SOME CONTRIBUTIONS TO QUEUEING THEORYshodhganga.inflibnet.ac.in/bitstream/10603/42045/1/introduction...
Transcript of SOME CONTRIBUTIONS TO QUEUEING THEORYshodhganga.inflibnet.ac.in/bitstream/10603/42045/1/introduction...
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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
distribution is Poisson ,service time distribution is arbitrary with mean service
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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
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 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
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
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)
Characteristics of the model
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
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.
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)
=
{ Combining all the terms of o(∆t)}
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= (
) (
) (
)
(
)
On subtracting, we get
(
) (
) (
)
(
)
(
)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)
Characteristics of the model
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
With mean service rate
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)
Characteristics of the model
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
distribution is Poisson ,service time distribution is arbitrary with mean service
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
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.
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[1 arrival 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
{Combining all the terms of o(∆t)}
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
{ Combining all the terms of o(∆t)}
Po(t+ o(t)[1- Δt+o(Δt)]+P1(t)[1-λΔt+o(Δt)][θΔt+o(Δt)]+o(Δt)
=
{ Combining all the terms of o(∆t)}
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[1 arrival 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
{Combining all the terms of o(∆t)}
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)
Characteristics of the model
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)
On subtracting, we get
( 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
With mean service rate
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
Characteristics of the model
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
With mean service rate
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.
Characteristics of the model
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
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