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SOME CONTRIBUTIONS TO

QUEUEING THEORY

A Thesis submitted to the University of Lucknow

For the Degree of

Doctor of Philosophy

in Statistics

By

NINI BURMAN

Under the Supervision

of

PROF. S. K. PANDEY

Department of Statistics

University of Lucknow

Lucknow – 226007, (U.P.)

INDIA

(2013)

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C E R T I F I C A T E

This is to certify that all the regulations necessary for the submission

of Ph.D. thesis “SOME CONTRIBUTIONS TO QUEUEING THEORY” by Ms. Nini

Burman been fully observed.

(Prof. S.K. Pandey) (Prof. A. Ahmad)

Professor Professor & Head

Department of Statistics Department of Statistics

University of Lucknow University of Lucknow

Lucknow- 226007 Lucknow- 226007

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C E R T I F I C A T E

This is to certify that the work contained in this thesis entiled “SOME

CONTRIBUTIONS TO QUEUEING THEORY” by Ms. Nini Burman has been

carried out under my supervision and that this work has not been

submitted anywhere else for Ph. D degree.

(Prof. S.K. Pandey)

Supervisor

Department of Statistics

University of Lucknow

Lucknow- 226007

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DD

Dedicated to

My husband

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Acknowledgement I feel great pleasure in submitting my research work entitled “SOME CONTRIBUTIONS TO

QUEUEING THEORY” which is possible because of GOD grace and many supporting hands

behind me .

I owe my heart-full gratitude and indebtedness to my esteemed supervisor Prof. S.K.

Pandey for his enlightening guidance and sympathetic attitude exhibited during the

entire course of this work. Whenever I encountered any difficulty with my research

work, he inspired me a lot for achieving related results with a great force.

I am highly indebted to Madam Dr. Manju Gupta for her valuable interest and

positive criticism and constant support throughout the course of investigation. Many

new ways to enrich the content have resulted from her constructive ideas.

I would like to thank the Principal and to all the colleagues of Navyug Kanya

Mahavidalaya for constant encouragement towards completion of work.

I express my deepest sense of gratitude towards my mother who has

always been a source of inspiration. My father had been guiding my path and showering

his blessings from heavenly abode. I wish the special word of thanks for my Daughter

Pavni, and son Ansh for extending every care, moral support and affection to enable this

work to become a reality.

(Nini Burman)

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INDEX

CHAPTER PAGE NO. CHAPTER 1-Introduction To Queueing Theory 1-22 CHAPTER 2-Queueing Model with Arbitrary Service Time 23-45 Distribution, Single Server and Unlimited Capacity CHAPTER 3- Queueing Model with Arbitrary Service Time 46-61 Distribution, Single Server and Limited Capacity CHAPTER 4- Queueing Model with Arbitrary Service Time 62-77 Distribution and Multiple Servers CHAPTER 5-Truncated Arrival Distribution 78-84 References 85-88

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Chapter 1

Introduction to Queueing Theory

In every day life it is seen that a number of people avail some service facility

at some service station, Imaging the following situation :-

1. Letters arriving at a typist desk

2. The process of calls in telephone exchange

3. The recalling message in a telegram office

4. Shoppers waiting in front of check out stands in a super market

5. Cars waiting at a stop light

6. Planes waiting for take off and landing in an airport

7. The arrival and departure of ships in a harbours

8. Broken machines waiting to be service by a repair man

9. Customer waiting for attention in a shop or super market

These situations have in common phenomenon of waiting. It would be

most convenient if we could be offered services and others like it

without the nuisance of having wait, but like it or not waiting is a part

of our daily life. Waiting line of queue are omnipresent. Business of all

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types, industries, schools, hospital, banks, theatres, petrol pumps etc. –

all having queuing problems. Waiting line problems arise either

because :-

i) There is too much demand on the facilities so that we may say

that there is an excess of waiting time or inadequate number of

service facilities or

ii) There is too less demand, in which case there is too much idle

facility time of too many facilities.

In either case, the problem is to either schedule arrivals or provide

facilities or both so as to obtain an optimum balance between the costs

associated with waiting time and idle time.

A group system of customers/items waiting at some place to receive

attention/service including those receiving the service is known as

queue or waiting line. Queuing theory is the main aspect of waiting

time models where discussion involves stochastic process. The main

purpose of the queuing theory is to briefly investigate those queuing

problems in the field of industries, Transportation and Business which

are concerned times due to man and machine involve and the

production of an item is done two or more distinct successive phases

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under the stochastic process by using Tandem, Bi-Tandem Queues.

The Pioneer investigator of Queuing theory was the Denish

Mathematician A.K. Erlang (1909) who published “The Theory of

Probability”. The great mathematician A.K. Erlang known as father of

teletraffic theory derived the formulae relating to traffic load. The

mathematical discussion on queuing theory made considerable progress

in early 1930,s though the work of Pollaczk (1930, 1934), Kolmogrove

(1931), Khintchine (1932,1955),and others. Kendall (1951, 1953) gave

a symmetric treatment of the stochastic process occur the theory of

Queues and Cox (1955) analyzed the congestion problems statistically.

Khinchive (1960) discussed the mathematical methods in the theory of

Queue. Morse (1958) discussed the wide variety of special Queuing

problems and applied queuing theory was given by Lee, A.M. (1958.

An element of queuing theory with applications was given by T.L.

Saaty (1961). On some problems in queuing and scheduling system has

given by Arpana Badoni (2001). The Trend towards the analytical

study of the basic stochastic process of the system has continued and

queuing theory has proved for researchers who want to do fundamental

research on stochastic process involving mathematical models. The

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process involved is not simple and for time dependent analysis more

sophisticated mathematical procedures are necessary. For example for

the Queue with Poisson arrival and Exponential service time under

statistical equilibrium the balance of the equation are simple and the

limiting distribution of queue size is obtaining by recursive arguments

and induction but for the time dependent solution the use of transform

is necessary. The first solution of this time dependent problem was give

by Bailey (1954) and Lederman and Reuter (1956) while Bailey used

the method of generating functions for the differential equation.

Lederman and Reuter used spectral theory in this solution. Later

Laplace transform have been used for the same problem and it has been

realized that Laplace transform is useful technique in the solution of

such difference differential equation. Other methods which rely on

heavy use of transforms are Takacs equation method (1955), the

supplementary variable techniques of Cox (1955), Keilson and

Kooharin (1960) where the Non- Markovian process are renderer.

Markvian by describing the process with sufficient number of

supplementary variable and operating on them, in addition to this

Koilson also in his later investigation uses sections of the process

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studied in isolation and the techniques of recurrence relations and

renewal theory used in varied forms by Gaver (1969), Takacs (1962)

and Bhatt (1964, 1968). Analytically these methods are powerful and

can be used in every complex situation. A significant contribution

towards the analysis of Queuing system was made by Kandall

(1951,1953) which demonstrated the use of Markovien chain technique

in the classification of the Markov chain imbedded in the queue length

process of the :-

i) Poisson arrival; general service distribution and single server.

ii) General arrival, Exponential service and multi server queuing

system.

The simple queue with Poisson Arrivals and exponential service time

on a random walk and derived the time dependent behaviour of the

queue length process in explicit from without resorting to transforms in

also considered by Champernonance (1956) Giving a brief history of

the development of the queuing theory, we now present the detailed

description of a queuing system.

DESCRIPTION OF THE QUEUING PROBLEM

A queuing system can be describes as customers arriving of service, waiting

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for service it it is not immediate, and it having waited for service, leaving the

system after being served. Such a basic system is depicted in

Fig 1.1 SCHEMATIC DIAGRAM OF A QUEUEING PROCESS

Although any queuing system may be diagrammed in this manner, it should

be rather clear that a reasonably accurate representation of such a system

would require a detailed characterization of the underlying processes. The

primary categories of such a characterization are dealt in detailed as

follows:-

COMPONENTS OF QUEUING PROCESSES

A little change in one or more of this basic process gives rise to difficult

queuing system. There are basic characteristics of queuing processes given

below:-

■ Arrival pattern of customers

CUSTOMERS ARRIVING SERVED CUSTOMERS 000-------- 000--------LEAVING DISCOURAGED CUSTOMERS LEAVING

SERVICE FACILITY

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■ Service Discipline of servers

■ Queue discipline

■ System capacity

■ Number of service channels

■ Number of service stages

■ Output or departure distribution

In most case, these basic characteristics provide an adequate of a queuing

system.

ARRIVAL PATTERN OF CUSTOMERS:-

The arrival pattern or input to a queuing system is usually measured in terms

of the mean number of arrivals, per some unit of time (mean arrivals rate) or

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

Since these quantities are closely related either one of these measures in

analyzing the system's input. In the event that the stream of input is

deterministic, then the arrival pattern is fully determined by either the mean

arrival rate or the mean inter-arrival time. On the other hand, if there is

uncertainty in the arrival pattern then there mean values give only measures

of central tendency for the input process and further characterization is

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needed in the form of the probability distribution associated with this random

process. Another significant factor referring to the input process is the

possibility that arrivals come in groups instead of one at a time. In the event

that more than one arrival can enter the system simultaneously, the input is

said to occur in bulk or batches. In the bulk arrival situation, not only the

time between successive arrivals of the batches are probabilistic, but also the

number of customers in a batch. The fundamental paper on "analysis of bulk

service queuing system subject to interruption" was given, by A.B.

Chandramouli (1997).

It is also desired to know the reaction of a customer upon entrance for service

in the system. A customer may decide to wait no matter how long the queue

becomes or if the queue is too long to suit him, may decide not to join it. If a

customer’s decides not to join the queue upon arrival customers is said to

have balked. Height (1957, 1960) was firstly introducing the concept of

balking in the problem for a single queue in equilibrium with Poisson input

and exponential holding lime. Finch (1959) studied the similar problem for

general input distribution. Recently Singh (1970), Yechiali (1971) and Black

Burn (1972) used the notion of balking in their respective studies in queuing

problems. The idea of "On bulk arrivals in bi-series channel links in series

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with common channel" has given by Heydari A.P.D. (1987). At the other

edge, a customer may join the queue, but after waiting time intolerable to

him, he loses patience and decides to leave. in this case he is said to have

reneged. The reneging process has been introduced by Barrer (1957), Disney

and Mitchell (1970). Bacheli, F. P. Boyler & G. Hebuterine (1984) discussed

"single server queue with impatient customer, in the event that these are two

or more parallel waiting lines; customers may move from one queue to

another queue, that is; jockey for position. Glazer was first to introduce this

concept of jockey into a queuing system.

One final factor to be considered regarding the arrival pattern is the manner in

which the pattern varies with time. An arrival pattern that does not vary with

time (i.e., the form and the values of parameter of the probability distribution

are time independent) is called a stationary arrival pattern and one that is not

time-independent in the aforesaid sense is named a non-stationary. Size of

customers population must be specified when the number of the customers in

the population is sufficiently large such that the probability of the customers

arriving for service is not significantly affected by the number of customers

already at the service facility. We refer the population size is infinite

otherwise a finite population is assumed and its size specified. It is used to

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describe the arrival process by prescribing the probability function or

probability density function of the inter arrival time T for customers. If the

p.d.f. is denoted by a (t) then

A(t) = Probability [inter arrival time = t]

If the function A (t) is differentiable then the probability density function a (t)

Is given by a (t) = d/dt{A (t )}

SERVICE DISCIPLINE

"The manner in which customers are served is referred to as the service

discipline for the queuing system. Server can be defined by a rate as number

of customers served per some unit of time or as time required to service of

customers. Thus if the duration of service is denoted by symbol x and its

probability distribution function by B(x).

Then B(x) = Probability (Service times =x)

Now if B(x) is differentiate then its probability density function is denoted by

b(x), where

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bx= (d/dx) B(x).

After this comes the description of the discipline by which customers are

selected from waiting queue for service. There are many disciplines followed

the most common discipline that can be observed in every day life First come,

First served or First In First Out. Others are 'Service in Random order',

'Last Come, First Served'

In many inventory system when there is no obsolescence of stared units as it

is easier to reach the nearer items which are the last in. Now when the

selection for service is made without any record to the order of the arrival or

the customers i.e. in the random order the selection referred to as SIRO

(Service in random order). Another important service discipline is priority

service discipline which means that there are identifiable groups in the

arriving stream of customers and there is a well defined order of priority given

to various groups.

The service rate may depend upon the number of customers waiting for

service. A server may work faster if he sees that the queue is building up or

conversely, he may get flustered and became less efficient. The situation in

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which service depends on the number of customers waiting is referred to as

state dependent service although this term was not used in describing arrival

patterns, the problem of customer's impatience can be looked upon as one of

state dependent arrival since the arrival behavior banks upon the amount of

congestion in the system.

Service, like wise arrivals, can be stationary or non-stationary with respect to

time. For example, learning may take place on the part of the server so that

he becomes more efficient, as he gains experience. The dependence upon

time is not to be confused with dependence upon state. The former does not

depend on the number of customers in the system but rather on how long it

has been in operation. The latter does not depend on how long the system has

been in operation, but only on the state of the system at a given instant of

time, that is, on how many customers are currently in the system. Of course,

any queuing system could be both non stationary and state dependent. Even if

the service rate is high, it is very likely that some customer will be delayed by

waiting in the line, in general, customers arrive and depart at irregular

intervals, and hence the queue length will assume no definite pattern unless

arrivals and service are deterministic. Thus it follows that the probability

distribution for queue length would be the result of two separate

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processes - arrival and services which are generally, though not universally,

assume mutually independent.

An example is the line formed by message awaiting transmission over a

crowed communication channel in which urgent message may take

precedence over a routing one. With the passage of time a given unit may

move forward in the line owing to the servicing of units at the front of the

line or may move back owing to the arrival of units holding higher priorities.

Further let two classes of customers might be defined C1 and C2 with

customers in the first class given preferential treatment to those in the

second. These are two major types in which two treatments are applied on

the arriving customers.

Suzuki (1963) studies the system with arbitrary distributed service time at

both servers and with zero capacity intermediate waiting time.

PREEMPTIVE SERVICE DISCIPLINE

One extreme kind of preferential treatment would be to ensure that a class C1

Customer never had to queue except when the service mechanism was busy

serving the customers of the same class. In this case the arrival of customer

with high priority when a low priority customer is in service would cause the

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interruption of the current service in order to serve the customers with high

priority

NON-PREEMPTIVE SERVICE DISCIPLINE:-

In this case an arriving class C1 is given a position in the queue in front of

any member of class C2 in the queue but behind any other member of class

C1 in the queue. In other words the arriving customer does not interrupt the

current service.

When a lower priority item has been preemptive returns to service, the

preemptive discipline must distinguish following two cases:-

Preemptive resume policy: -

The service of the lower priority unit is continued from the point at which it

was interrupted- Preemptive Resume Priority Queue, discussed by Jaiswal,

N.K. (1961)

ii) Preemptive - respect identical policy:-

-• >

When the interruption is cleared the service begins again from scratch but

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with a new independent service period. Time independent solution of the

preemptive priority problem under exponential service time distribution was

presented by Heath Cote (1959). White and Christie (1S58) give the steady

state solution for the preemptive resume problem as an approximate solution

for the repeat discipline with under exponential service time distribution. For

the preemptive resume rule, Miller (1960) obtained the various queue

parameters but the queue length probability generating function could not be

derived. Since one more time quantity namely the preemptive time of non

priority item is necessary to make the process Markovian the incorporation of

this extra time variable in to the definition of state probabilities make the

problem difficult to solve by the 'Imbedded Markov Chain' technique. The

preemptive priority queueing problems with exponential arrival and service

time have been discussed by several authors. In particular White and Christie

(1958) and Stephan (1956) have considered the 'steady state' distribution for

two classes of customers i.e., for one priority class.

TRANSIENT AND STEADY STATE

Queuing theory analysis involves the study of a system's behavior over time.

A system is said to be in Transient State when its operating characteristics are

dependent on time, in this state the probability distribution of arrivals, waiting

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time and service time of the customers are dependent on time. This usually

occurs at the early stages of the operation on the initial conditions. In the long

run of the system the behavior of the system becomes independent of the

time. This state is referred to as steady State.

A queueing system acquires a steady state when the probability distribution

of arrivals, waiting time and service time of the customers are independent of

time. A necessary condition of the steady state to be reached is that total

elapsed time tends to infinity since the start of the operation must be

sufficiently large. Let Pn(t) denote the probability that there is n Unit in the

system at time t, then the system acquires steady state as

This condition is not however sufficient since the parameter of the system

may permit the existence of a steady state.

QUEUE IN TANDEM/SERIES:-

A queue service system may consist of a single phase service or it may

consist of multiphase service. An example of multiphase queue service

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system would by physical examination where each patient pass through

several phases such as medical history, ear, nose and throat examination,

blood tests, electrocardiogram , eye examination and so on. A system of

queue in which service is done in successive but finds distinct phases is called

a system of queues in series.

Jackson (1954), Johnson (1954) and Maggu (1972-73) have laid down the

foundation of queues in series/Tandem. The multistage emending system

have studied by Burke (1956, 1968) and Reich (1956, 1957), Barcin (1954)

is the first person to tackle the problem of queue in Tandem/Series taking

Poisson input and exponential holding time.

Avi Itzahak and Yadin (1865) studied the tandem queue with no intermediate

queue and with arbitrary distributed service time at both servers. Friedman

(1965) studied finite tandem queue with multiple parallel server in which

inter-arrival distribution was arbitrary and service times were constant and

identical within each station. Hiller (1967) was concerned with numerical

work on tandem queue. Prabhu (1967), Neuts (1968) restudies the series

queues of Suzuki (1963) but with finite inter-mediate queue capacity.

Sharma (1373) presented the most recent work on serial queues. A general

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network with a finite number of queues in Tandem with multiple servers

within each station and Poisson arrival and exponential service times was

considered. He obtained the queue length distributions for both finite and

infinite queue capacities. In his case, arrivals not gaining access to the second

server were assumed lost. Burke (1972) reviewed the results on serial enema

from 1954 to 1972. Recently Sharma, O.P. (1990) discussed in detail in his

book "Markovian Queues" about the tandem queue.

CUSTOMER BEHAVIOUR

Queuing models representing situations in which human beings take the roles

oi customers and/or servers must be designed to account for the effect of

customer behavior. A customer may behave in the following manner.

i) Balking: -A customer may leave/ not like to join the queue

because the queue is too long and he has no time to wait or there is not

sufficient space. Such a behavior is determined as balking.

ii) Reneging: - A customer joins the queue but after some time he loses his

patience and leaves the queue. Such behavior is called reneging.

iii) Jockeying: - in multi-channel queueing system there may be more than

one queue. Then a customer may leave one queue and join the other. It is

called jockey for position and such behavior is called jockeying.

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iv) Collusion: - Several customers may collaborate and only one customer

may join the queue while the rests are free to attend other things. Some may

even arrange to wait in turns such type of behavior is called as collusion.

OUTPUT OR DEPARTURE DISTRIBUTION

The departure distribution determines the pattern by which the number of

customers leaves the system. Departure can be described using the service

time which defines the time spent on a customers in services. This

distribution is usually determined by sampling from the actual situation. The

output of a Queuing system, was given by Burke, P. J. (1978).

Though enough work has been done by various persons, there are possibilities

of improving upon them.

Till now maximum work has been done by taking the situations when arrival

follows Poisson process and service time follows exponential distribution.

Here the work has been designed by taking the general service time

distribution and then by considering the particular cases. Also the case when

there is no unit in the system at time t=0 is considered and arrival distribution

has been derived.

Given an arrival process [N(t); t≥0], where N(t) denotes the number of arrival

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upto time t, N(0)=0, an arrival can be characterized by the following three

axioms can be described as a Poisson process:

Axiom 1. Number of arrivals in non overlapping intervals are stochastically

independent.

Axiom 2. The probability of more than one arrival between time t and t+Δt is

o(Δt),i.e probability of two or more arrivals during small time interval Δt is

negligible. i.e. Po (Δt)+ P1 (Δt)+ o(Δt)=1

Axiom 3. Probability that an arrival occures between time t and t+ Δt is given

by,

P1 (Δt)=λΔt+o(Δt)

So that

Under these assumptions probability distribution of number of arrivals in a

fixed time interval follows Poisson distribution.Let the service time

distribution be s(t) with mean service rate θ per unit.Axioms of service time

distribution are similar to those of axioms of arrival distribution.Also it is

assumed that arrivals and services are distributed independently.

In chapter 2,we have considered the queuing system when arrival

distribution is Poisson ,service time distribution is arbitrary with mean service

time θ per unit of time,there is single service channel ,system capacity is

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unlimited and service discipline is first come first serve basis.For the model

the steady state equations governing the queue are obtained .Also various

characteristics for the model, probability density function for waiting time

distribution ,busy period distribution,etc. are obtained.Particular cases when

service time distribution are double Gamma and Weibull are considered and

for both the cases the steady state equations and various characteristics for the

model are obtained.

In Chapter 3, we have considered the queuing system when

arrivaldistribution is Poisson ,service time distribution is arbitrary with mean

service time θ per unit of time,there is single service channel ,system capacity

is limited to N and service discipline is first come first serve basis. For the

model the steady state equations governing the queue are obtained . Also

various characteristics for the model,like expected number of customers in the

system,expected queue length etc. are obtained. Particular cases when service

time distribution are double Gamma and Weibull are considered and for

both the cases the steady state equations and various characteristics for the

model are obtained.

In chapter 4, we have considered the queuing system when arrival


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time θ per unit of time,there are c parallel service channel ,system capacity is

unlimited and service discipline is first come first serve basis. For the model

the steady state equations governing the queue are obtained . Also various

characteristics for the model,like expected number of customers in the




model are obtained.

In chapter 5,we have considered the situation when there is already one unit

in the system at t=0.Arrival distribution is obtained .Also expected number of

customers in the system and variance of n are obtained.

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Chapter 2

Queuing Model with Arbitrary Service Time

Distribution,Single Server and Unlimited Capacity

In this chapter,we have considered the queuing system when arrival


time θ per unit of time,there is single service channel ,system capacity is

unlimited and service discipline is first come first serve basis.For the model

the steady state equations governing the queue are obtained .Also various

characteristics for the model, probability density function for waiting time

distribution ,busy period distribution,etc. are obtained.Particular cases when

service time distribution are double Gamma and Weibull are considered and

for both the cases the steady state equations and various characteristics for the

model are obtained.

Steady state equations for the model:

Let Pn(t) be the Probability that there are n units in the system at time t and

Pn(t+Δt) be the probability that the system has n units at time t +Δt.

This event can occur in following four mutually exclusive and exhaustive

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ways.

1. There are n units in the system at time t and there is no arrivals and no

service completion during time interval Δt

2. There are (n-1) units in the system at time t and small time Δt have 1

arrivals but no service completion.

3. There are (n+1) units in the system at time t and small time Δt have no

arrival but one service completion.

4. The are n units in the system at time t and time Δt have no arrival and

no service completion.

Thus for n≥1;

Pn (t+Δt) = Pn(t). P(no arrivals is Δt). P(no services in Δt)

+ Pn-1 (t). P(1 arrival in Δt) P (no service in Δt)

+Pn+1(t).P (no arrival in Δt). P(1 service in Δt)

+ Pn (t). P (one arrival in Δt). P(1 service in Δt)+ o(Δt)

[Since arrivals and service occur randomly and independently and probability

of more than 1 arrival or 1 service is negligible in small time interval Δt]

Pn(t+Δt) = Pn(t){1-λΔt+o(Δt)}{1-θΔt+o (Δt)}+

Pn-1(t){λΔt+o (Δt)}{1-θΔt+o (Δt)}+

Pn+1(t){1-λ t+o (λΔt)}{θλt+o (Δt)}+o (Δt) ; n ≥ 1

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Or, Pn (t+Δt) = Pn (t) – (λ+θ) Pn (t) Δt + λ Pn-1 (t) Δt + θPn+1 (t)Δt

+o(Δt) ; n ≥ 1

{On combining all the terms of o(Δt)}

Or,

+

;n≥1

Or,Pn’(t) ; n ≥1….(2.1)

Since

For n=0, equation (2.1) is not valid.

For n=0, the event that there is no unit in the system at time t+Δt occur in two

mutually exclusive and exhaustive ways ,i.e.,

1. There is no unit in the system at time t and no arrival during Δt [as the

system is empty therefore question of any service does not arise].

Or

1. There is one unit in the system at time t and during Δt there is no

arrival but 1 service completion.

Thus,for n=0

Po(t+Δt) = Po(t).P [no arrival during Δt] + P1(t) P [no arrival during

Δt].P[one service completion during Δt]+ o(Δt) (since arrival and services

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are independent).

=Po(t){1-λΔt+o(Δt)}+P1(t){1-λΔt+o(Δt}.

{θ Δt+o(Δt)}+o(Δt)

=P0(t) – λ P0 (t) Δt + θP1(t)Δt + o(Δt)

{Combining all the terms of o(Δt)}

λPo(t) + θP1 (t)+

Or , Po’ (t) = λPo (t) + θP1(t) ; - ….(2.2)

since

Under steady state conditions Pn (t) Pn for large t so that Pn’(t) 0 for all n.

Thus equation (2.1) and (2.2) reduces to

– (λ+θ) Pn + θPn+1 + λPn-1=0

Or,

Pn+1=

….(2.3)

And,

θP1-λPo=0

which implies P1 =

P0 ….(2.4)

Putting n=1 in equation (2.3) and using (2.4) we get

P2 =

[(λ+θ) P1 – λP0]

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=

[(λ+θ)

P0 – λP0]

=(

)2 Po ….(2.5)

Putting n=2 in equation (2.3) and using (2.4) and (2.5) gives,

P3 =

[(λ+θ)P2-λP1]

=

[(λ+θ)

P0-λ

P0]

=(

)3 Po

Let the result is true for n= r i.e.

Pr = (

)r Po ….(2.6)

Then putting n = r in (2.3)

P r+1 =

[(λ+θ)Pr - λPr-1]

=

[(λ+ θ) (

)

r PO - λ(

)

r-1 P0]

=(

)r+1

PO

Hence the result is true for n= r+1.Since it is true for n=1,2 , therefore by

method of mathematical induction it holds for all positive integral values of n

. i.e.

34

Pn = (

)n

Po ….(2.7)

since , ∑ = 1

This implies

Po∑

=1

[

⁄ ]=1 (Sum of infinite terms of G.P.)

Or Po = (

) ….(2.8)

Hence the steady state equation for the model is ,

Pn = (

)n

(

) ….(2.9)

Characteristics of the model

1. Probability that queue length ≥ k

P [ queue Size ≥ k] = ∑

=∑ - ∑

= 1- ∑

(

)

= 1-(1-

) ∑

35

=1- (

⁄ ) ⁄

⁄

{Sum of k terms of G.P}

=1 – 1 +(

=(

….(2.10)

2. Average number of customers in the system is given by :

E(n)= ∑

=∑ (

)n

(1-

)

=(1-

) ∑ (

)n

= (1-

)S

Where S= ∑ (

)n

is sum of miscellaneous series, whose first terms are

in A.P. and second terms are in G.P. with common ratio

Or, S =

+ 2 (

)2

+ 3 (

)3 +……………

S = (

)2

+ 2 (

)3 +……………

36

On subtracting, we get

(1-

)S =

+ (

)2

+ (

)3 +……………

(1-

)S =(

)

(

)

Or , S =

(

)

Thus average number of customers in the system is ,

E(n)= (1-

)S

= (1-

)

(

)

= (

)

(

)

=

....(2.11)

3. Average queue length : Queue length is the length of queue

excluding the person being serviced. Thus queue length is given by,

m = n - 1

37

Average queue length is given by,

E(m) = ∑ ; m is a function of n

=∑

=∑ -∑

=∑ -∑

-Po+Po

= E (n) - ∑ Po

= (

)

(

) (

)

= (

)

(

)

= (

)

(

)

= (

)

Or;

Average queue length

E(m) = (

)

….(2.12)

4. Average length of non empty queue :

It is given by E[m/m>0]

=

38

=

=

=

=

=

=

( )

(

) ,

(

)-

=

( )

(

)

Or; E[m/m>0] .

/

….(2.13)

5. Variance of queue length is given by

V(n) = E[n-E(n)]2

= E[n2]-{E(n)}

2

= .

/

39

E(n2) = ∑

= ∑

= ∑ ∑

=∑

=∑ (

)

,

- 2

3

=(

) (

) ∑

2

3

where

=(

) (

)

∑

2

3

=(

) (

)

(

)

=(

) (

)

=(

) (

)

Or; (

)

(

)

(

)

(

)

Therefore,

40

= { (

)

(

)

} 2

3

=.

/

=

(

)

v(n)=

(

) ….(2.14)

6. Probability density function of waiting time (excluding service time)

distribution.

In steady state, each customer has the same continuous waiting time

distribution with probability density function ψ (t). Let ψ(t)dt denote

the probability that a customer begins to be served in the interval (t,

t+dt) where t is measured from the time of his arrival.

Let a customer arives at time t=0 and service begins in the

interval (t, t+Δt), then

1) If system is empty, then waiting time is zero with probability Po

)

2) If there are n customers already in the system when ̅̅ ̅̅ ̅̅ ̅th customer

arrives, n customers must leave the system before service of ̅̅ ̅̅ ̅̅ ̅th

41

customer begins or (n-1) customers must served during the time

interval (o, t) and nth

customer during (t, t+Δt)

As mean service rate is assumed to be θ per unit of time or θt in time t and

probability of (n-1) departures in time t is given by poisson distribution

=

It there are n customers in the system, then

Ψn(t) = P[ ̅̅ ̅̅ ̅̅ ̅ customers are served by time t]

×P [ nth

customer is served during Δt]

=

.θ Δt

The probability density function of waiting time of a unit is given by

=∑

=∑ (

)

(

)

Δt

= (

) (

) ∑

(

)

= (

) (

) ∑

(

)

= (

)

= (

)

42

ψn(t)= (

) . ; t>0 ….(2.15)

Now

∫

= (

) ∫

= (

)

=1

Which inplies that the complete distribution of waiting time is partly

continuous and partly discrete.

i.e. it is continuous for (t, t+ ∆t)

with probability density function

=

= (

)

(ii) and discrete for t=0 with probability function

Po = 1-

7. The probability that waiting time exceeds t is given by

∫

= (

)∫

43

=

∫

=

=

….(2.16)

8. Probability distribution of time spent in the system (i.e. busy period

distribution)

Let ψw(t/t>0) be the probability density function.

For waiting time such that a person has to wait ,i.e. the server remains

busy in the busy period.

=

= (

)

∫

= (

)

∫ (

)

∫

….(2.17)

44

9.

This is expected waiting time in the queue excluding service time and is

given by

∫

∫

On Integrating by parts

=

) *

+

= (

)

=

….(2.18)

Particular cases

2.1 Let the service time distribution is double Gamma with parameters

(µ, ν) having probability density function,

s(t) =

….(2.1.1)

with mean service rate,

E(t) = ∫

= ∫

=

∫

µt =

t = /µ

dt = d /µ

45

=

∫

=

∫

=

=

….(2.1.2)

Hence for double Gamma distribution mean service rate is

Putting

in (2.9) the Steady state equation for the model is,

(

)

And Pn= (

) *

+ ,n=0,1,2,… ….(2.1.3)

Characteristics of the model are

2.1 (a) P [ queue size ≥K] =

= (

….(2.1.4)

2.1(b) Average number of customers in the system is given by

(

)

….(2.1.5)

2.1 (c) Average queue length is given by

(

)

(

) ….(2.1.6)

46

2.1(d) Average length of non empty queue is given by

⁄

=

….(2.1.7)

2.1(e) Variance of queue length is given by

(

)

=

….(2.1.8)

2.1(f)Probability function of waiting time (excluding service time) is given by

Ѱw(t)= 2

(

)

=8(

)

(

)

(

)

….(2.1.9)

2.1 (g) P[ waiting time exceeds t] =

(

)

….(2.1.10)

2.1(h) Probability function of busy period is given by

47

=

=(

)

(

)

….(2.1.11)

2.1(i) Expected waiting time an arrival spends in the system (excluding

service time) is given by

E[W/W>0]=

=

….(2.1.12)

For µ=1 , all the results resembles to the cases when service time distribution

is exponential.

2.2 Let the service time distribution is weibull with parameters α and β i.e.

s(t)= ; ….(2.2.1)

with mean service rate

E(t) = ∫

= ∫

Let = t2

so that = ⁄

or =( ⁄

48

and

or

at

Thus mean service rate is

∫

∫

θ =

since ∫

= =Γn

Hence for Weibull distribution mean service rate is

θ =

….(2.2.2)

Putting θ =

in equation (2.9)

the steady state equation for the model when arrival distribution is Poisson,

service distribution is Weibull, capacity of system is unlimited, there is single

service channel and service discipline is FCFS basis is given by

(

)

49

and Pn (

) *

+

.

/

0

1 ….(2.2.3)


2.2.a) P[queue size≥K]

.

/

….(2.2.4)

2.2 b) Average number of customers in the system is given by

E[n] =

(

)

….(2.2.5)

2.2c) Average queue length –

E (m) = (

)

(

)

….(2.2.6)

2.2d) Average length of non empty queue is given by

E[m/m>0] =

50

=

(

)

….(2.2.7)

2.2(e) Variance of queue length is given by

V(n) =

(

)

=

(

)

(

)

….(2.2.8)

2.2 f) Probability function of waiting time (excluding service time) is given

by

= 2

(

)

=

{

.

(

)

/ (

(

) )

….(2.2.9)

2.2 (g) P[waiting time exceeds t] =

=

(

(

) )

….(2.2.10)

2.2 (h) Probability function of busy period is given by

51

(

)

(

(

) )

….(2.2.11)

2.2 (i) Expected waiting time an arrival spent in the system (excluding service

time) is given by

E(W/W>0) =

{

}

….(2.2.12)

In Particular, for β=1 all the above results are same when service time

distribution is exponential.

52

Chapter 3

Queueing Model with Arbitrary Service Time

Distribution,and Limited Capacity

In this chapter , we have considered the queuing system when

arrivaldistribution is Poisson ,service time distribution is arbitrary with mean

service time θ per unit of time,there is single service channel ,system capacity

is limited to N and service discipline is first come first serve basis. For the

model the steady state equations governing the queue are obtained . Also

various characteristics for the model,like expected number of customers in the




model are obtained.

Steady state difference equation

Let Pn (t) be the probability that there are n units in the system at time t and

Pn (t+∆t) is the probability that there are n units in the system at time (t+∆t)

then

53

Pn(t+Δt) = Pn(t).P[no arrival in the system during Δt].P[no service during Δt]

+Pn-1(t).P[1 arrival during Δt].P[no service during Δt]

+Pn+1(t).P[no arrival during Δt].P[1 service during Δt]+o(Δt) ;n≥1

{Since cases are mutually exclusive and exhaustive and arrival and service

are distributed independently}

Or;

Pn(t+∆t) = Pn(t)[1-λΔt+o(Δt)][1-θΔt+o(Δt)]

+Pn-1(t)[λΔt+o(Δt)][1-θΔt+o(Δt)]

+Pn+1(t)[1-λΔt+o(Δt)][θΔt+o(Δt)]+o(Δt) ; 1 ≤ n ≤ N-1

Or;

Pn(t+∆t) = (t)∆t+ θ ∆t+o(∆t);

1 ≤ n ≤N-1

{Combining all the terms of o(∆t)}

Or;

→

(λ+θ)Pn(t) + λPn-1(t) + θPn+1(t)

+ →

Or ;

;1 ≤ n ≤ N-1 ….(3.1)

54

→

For n=0

Po(t+∆t) = Po(t).P[no arrival during

+P1(t).P[no arrival during +o(

{ Combining all the terms of o(∆t)}

Po(t+Δt) = Po(t)[1- λΔt+o(Δt)] + P1(t)[1-λΔt+o(Δt)][θΔt+o(Δt)]+o(Δt)

=


Or ;

→

= (t)+

→

Or Po’(t)= (t)+ ….(3.2)

And for n=N

PN(t+

=PN(t).P[no service during Δt]

+ PN-1(t).P[1 arrival during Δt].P[no service during Δt]+o(Δt)

{Since for n=N, arrival rate λ=0}

= PN(t) [1-θΔt+o(Δt)] + PN-1(t) [λΔt+o(Δt)][1-θΔt+o(Δt)] + o(Δt)

55

Or

(t+∆t)= (t)∆t+ o(∆t)

→

→

Or ….(3.3)

under steady state conditions for large t and Pn’(t) for all n.

Thus equation (3.1) , (3.2 ) and (3.3) reduces to

0 = +

Or;

….(3.4)

….(3.5)

And

….(3.6)

Putting n=1 in eq. (3.4), we get

*

+

(

)

n=2 in eq. (3.4) gives

56

(

)

And so on

n= N-2 gives

[

(

)

]Po

=(

)

And ;

(

)

=

(

)

=(

)

N .Po ….(3.7)

Thus is general , (

) , n=0,1,………….N

And

For finding the value of , we know that

∑

Or;

………..

57

Or;

[

(

) (

)

]

Or;

2 (

)

3

, (

)-

;{sum of N+1 terms of G.P with common ratio

Or;

Po=

{

(

)

(

)

….(3.8)

Thus the steady state equation for the model is given by

Pn=

{

{

(

)

(

) }

….(3.9)

; 0≤ n ≤ N

Measures of the model

1. P[queue Size ≥ K] = ∑

= ∑ ∑

58

=1 - ∑ (

)

=1 - ∑ (

)

=1 - 62 (

) 3

7

{sum of k term of G.P. with common ration

} ;

(

) 2 (

) 3

{ (

)

},

- ;

(

)

(

)

(

)

(

)

{ (

)

} 2 (

) 3

{ (

)

} ;

= { (

)

} 2 (

) 3

{ (

)

}

(

) (

)

(

)

59

(

) (

)

(

)

….(3.10)

2. Average number of customers is the system is given by

∑

∑ (

)

∑ (

)

Where S=∑ (

)

Or S=(

) (

) (

) (

)

S= (

) (

) (

)

(

)


(

) (

) (

)

(

)

(

)2 (

)

3

, (

)-

– (

)

60

=

. (

)

/ (

)(

)

(

)

= (

) (

)

(

)

(

)

Or (

)0 (

)

(

)

1

(

) ;

Therefore,

Average no of customers in the system is given by,

E(n)=

(

)

2 (

)

3

(

)

(

)

,

-

(

)

(

)

,

- 2 (

)

3

;

….(3.11)

3. Average queue length is given by

E(m) ; where m=n-1

i.e. Lq= E(m)= ∑

= ∑ ∑

61

= ∑ ∑

= E(n)- ∑

= E(n)-

Or E(m)= E(n)-

= E(n)- 6 (

)

2 (

)

3

7

= E(n) 6 (

)

( )

2 ( )

3

7

= E(n) (

) 6

( )

2 ( )

3

7

=(

)

0 (

)

(

)

1

,

-2 (

)

3

-(

) 6

(

)

2 (

)

3

7

=(

)

02 (

)

(

)

,

-2 (

)

31

(

)2 (

)

3

=(

)

02 (

)

(

)

(

)

(

) (

)

1

(

)2 (

)

3

62

= (

)0(

) (

)

(

)

1

(

)2 (

)

3

Or;

(

) 0 (

)

(

)

1

, (

)-2 (

)

3

….(3.12)

4. Average length of non-empty queue is given by,

E[m/m>0] =

, m= n-1

P[m>0] = P (n-1>0)

= P (n>1)

= 1 P(n≤1)

= 1

= ,(

)-

2 (

)

3

, (

)-,

-

2 (

)

3

=

2 (

)

3 ,

- (

) (

)

2 (

)

3

;

63

=

(

) 2 (

)

3

2 (

)

3

Thus, E[m/m>0] =

( ) 2 (

)

( )

3

{ ( )}2 (

)

3

( ) 2 (

)

3

2 ( )

3

=

2 (

)

(

)

3

, (

)-2 (

)

3

….(3.13)

Particular Cases

3.1 Suppose the service time distribution is double Gamma with parameters

(µ,ν)>0, and its probability density function is given by

S(t)=

; t >0 ; (µ,ν) >0

With mean service rate

θ = E(t) =

Putting θ=

in equation (3.9) , the steady state equation for the model is given

by

64

Pn =

{

,

-

2 (

)

3

for 0 ≤n ≤ N

….(3.1.1)


3.1.(a) P[ queue size ≥ k] =

(

) 0 (

)

1

0 (

)

1

;

θ =

P[ queue size ≥k]=

(

) 0 (

)

1

0 (

)

1

;

….(3.1.2)

3.1.(b) Expected number of customers in the system is given by

(

) 0 (

)

(

)

1

(

)2 (

)

3

65

(

) 0 (

)

(

)

1

(

)2 (

)

3

;

….(3.1.3)

3.1.(c) Average queue length is given by

Lq = E (m)

(

) 0 (

)

(

)

1

(

)2 (

)

3

(

) 0 (

)

(

)

1

,(

)-2 (

)

3

;

….(3.1.4)

3.1.(d) Average length of non empty queue is given by,

E[m/m>0] =

2 (

)

(

)

3

,(

)-2 (

)

3

;

….(3.1.5)

In particular for µ=1, all the result are same as those when the service time

distribution is exponential with mean service rate ν per unit of time

Particular Case 2

3.2 If service time distribution is weibull with parameters α and β, then

66

probability density function of service time is given by,

S(t) = (α β) ; t ≥ 0 (α, β) >0


θ = E(t) =

Putting θ =

in (3.9), the steady state equation for the model is

Pn =

4

5

8 4

59

{ 4

5

}

;

;

For 0 ≤ n ≤ N ….(3.2.1)


3.2 (a) P[queue Size ≥ k]

(

√

)

[ 4

5

]

{ (

)

}

;

….(3.2.2)

3.2 (b) Expected number of customers in the queue is given by

67

E(n) =

4

5{ 4

5

4

5

}

8 4

59{4

5

}

;

….(3.2.3)

3.2 (c) Average queue length is given by ;

Lq = E (m)

=

4

5

[ 4

5

4

5

]

8 4

59{4

5

}

; ….(3.2.4)

3.2 (d) Average length of non-empty queue is given by

E(m/m>0) =

[ 4

5

4

5

]

8 4

59{4

5

}

;

….(3.2.5)

For =1, all the results are same as those when the service time distribution is

exponential.

68

Chapter 4

Queueing Model with Arbitrary Service Time

Distribution and Multiple Servers

In this chapter, we have considered the more realistic situation when arrival


time θ per unit of time,there are c parallel service channel ,system capacity is

unlimited and service discipline is first come first serve basis. For the model

the steady state equations governing the queue are obtained . Also various

characteristics for the model,like expected number of customers in the




model are obtained.

There are c(fixed) parallel service channels and a customer can go to any of

the free counter for his service,where the service time is identical and have

the same probability density function s(t) with mean service rate θ per unit of

time per busy server. Thus, overall service rate when there are n units in the

69

system is given by:

1. If n≤c, all the customers may be served simultaneously and in such

cases there will be no queue.(c-n) servers may remain idle and then

mean service rate is

nθ, for n= 0,1,…..,c-1,c.

2. If n≥c, all the servers will remain busy, number of customers waiting in

the queue will be (n-c)and then mean service rate is cθ

,i.e., mean service rate is given by

{

Steady state difference equations

Let Pn (t) be the probability that there are n units in the system at time t and Pn

(t+∆t) is the probability that there are n units in the system at time (t+∆t) then

Pn(t+Δt)=Pn(t).P[no arrival in the system during Δt].P[no service during Δt]


+Pn+1(t).P[no arrival during Δt].P[1 service during Δt]+o(Δt);1≤n≤c-1

{Since cases are mutually exclusive and exhaustive and arrival and service

are distributed independently}

Or;

70

Pn(t+∆t) =Pn(t)[1-λΔt+o(Δt)][1-nθΔt + o(Δt)]

+Pn-1(t)[λΔt+o(Δt)][1- (n-1)θΔt+o(Δt)]

+Pn+1(t)[1-λΔt+o(Δt)][(n+1)θΔt+o(Δt)]+o(Δt) ; 1 ≤ n ≤ c-1

Or;

Pn(t+∆t)= Pn(t) – (λ+nθ)Pn(t)Δt + λPn-1(t)Δt+(n+1)θPn+1(t)Δt+o(Δt)

; 1 ≤ n ≤c-1


Or;

= (λ + nθ)Pn(t) + λPn-1(t)

+ (n+1)θPn+1(t)+ →

Or ;

; 1 ≤ n ≤c-1 .…(4.1)

since →

For n=0

Po(t+∆t)=Po(t).P[no arrival during

+P1(t).P[no arrival during +o(

71


Po(t+ o(t)[1- Δt+o(Δt)]+P1(t)[1-λΔt+o(Δt)][θΔt+o(Δt)]+o(Δt)

=


Or ;

→

= (t)+

→

Or Po’(t)= (t)+ ….(4.2)

And for n≥c

Pn(t+Δt)=Pn(t).P[no arrival in the system during Δt].P[no service during Δt]


+Pn+1(t).P[no arrival during Δt].P[1 service during Δt]+o(Δt)

=Pn(t)[1-λΔt+o(Δt)][1-cθΔt + o(Δt)]

+Pn-1(t)[λΔt+o(Δt)][1- cθΔt+o(Δt)]

+Pn+1(t)[1-λΔt+o(Δt)][cθΔt+o(Δt)]+o(Δt) ; n≥c

Or;

Pn(t+∆t)= Pn(t) – (λ+cθ)Pn(t)Δt + λPn-1(t)Δt+cθPn+1(t)Δt+o(Δt) ; n≥c


72

Or;

= (λ + cθ)Pn(t) + λPn-1(t)

+ cθPn+1(t)+ →

Or ;

;n≥c ….(4.3)

Under steady state conditions for large t and

Pn’(t) for all n.

Thus equation (4.1) , (4.2 ) and (4.3) reduces to

0 = + ;1≤n≤c-1

Or;

;1≤n≤c-1 ….(4.4)

0=

Or;

P1 =

….(4.5)

And,

Pn+1 =

;n≥c ….(4.6)

Putting n=1 in equation (4.4), we get

73

*

+

(

)

n=2 in eq. (4.4) gives

=

(

)

(

)

and so on.

In general,

Pn =

=

;1≤n≤c-1

n= c-1 gives

Putting n=c in equation (4.6) gives,

Pc+1 =

74

=

(

)

Po

=

=

Similarly n=c+1 in equation (4.6) gives

Pc+2 =

And so on.

In general for n≥c

Pn = Pc+(n-c)=

We know that,

∑

Or;

Po+∑ ∑

Or;

∑ *

(

) +

∑ *

(

) +=1

Or,

∑

(

)

*∑

(

) +

75

Or,

∑

(

)

*∑

+

Or,

[∑

{

⁄

⁄ }]Po=1

Or,

0∑

(

)

2

(λ

θ)

(λ

θ) 3

1

; λ

θ ….(4.7)

Thus steady state equation for the model is given by,

=8

(

)

(

)

….(4.8)

Where Po is given by equation (4.7)


1. If n ≥ C, queue of n customers would consist of c customers being

served together with (n-c) waiting customers

76

Hence,

Expected queue length is given by ,

E(m) = ∑

= ∑ n-c = j

⇒

n= ,

= ∑

(λ

θ)

E(m) =

(λ

θ) ∑ (

λ

θ)

=

(λ

θ) ....(4.9)

Where,

77

S = (λ

θ) (

λ

θ) (

λ

θ)

λ

θ (

λ

θ) (

λ

θ)


( λ

θ) S =

λ

θ (

λ

θ) (

λ

θ)

= (λ

θ)

, (λ

θ)-

Therefore,

S = (λ

θ)

, (λ

θ)-

Which implies,

E(m) =

(λ

θ) (

λ

θ)

, (λ

θ)-

....(4.10)

Where Po s given by equation (4.7)

2. Average number of customers in the system is given by :

78

E(n) = ∑ Pn

=E(m)+ λ

θ ….(4.11)

3. Wq = Expected waiting time per customer in the queue

….(4.12)

4. Expected waiting time per customer in the system is given by,

Ws

=Wq+

. …. (4.13)

5. Probability distribution of busy period is given by :

An arrival must wait in the queue if there are c or more customers in the

system. Thus,

Prob [Busy Period]= P [n ≥ c]

= ∑ Pn

= ∑

(

)

=

∑ (

)

=

(

)

(λ

θ)

79

=

(

)

(λ

θ)

….(4.14)

Where Po is given by equation (4.7)

Particular cases

4.1 Let the service time distribution be double Gamma with parameters

(µ,ν)>0, having probability density function,

S(t) =

; t>0; (µ,ν) >0


θ = E(t) =

Putting θ=

in equation (4.8) , the steady state equation for the model is given

by

Pn=8

(λµ

ν)

(λµ

ν)

Where

0∑

(λµ

ν)

2

(λµ

ν)

(λµ

ν) 3

1

; λµ

ν ….(4.1.1)

80


1) Expected queue length is given by,

E(m) =

(λµ

ν) (

λµ

ν)

, (λµ

ν)-

….(4.1.2)

2) Average number of customers in the system is given by,

E(n) = E(m) + λµ

ν ….(4.1.3)

3) Expected waiting time per customer in the queue is

….(4.1.4)

4) Expected waiting time per customer in the system is given by

Ws =

….(4.1.5)

5) Probability distribution of busy period is given by ;

=

(λµ

ν)

, (λµ

ν)-

….(4.1.6)

Where E(m) is given by equation (4.1.2)

For μ = 1,all the above results are same as those when service time

distribution is exponential.

Particular case 2

81

4.2 Suppose the service time distribution is Weibull with parameters (α,β)> 0

and p.d.f. is given by

S(t) = (αβ) ; t > 0;(α, β) >0


θ = E (t) –

Γ (

)

Putting θ =

Γ (

) in (4.7) and (4.8)

gives the steady state equation for the model as

={

(

)

(

)

....(4.2.1)

Where Po is given by

[

∑

.

λ

Γ(

)

/

{

4

λ

( )

5

4λ

( )

5

}

]

….(4.2.2)

82

Measures of the model

1. Average queue length is given by

E(m) =

.

λ

(

)

/

4λ

( )

5

8 λ

( )

9

….(4.2.3)

Where Po is given by equation (4.2.2)

2. Average number of customers in the system is given by,

E(n) = E(m) +λ

(

)

….(4.2.4)

Where E(m) is given by equation (4.2.3)

3. Expected waiting time per customer in the queue is given by,

….(4.2.5)

4. Expected waiting times per customer in the system is given by,

….(4.2.6)

83

5. Probability distribution of busy period is given by,

=

4λ

( )

5

8 λ

( )

9

….(4.2.7)

For β=1 all the above results are same as those,when service time distribution

is exponential.

84

Chapter 5

Truncated Arrival Distribution

In this chapter,we considered the situation when there is already one unit in

the system at t=0.Arrival distribution is obtained .Also expected number of

customers in the system and variance of n are obtained.

Here the situation can be interpreted that at each time interval just a new unit

arrives in the system with same arrival rate λ for all individuals and

probability of existing of a unit in the system at time ∆t is λ∆t+0(∆t)

Let Pn(t) be the probability that these are n units in the system at time t and

Pn (t+∆t) be the probability that system has n units at time t+∆t

Thus;

Pn( t+∆t) = P [ n units in the system at time t].P[no arrival in ∆t]

+ P[ ̅̅ ̅̅ ̅̅ ̅̅ units in the system at time t]. P[ 1 arrival in ∆t]+ o(∆t)

; n>1

; n>1

85

→

→

; n>1

Or ; n>1

since →

0

Or;

; n>1 ….(5.1)

For n=1

P1(t+∆t) = P[there is one unit in the system at time t+∆t)

Or

→

→

Or

Or

= - ….(5.2)

Integrating equation (5.2) on both side with respect to t, we get

log ….(5.3)

Where A is the constant of integration.

Using condition

86

,

….(5.4)

We get

On putting t=0 in eq. (5.3), we get

log

⇒ ⇒

Putting A=0 in eq. (5.3) we get

Which implies

….(5.5)

Puttting n=2 in eq (5.1) gives

(from equation 5.5)

Or

Or

Or

{

}

On integrating both sides with respect to t, we get-

∫

….(5.6)

87

Putting t=0 in (5.6), we get

⇒ Pn(t)=0 for n>1

from eq (5.4)

Thus equation (5.6) reduces to,

=

Or { }

= (1- ….(5.7)

n=3, eq (5.1) gives

{ }

Or,

{ }

Or

{

} { }

On integrating both sides with respect to t, we get

∫

∫

88

Or

t=0⇒

⇒ 0 = (1-2)+c using (5.4)

⇒ c = 1

⇒ ( )

Or [ ]

[ ]

[ ] ….(5.8)

And so on

In general

[ ]

….(5.9)

Which is the probability function of geometrical form with

As t

Thus if there is already one unit in the system at the starting then arrival

distribution is of geometrical form.


1. Average number of customers is given by

E(n) = ∑

89

E(n) = ∑

n-1

= ∑ ; q=1-

∑

0

1

*

+

Substituting , we get

{ ( )}

.…(5.10)

2) v(n)=

=

= ….(5.11)

From (5.10)

∑

= ∑ ∑

= ∑

90

∑ ( )

( )∑

Where q =

Or, ( )∑

= ( )

∑

( )

(

)

( )

….(5.12)

Putting q=

E ( )

( )

E ( )

( )

= ( )

=

= ….(5.13)

From (5.11) and (5.13), we get

V(n)=

=

=

91

References

[1] Allen, A.: Probability, statistics, and queueing theory : with computer

science applications, Academic Press, 1990

[2] Artalejo, J.R – Gomez-Corral, A.: Retrial queueing systems : a

computational approach, Springer, 2008

[3] Bingham, N.H.: The work of Lajos Takacs on Probability Theory Journal

of Applied Probability 31A(1994) 29-39

[4] Bolch, G. – Greiner, S. – de Meer, H. – Trivedi, K.S.: Queueing networks

and Markov chains : modeling and performance evaluation with computer

science applications, John Wiley and Sons, 1998, 2006

[5] Cooper, R.: Introduction to queueing theory, CEEPress Books, 1990

[6] Daigle, J.: Queuing theory for telecommunications, Addison-Wesley

Publisher, 1992

[7] Dshalalow, J.H. – Syski, R.: Lajos Takacs and his work, Journal of

AppliedMathematics and Stochastic Analysis 7(1994) 215-237

[8] Dshalalow, J.: Advances in queueing : theory, methods, and open

problems, CRC Press, 1995

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and engineering , CRC Press, 1996

92

[10] Erlang, A.K.: The Theory of Probabilities and Telephone Conversations,

Nyt Tidsskrift for Matematik B 20(1909)

[11] Erlang, A.K.: Solutions of Some Problems in the Theory of Probabilities

of Significance in Automatic Telephone Exchanges, Post Office Electrical

Engineering Journal 10(1917) 189-197

[12] Galambos, J. – Gani, J.: Studies in Applied Probability, Papers in honour

of Lajos Takacs, Journal of Applied Probability 31A(1994)

[13] Giambene, G.: Queuing theory and telecommunications : networks and

applications,Springer, 2005

[14] Gnedenko, B.V. – Kovalenko, I.N.: Introduction to Queueing Theory,

Birkhauser, 1968, 1989

[15] Gross, D. – Shortle, J.F. – Thompson, J.M. – Harris, C.M.: Fundamentals

of Queueing Theory, Wiley, 2008

[16] Gyorfi, L.: Tomegkiszolgalas informatikai rendszerekben, Műegyetemi

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[18] Haribaskaran, G.: Probability, queueing theory and reliability

engineering, Laxmi, 2006 30 J. Sztrik

93

[19] Jewel, W.S.: A Simple Proof of L = λW , Operations Research 15(1967)

1109-1116

[20] Khintchine, A.Y.: Mathematical Methods in the Theory of Queueing,

Griffin, 1960

[21] Kleinrock, L.: Queueing Systems I-II, John Wiley, 1975,1976

[22] Kleinrock, L.: Sorbanallas, kiszolgalas : Bevezetes a tomegkiszolgalasi

rendszerek elmeletebe, Műszaki Kiado, 1979(Edited).

[23] Kobayashi, H. – Mark, B.L.: System Modeling and Analysis, Pearson

International Edition, 2009

[24] Lakatos, L. – Szeidl, L. – Telek, M.: Tomegkiszolgalas, Informatikai

Algoritmusok, Edited by Ivanyi, A. ELTE Eotvos Kiado 1298-1347, 2005

[25] Little, J.D.C: A Proof for the Queuing Formula L = λW, Operations

Research 9(1961) 383-387

[26] Ramalhoto, M.F. – Amaral, J.A. – Cochito, M.T.: A survey of J. Little’s

formula, International Statistical Review 51(1983) 255-278

[27] Stidham, S.: A Last Word on L = λW, Operations Research 22(1974)

417-421

[28] Sztrik, J.: On the finite-source ~ G/M/r queu, European Journal of

Operational Research 20(1985), 261-268

94

[29] Sztrik, J.: Bevezetes a sorbanallasi elmeletbe es alkalmazasaiba,

Debreceni Egyetemi Kiado, 1994 (Edited).

[30] Takagi, H.: Queuing analysis : a foundation of performance evaluation,

I-III,North-Holland, 1991-1993

[31] Takacs, L.: Stochastic Processes, Problems and Solutions John Wiley

and Sons,1960

[32] Takacs, L.: Introduction to the Theory of Queues, Oxford University

Press, 1962

[33] Takacs, L.: Combinatorial Methods in the Theory of Stochastic

Processes, John Wiley and Sons, 1977

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14th Meeting of Statisticians, Poland, 399-402, 1981

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computer science applications, Prentice-Hall, 1982, 2002

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