋯ ⋯

1.9.2 Classification of States and Chains

A property defined for all states of a chain is a class property if its

possession by one state in a class implies its possession by all states of the

same class. One such property is the periodicity of a state.

1.9.2.1 Periodicity : State i is a return state if,

( ) 0n

iip > , for some n ≥ 1. (1.34)

25

The period di of a return to state i is defined as the g.c.d., of all m such that

( ) 0>m

iip (1.35)

i.e., ( )gcd{ : 0}m

i iid m p= >

1.9.2.2 Aperiodic: State i is said to be aperiodic if, di = 1.

1.9.2.3 Periodic: State i is said to be periodic if, di > 1.

1.9.2.4 Absorbing state: The state j is said to be absorbing if,

1, 0,= = ≠jj jkp p k j

1.9.3 Reducible and non-reducible Markov Chains

In an irreducible Markov chain, every state can be reached from every

other state and chains which are not irreducible are called reducible.

In the next section 1,10, we shall describe M / M / 1: (The Classical

Queueing System).

1.10 M / M / 1: The Classical Queueing System

M/M/1 queue is the simplest non-trivial interesting queueing system and

described by selecting the birth-death coefficients as follows:

; 0,1,2,....n nλ λ= =

; 1,2,3,....n nµ µ= =

Figure 1.8: State-transition rate diagram for M/M/1 queue.

26

We assume that customers are served in FCFS (first comes-first served)

basis. Balance equations are given as follows:

0 0 1 10 ; 0p p kλ µ= − + = (1.36)

1 1 1 10 ( ) ; 1n n n n n n np p p kλ µ λ µ− − + += − + + + ≥ (1.37)

Applying 1

00 1

; 0,1,2,...n

in

i i

p p nλµ

−

= +

= =∏ (1.38)

Using equation (1.38), we obtain, 1

00

n

n

i

p pλµ

−

=

= ∏ .

Or, 0 ; 0

n

np p kλµ

= ≥ (1.39)

And, 0

1

1

1

n

n

pλµ=

= + ∑(1.40)

The necessary and sufficient condition for ergodicity in the M / M / 1

queue is λ < µ.In equation (1.40) sum converges, since λ < µ therefore

0

1p 1

/1

1 /

λ= = −λ µ µ+ − λ µ(1.42)

and 0 1λρ = ≤ ρ <µ

Finally, we have from equation (1.39),

nnp (1 ) ; n 0,1,2,....= −ρ ρ =

1.11 Machine Interference in Queueing Theory

With the advancement of latest technology, machining system has

pervaded in every nook and corner of our lives. Thus, ensuring our almost

27

dependence on machining system. As the time progresses a machine may fail

due to wear and tear or due to some unpredicted fault and thus requires a

corrective measure by providing the repair facility, after which it can be again

start functioning properly. If at any time, failed machines need the repairman's

attention, a queue of failed machines may develop due to unavailability of idle

repairmen. This phenomenon needs attention of mathematicians as such

machine repair models have captivated the interest of many renowned

researchers working in the area of queueing theory.

Thus, the loss of production due to the period of machines having to

wait for repair or service is called machine interference. Machine interference

occurs when a single repairman repairs two or more machine or a group of R

servers services a lot of N machines in case R < N. The time that is spent

between the call for service and the start of service is known as interference

time. Some notable work was done by Elsayed, E.A. [1981], Jain, M. [1998],

Jain, Singh and Beghel [2000] and Showky, A.I. [2000].

In many fast growing industries the operations may be interrupted due to

failure of machines involved in the system. The service facility should be so

adjusted that the machines are repaired immediately, so as to work

continuously. The factors, which effect the solution of machine interference

problems are running time of machine before break down, or while the

repairman is engaged, the second machine will remain in waiting until the

service of the first machine is completed. In case of several repairmen, if the

machines are failed these are repaired by repairman, the excess number of

machines will have to wait until repairmen are available.

1.12 Machine Repairmen Model: (M/M/1/m/m)

This is an interesting special case of birth-death process occurs when the

birth rate λn is of the form (M – n)λ; n = 0, 1, …, M and the death rate is

µn = µ.

28

This type of situation occurs in the modelling of a server where an

individual client issues a request at the rate λ whenever it is in the "thinking

state". If n out of total M clients are currently waiting for a response to a

pending request, then the effective request rate is (M – n)λ. Where µ is the

request completion rate.

( ) ; 0

0 ;

− λ ≤ ≤λ = ≥n

M n n M

n M (1.43)

; 1

0 ;

µ ≤ ≤µ = >n

n M

n M. (1.44)

A similar situation arises when M machines share a repair facility. The

failure rate of each machine is λ and repair rate is µ.The state diagram of such

finite population system is given in the following figure 1.9.

Figure 1.9: The state diagram of a finite population queueing system.

The expression for steady state probabilities, which are obtained using

above equations, we have

1

00 1

; 1,2,...−

= +

λ= =µ∏n in

i i

p p n (1.45)

( )1

00

; 0−

=

λ −= ≤ ≤µ∏nn

i

M ip p n M (1.46)

or ( )0

!

!

λ= µ − n

n

Mp p

M n(1.47)

( )0

!

!= ρ

−n M

pM n

29

Hence,

( )0

0

1

!

!=

=ρ

−∑M n

n

pM

M n

( )1

00

!

!

−

=

= ρ − ∑M n

n

MP

M n (1.48)

1.13 M/M/2 Queuing System in Steady-State

0 1λ = µP P

( ) 1 0 22λ +µ = λ + µP P P

( ) ( )1 12 2 ; 2,3,...− +λ + µ = λ + µ =n n nP P P n

Therefore 02= ρn

nP P ( )1,2,...=n . (1.49)

Where, 2

λρ = µ and

0

1

1

−ρ=+ ρ

P

Also, 2

2

1

ρ=−ρ

L

3

2

2

1

ρ=−ρqL

(1.50)

And probability of having to wait = 22

1

ρ+ ρ

. (1.51)

Now, we consider finite population queueing model.

1.14 Finite Population Model (M/M/C/m/m):

This is a simple matter to extend the limited population model to

cover systems with multiple servers. The parameters of this model are given

below:

30

( ) ; 0

0 ;n

m n n m

n m

− λ ≤ ≤λ = ≥ (1.52)

; 1

;

0 ;

µ ≤ ≤µ = µ < ≤ >n

n n c

c c n m

n m

(1.53)

Substituting these values in the following equations:

1

00

1

−

=

=

λ=

µ

∏∏

n

i

in n

i

i

p p (1.54)

And

11

00

1

1

1

−−

∞=

=

=

λ = + µ

∏∑

∏

n

i

i

nn

i

i

p (1.55)

We obtain,

( ) ( )1

1

00

! !

! ! ! !

−

−

−

= =

λ λ= + − µ − µ ∑ ∑i i

c m

j cj j c

m mp

m j j m j c c (1.56)

And,

( )

( )

0

0

!; 0

! !

!;

! ! −

λ < ≤ − µ = λ < ≤ − µ

n

n n

n c

mp n c

m n np

mp c n m

m n c c

(1.57)

The expected number in the system and the expected number in the

queue and calculated from the following definitions.

( )=

= −∑mq n

n c

L n c p (1.58)

31

0=

=∑m n

n

L np (1.59)

0

−

=

= −∑m m r

r

m r p (1.60)

The effective arrival rate is the product of the failure rate of

individual machines and the expected number of machines running.

( )λ = λe E r

0

−

=

= λ∑m m r

r

r p (1.61)

Waiting times are

q

q

e

LW =

λ

( )

0

=

−

=

−=

λ

∑∑

m

n

n c

m

m r

r

n c p

r p

(1.62)

And,

e

LW =

λ

0

1

−

=

= −λλ∑m m r

r

m

r p

(1.63)

In case of single channel, the arriving customer distribution is equal

to (m – 1) customer system.

Then

( ) ( )1= −n nq m p m (1.64)

32

The corresponding waiting times for an arriving customer are

( )1 1m

q n

n c

n cW q m

c

−

=

− +′ = µ∑ (1.65)

And, 1

qW W′ ′= + µ (1.66)

Example

In a networking conference, each speaker has 15 min to give his talk,

otherwise, he is rudely removed from podium, Given that time to give a

presentation is exponential, With mean 10 minutes. What is the probability,

that a speaker will not finish his talk.

Solution: Given that

( ) 1=λ

E X = 10 minutes ⇒1

10λ =

Let t be the time required to give a presentation that a speaker will

not manage to finish his presentation if t exceeds 15 minutes.

( ) 1.515 0.22−> = =P t e

22

22%100

= =

This means speaker will not finish his presentation with in time is 22%

while he/she will finish his/her presentation with in time, that probability is

78%.

In the next section 1.15, we shall explain some applications of queueing

theory in industry, organization and in our daily life.

33

1.15 Some Applications of Queueing Theory

“Each of us had spent a great deal of time waiting in lines”

Where the time goes:

In a life time, the average person will spend (Approx.)

Six months : Waiting at stoplights.

Eight months : Opening junk mail.

One year : Looking for misplaced objects.

Two years : Reading - e-mail.

Four years : Doing house work.

Five years : Waiting in line.

Six years : Eating, etc.

Queueing theory or waiting lines is useful crowed safety

management, entry and exist systems, concession planning and crows flow

assessment, venue ticket sales, transport loading (to and from avenue),

traffic control and planning, finding the sequence of computer operations,

telecommunication like waiting calls when server is busy, air port traffic,

layout of manufacturing systems, capacity planning for buses and trains,

health services (e.g. control of hospital bed assignments). Also queueing

theory is useful in game e.g. when to remove a goalie in a hockey game. We

have application of queueing theory in machine repairing system.

1.16 Structure of the Rest of the Thesis

This thesis is organised, according to the specific models that we

study. More concretely, in every chapter we first motivate the model under

consideration and then describe the previous contributions reported in the

literature. In chapter 2, we present an overview of the general approaches

34

proposed in the literature for queueing models and machine repairing

system. In this chapter, firstly, we give some general approaches undertaken

by Operational Researchers in dealing with queueing theory and machine

repairing systems and related areas. Secondly, some texts past and present

work have been incorporated.

In chapter 3, we discuss detailed explanations of machine repairing

system for queuing models. In this chapter we have explained some

performance measures, distributions and illustrated with numerical

examples.

In chapter 4, we present mathematical formulation and analysis of

different queueing models. In this chapter we performed birth and death

model and related results. Then we explain M / M / C (C > 1) model,

M / M / 1 / N (single-server) model. Also we have find results on M / G / 1

queue, M / Ek / 1 queue and M / D / 1 queue.

. In Chapter 5, we focus on comparative study of different queueing

models for machine repairing system. In this chapter firstly, we have

developed M/M/R machine repair problem with spares secondly we gave

machine repair problem with spares and N-policy vacations. Finally, we

analyzed queuing model for machine repairing system with Bernoulli

vacation schedule.

In chapter 6, we develop queueing networks modelling software and

computing language(s) and we present software like MATLAB which has

been used for our calculation of numerical results.

Finally, in chapter 7, we summaries the conclusion of this research

and suggests issues for further research. Bibliography (alphabetical order)

and list of publications and presentations are placed at the end of the thesis.

35

1.17 Summary

In this chapter, we undertook a theoretical investigation of queueing

model and machine repair model. We have explained basics of queueing

systems and performance measures of queues, general notations, classical

queueing system and machine repairmen model. We have considered some

practical and general life examples and applications of queueing theory.