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CHAPTER- I

INTRODUCTORY BASIC CONCEPTS IN QUEUEING

AND SCHEDULING THEORY

Research work has been designed so as to find solution of certain typical

problems which crop up in the industry and business houses in the theory of

Queueing and Scheduling. Efficient computer programmed and software have

been made keeping in mind that time and accuracy are the code words for

today.

The complete picture of thesis may be considered in two folds. First to modify

the existing tools of Queueing and Scheduling theory in a way that permits

them to be applied to real life problem and second to make an extensive

application of these tools to various important day systems in which concepts

related to tandem, bi-tandem Queues using preemptive and non- preemptive

scheduling disciplines will be utilized. Again in accordance to the problem, if

required, concepts of transportation time job block and breakdown times will

also be considered depending on the problem the most heuristics or optimal

and feasible solution methods will be obtained. Today we are using

scheduling methods in almost every field of life to provide facilities to

effectively process the items or jobs. It is necessary to select from the

situations and its environment a criterion Objective function.

Sequencing/scheduling is used to the order in which units require service and

served. The objective of studying this problem is to assist a decision maker in

his business and industry. One of the perhaps most disagreeable activities,

namely waiting in lines, is a delightful subject of the day. Queues as

civilization progresses and must find ways to tolerate these unpleasant

situations. In queueing theory we deal with those waiting line problems in

which customers waiting line will be minimized at the minimum cost of service.

Now we discuss each section one by one as follows:-

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Section A

Queueing Theory

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

facility at some service station imagine 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 lending in an airport.

7- The arrival and departure of ships in a harbors.

8- Broken machines waiting to be serviced by a repair man.

9- Jobs waiting for processing by a computer.

10- 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. 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. Queueing theory is the main aspect of waiting time models where

discussion involves stochastic process. The main purpose of the queueing

theory is to briefly investigate those queueing 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 under the stochastic process by using Tandem, Bi-

Tandem Queues. The Pioneer investigator of Queueing 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

telegraphic theory derived the formulae relating to traffic load. The

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mathematical discussion on queueing Theory made considerable progress in

early 1930’s through 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 Queueing problems and Applied

queueing Theory was given by Lee, A. M. (1958). An element of queueing

theory with applications was given by T.L. Saaty (1961). On some problems

in queueing 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 queueing theory has proved for researchers who

want to do fundamental research on stochastic process involving

mathematical models. The process involved is not simple and for time

dependent analysis more sophisticated mathematical procedures are

necessary. For example for the Queue with Poisson arrivals 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 rendered.

Markvian by describing the process with sufficient number of supplementary

variables and operating on them, in addition to this Keilson also in his later

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investigation uses sections of the process 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 Queueing system was made by

Kandall (1951,1953) which demonstrated the use of Markovian 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 queueing system.

The simple queue with Poisson Arrivals and exponential service time on a

random walk and derived the time dependent behavior of the queue length

process in explicit from without resorting to transforms is also considered by

Champernonane (1956) Giving a brief history of the development of the

queueing theory. We now present the detailed description of a queueing

system.

1.1 - DESCRIPTION OF THE QUEUEING PROBLEM A queueing system can be describes as customers arriving for service, waiting

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

system after being served. Such a basic system is depicted in Fig 1.1

CUSTOMERS ARRIVING SERVED CUSTOMERS

0000 --------- → SERVICE → 000 ---------LEAVING FACILITY

DISCOURAGED CUSTOMERS LEAVING

FIG 1.1

SCHEMATIC DIAGRAM OF A QUEUEING PROCESS

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Although any queueing 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.

1.2 - COMPONENTS OF QUEUEING PROCESSES A little change in one or more of this basic process gives rise to difficult

queueing system. There are basic characteristics of queueing processes

given below:-

§ Arrival pattern of customers

§ Service Discipline of servers

§ Queue discipline

§ System capacity

§ Number of service channels

§ Number of service stages

§ Out put or departure distribution

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

system. 1.2.1 - ARRIVAL PATTERN OF CUSTOMERS:- The arrival pattern or input to a queueing 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 queueing 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

customers decides not to join the queue upon arrival customers is said to

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

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

exponential holding time. 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 queueing

problems. The idea of “On bulk arrivals in bi-series channel links in series 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). Bachelli, 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 queueing 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

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describing the input process 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 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

dt

tdAta )()( =

1.2.2 - SERVICE DISCIPLINE The manner in which customers are served is referred to as the service

discipline for the queueing 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 differentiable then its probability density function is denoted by

b(x), where

Bxdxdbx =

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.

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As it is usually referred to another queue discipline to many inventory system

when there is no obsolescence of stared units as it is easier to reach the

nearest items which are the last in. Now when the selection for service is

made without any regard to the order of the arrival of 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

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 queueing system could by 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 definitive

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 this presidential treatment is 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.

1.2.2(a) - 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

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

priority

1.2.2(b) - 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:-

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i) 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

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 (1958) 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.

1.2.2(c) - TRANSIENT AND STEADY STATE Queueing 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 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

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of the time. This state is refused 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 elapsed

time tends to infinity.

Let Pn(t) denote the probability that there is n Unit in the system at

time t, then the system acquires steady state as t→ ∞ i.e.

Lim t→ ∞

dtd Pn(t)

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

may permit the existence of a steady state.

1.2.2(d) - 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

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.

Arrival Service Service Departure.

Queue → Stage I → Stage II → Departure Fig- 1.2

. A queue in series system can be depicted as above.

Jackson (1954), Johnson (1954) and Maggu (1972-73) have land 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

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Poisson input and exponential holding time.

Avi Itzahak and yadin (1965) 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 (1973) presented the most recent work on serial queues. A general

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. 1.2.2(e) - QUEUES IN BI-SERIES Queues in bi-series were first studied by Maggu (1970). Bi-series means that

a customer can get his services first in station 1 and then in station 2 or he

can get his service first in station 2 and then in station 1 and then leaves the

system. In another words the service of a customer consists of two parts that

can be done in respect of the order of performing service. He derived the

steady state equation describing the model. A queueing system consist two

queues Q1 and Q2 with servers S1 and S2. We come across situation where

the output from Q1 or part of it comprises the input to Q2 and vice versa at the

same moment. This system of queue is called in Bi-series. An Example for

queue in Bi-series model can be viewed in a barber shop where there are two

servers dealing with both is a part of the bi-series service system. A customer

having being served with hair cut may or may not join the second queue

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giving shaving service. Similarly a person after being should may or may not

require hair cut. The basic idea of bi-series /bi-tandem queues was introduced

by Maggu (1970), P.K. Gupta (1975), M.S. Alam (1978) and Mahabir singh

(1979). Mohammad, Hussain & Ikram (1983) showed that two bi-series

queues with batch arrival at each stage. The idea of “On multi-input bi-tandem

queue modeling has given by Heydari A.P.D. (1987). On feed back in the

network of queues with service rats proportional to queue numbers have

discussed by Kumar Ayu (1990). Queues in bi- tandem channel has given by

Chandramouli (1996)

Service Q1 S1 Q2 Service S2

Fig 1.3

In the present study of the concept of bi-series queues has been frequently

used in chapter 2 therefore it seems appropriate to discuss in brief the basic

model studied by Maggu (1970) as follows.

Let the bi-serial queueing system consists of two queues Q1 and Q2 which

being attended by servers S1 and S2 respectively. Poisson arrivals from

outside the system occur with mean rates λ1 and λ2 and join the respective

queues Q1 and Q2 before the servers S1 and S2 respectively. It is assumed

that the service time distribution for the queues Q1 and Q2 is mutually

independent, negative exponentially distributed with respective

parametersµ1and µ2. Further, it is assumed that P1, P2, (P1, P2, ≥ 0, P1 +

P2 =1) denote respective probabilities of a unit completing service at S1 to

either leave the system or enter the subsequent phase channel S1. Maggu is

his notations obtained the following differential difference equations:

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P’(m,n,t) = -(λ1+λ2+µ 1+ µ 2 ) P(m,n,t) + λ1 P(m-1,n,t)

+ λ2 P(m,n-1,t) + µ 1 P1P(m+1,n,t) + µ 1 P2P(m+1,n-1,t)

+µ 2q1P(m,n+1,t) + µ 2q2P(m-1,n+1,t)

For m, n ≥ 0

By using generating functions technique, he obtained the implicit form of

probability generating function of queue lengths in terms of Laplace transform.

In the steady state case he obtained the following expression for the expected

queue length of the system

EQ = λ1 + λ2 q2 + λ2 +λ1P2

µ 1(1-p2q2) - (λ1+ λ2 q2) µ 2(1-p2q2) - (λ2 +λ1P2)

1.2.2(f) - TANDEM QUEUE WITH BLOCKING Consider a sample sequencing two station single servers at each station

model where no queue is allowed to form at other station. If a customer is in

station II and service is completed at station I, the station I must wait there

until the station II customer is completed, that is system is blocked. The main

studies in tandem queue involving blocking concept are due to G.C. Hunt

(1956), T Singh and PL Maggu (1977), F. P. Kelly (1984), these stages

tandem queue with blocking discussed by C. Longasis and B. conally (1981).

1.2.2(g) - CYCLIC QUEUE In cyclic queues problem there is a set of queues in tandem serving a fixed

population where a unit leaving the last phase of service returns and waits in

an ordered fashion for service in the queue of the first phase, going through

the queue and the service channel of the phase in sequence and so on e.g.

programmed submitted for run on a computer since a programmed would

normally be required to run through a sequence of operations.

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1.2.3 - CUSTOMER BEHAVIOUR Queueing models representing situations in which human beings take the

roles of 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 looses

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.

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.

1.2.4 - 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 Queueing system, was given by Burke, P. J. (1978).

Suppose the out put in to a service channel occurs at times

t 0 < t1 < t 2 – - - < t k < t k+1 < - - - the random variables.

bk = t k – t k-1 (K 1≥ ) are called inter service times. Suppose

b1, b2 - - - - bk be the duration of successive service and let us assume that

the service times are mutually independent, identically distributed positive.

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Random variables B(x) = Pr (bk x≤ ) is called the service time distribution.

The departure is usually considered to be independently of each other and

of arrivals.

Let E(ak) = y λ and E (bk) = y µ denote the mean inter arrival times and

the service time of the successive units.

For the single server system the ratio ρ = Y µ / Y λ (O < P < ∞ )

where ρ is called the relative traffic intensity or utilization factor and it is a

measure of congestion in the system λ is the input andµ is the out put

Parameter. Jackson (1954), Hunt (1956) studied the problem with varying

Parameters.

1.2.5 - QUEUEING NETWORK THEORY

The initial development of queueing network theory may be seen since 1954

(Series queues) and 1957 (Queueing Networks). The literature on the subject

varies rapidly. Here we will discuss about development of queueing networks

literature in recent years. There can be four parts of queueing networks theory

as: -

1) Open unrestricted networks

2) Closed unrestricted networks;

3) Open restricted networks

4) Closed restricted networks.

We briefly describe the development of multistage queueing networks models

for all the above four groups separately.

Some terms used in queueing networks theory are defined below

1.2.5(a) - Blocking

Mean of the blocking is that when a customer has completed service in the

first stage but can not proceed because the second station and queue are

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completely filled. There is different type of blocking but the most often dealt

with in the literature are:-

Blocking after service, blocking before service these are also called

manufacturing and communication blocking. Blocking after service occurs

when a station has completed the service of a customer while there is no

space in the next queue for it. So the customer will remain at the first station

until next will be free. 1.2.5(b) - A Cyclic Network A cyclic network is open network in which each customer can visit any mode

only once. Queueing networks models appears in many important areas such

as manufacturing system, communication system, Multi-programmed

computer system, congestion, hospital facilities, machines, serial production

lines, satellite networks, air traffic control, maintenance and repair facilities,

scheduling networks of queues etc. The development of this area can be seen

in Klein rock (1963, 1975), Gelenbe and Pujolle (1986) and War land

(1988). There are many methods used for solving queueing networks models.

Approximation methods, decomposition methods, reduction method, method

of stages, isolation methods, parametric methods, Markov renewal approach ,

perturbation analysis central limit theorem, moment generating function

approach, matrix-geometric and recursive algorithms, heuristic algorithm, etc.

First, we briefly review the development of the two-stage queueing networks

models. Due to the fact that nearly all the multi stage queueing networks

models are extensions of these or at least based on them. 1.2.6 - TWO STAGE QUEUEING NETWORKS Tandem queueing systems in which the output from one queueing process

serves as the input to another have been studied by many authors. The first

work on sequences of queues in tandem starts with the work of Jackson

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(1954, 1956) and Taylors and Jackson (1954) Taylors and Jackson studies an

open queue in tandem system with finite number of customers. They were

firstly introduced the idea of cyclic queues. Jackson (1954) designed the

differential-Difference equations characterizing the system of two queues in

series and obtained the steady state solution: -

n1 n2

λ λ

P (µ1, µ2) = µ1 µ2 P0, 0

Where P0,0= 1- λ 1- λ

µ µ - (1)

The average number of customers in the system is

λ/µ1 + λ/µ2

1 – λ / µ 1 –λ/µ2 - (2)

The average number of customers waiting for service in the system

(λ / µ 1) 2 + (λ / µ 2)2

(1 – λ / µ 1) (1 – λ /µ 2) - (3)

Probability that there are n customers in phase one is

n

λ (1 – λ / µ 1)

λ/ µ 1 - (4)

Probability that there are n customers in phase two is

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n

λ (1 – λ/ µ 2)

µ 2

Jackson (1954) was considered the first to introduce the concept of “Product

form” in the theory of queue. He also studied the problem of two queues in

tandem with restricted number of customers.

Hunt (1956) studied some exponential service cases for a limited number of

servers and obtained few results concerning the whole system.

Suzuki (1963) studied a queueing system consisting of two servers in tandem

with infinite number of customers before the two servers. He derived the

queue size and waiting time distributions at the second server. He also

considered in (1964) a tandem queueing model with blocking where he

studied a Markov chain imbedded in the process. The transient solutions of

the above models were obtained by Prabhu (1967). Tandem queues with

blocking can be studied in terms of an imbedded semi-markov process by

Neuts (1968). Further studies for such types of systems with different

assumptions for the arrival processes and the service times were studied by

Langaris (1986).

There are some systems which consist of two service stations in tandem and

the service in both stations is performed by a single server. The server

switches from one station to the other according to certain rules. Nelson (1968) used simulation to find the mean and variance of the time in the

system for various values of the traffic intensities of each station. The steady

state solution of such system was obtained by Taube-Netto (1977).

Rosenshine and chandra (1975) developed approximation solution for a

two-stage tandem system and obtained the average steady-state queue

length in four different but related tandem queues that arise in connection with

20

the service system of an air terminal complex. The approximation expression

obtained is validated by simulation Shimshak and Sphicas (1982) studied

covariance that exists in the dependent departure interval and its effect on

waiting times. They also studied a two station tandem system where the

convenience of the departure interval from the first station is known a prior.

Kumar (1980) studied a system consisting of two Bi-series queues in which

the service of a customer can be done in one or both the service stations and

the order of service is immaterial. He assumed Poisson arrival and

exponential processing times at both stations. He obtained the P.d.f. of the

queue length in terms of Laplace transforms with the aid of generating

function technique.

Friedman (1965) considered tandem system with multiple servers at each

station and infinite queues between the stations. He proved that service time

at each station is deterministic for an arbitrary arrival process; the epoch at

which the customer departs from the system is dependent of the order of the

station. A single server at each station and the service time at all stations are

exponentially distributed is considered by Weber (1979). Layton (1986) gave

a different proof for the interchangeability of tandem queues using a

sophisticated coupling technique. Anathema (1987) gave a different proof

using stochastic intensity and non liner filtering theory.

Another proof using random walk arguments is given by Tsoucas and Warland (1987), Kijima and MakiMoto (1990) considered the case of two

exponential single server queues in tandem and they gave a direct proof of

interchangeability with slight extension of the result. All these authors

assumed infinite queues between stations and there no possibility of blocking,

Chaos and Pinedo (1992) studied the effect of the order of service stations

on the departure process in a tandem system with blocking. They proved a

similar result for the reversibility of these stations tandem system with

blocking. They proved a similar result for the reversibility of a three stations

tandem system with blocking.

21

1.3 - NOME CLATURE Initially the standard notation was given by D.G. Kendall (1953) in the form

a/b/c and is known as Kendall notation. Later A. M. Lee (1966) added the

symbol d and e in Kendall notation. Thus a queueing system is described by a

series of symbols and slashes seen as (a/b/c); (d/e/f) where ‘a’ indicate the

arrival on (inter arrival) distribution; ‘b’ indicates the departure (or service time)

distribution, ‘c’ indicates the number of parallel service channels in the

system. ‘d’ indicates the maximum number allowed in the system (in service

and in queue). ‘e’ indicates the queue discipline and ‘f’ indicates the calling

sources. For example:

M/D/1: FCFS / N/∞ means a system with Poisson arrival deterministic

service time and one server in the facility. The service discipline is first come

first serve. The capacity of the system is equal to N (queue + Service). The

source generating the customers has an infinite capacity.

Table 1.1 Some Queueing Notation Table

Arrival time distribution M Passion or exponential

‘a’ distribution

D Deterministic or constant time

EK Erlangian distribution with parameter

K (K=1.2) or gamma distribution

GI General independent distribution

Service- time distribution M Exponential or Poisson

‘b’ distribution

D Deterministic on constant time

Ek Erlangian distribution with parameter

Characteristics Symbol Explanation

22

K (K=1, 2) or gamma distribution

G General Distribution

Number of parallel Servers 1,2, - -,∞ 1,2,- -∞ server in the system

‘c’

Queue Discipline FCFS First come First serve

‘d’ LCFS Last come First serve

SIRO Service in random order

FCLS First come Last serve

Pri Priority

G.D. General Discipline

Maximum number N Finite N (N = 1, 2, 3, - - -)

Allowed in system

(in service + in queue) ∞ Infinite ∞ or capacity of the system ‘e’

calling

(in put) N Finite or Infinite

Source ‘f’ ∞ 1.4 - QUEUEING METHODOLOGY Stochastic process are commonly used for modeling queues, which are

formed in a number of real life situation namely in communication system,

Cellular radio system, flow of traffic in computer system etc. Brief accounts of

stochastic process are as follows: -

1.4.1- STOCHASTIC PROCESS A stochastic process is a family of random variables indexed by a parameter

set realizing values on another set known as the state space. Stochastic

process is process in which the state of changes with a parameter.

23

Mathematically, a stochastic process x(t), is a collection of random variables

defined for every t∈ T i.e. where T is the index set of the stochastic process.

In discrete time stochastic process T can be taken as the set on integer

values or in the continues time stochastic process T can be taken as the set

of continuous values

T = (- ∞ < t < +∞) in which case we have a continue time stochastic process.

The particular value of x (t) at time, t1p, x (t1) is a random variable, it has

associated with it some probability distribution function of the form, F(x1, t1) =

P [x (t) < x 1]. The stochastic Process is completely described, if for every t1, t2,

…. tk, we know the joint distribution functions.

F (x1, x2 - - xt, t1, t2, tk) = P [x(t1), < x(t2) - - x(tk) < xn]

We shall see below that for a certain general class of processes a limited

knowledge of the past history of the process is sufficient to describe F(x1, x 2 -

- x k, t1, t2 - - tk). A stochastic to process is said to be stationary, if for every t1,

t2 - - tk and any h in T, the K- dimensional vector x (t1), x (t2), - - x (tk)] [x (t1 +

h), x (t2 +h) ------- x (tk + h)] are identically distributed. The process is strictly

stationary if the collection remains invariant to an arbitrary Shift in time or

equivalently if all possible measure remains invariant to this shift. In essence,

we are saying F (x1, x2 - - xt, t1, t2, tk) = F (x (t1 + h), x (t2 +h) ------- x (tk + h)

A stochastic process is weakly stationary or second order stationary if the

covariance

Cov[x (t), x (t+h)] = R (h)

For any t and h, where–

Cov [x t), x (t+h)] = E [ x (t) - x ] [x (t + h) – x (t + h) ]

x (t) = E [x (t + h)- E (x (t +h) ]

24

Chandramouli A.B. (1993) discussed on some applied stochastic

deterministic Queueing and Scheduling models

An idea about “some stochastic models relating to engineering system has

given by Alka choudhary (1997) 1.4.2 - MARKOV PROCESS Although initially introduced & studied in the late 1960’s & early 1970’s

statistical methods Markov source. Markov modeling has increasingly popular

in the last several years. There are strong reasons why this occurred the

models are very rich in mathematical structure. Hence can from the

theoretical basis for use in a wide range of applications. A Stochastic process

is also known as Markov process. A stochastic process is said to be a Markov

process if the present state of the system is sufficient to predict the future

without knowledge of the past history of the system. Markov process whose

state space is discrete is called a Markov chain. Mathematically, a stochastic

process x(t), t ≥ 0, is said to be a Markov chain, if, for a set of points

t1, t2 - - tk - - < tn in the index set of the process, the conditional

distribution of x (tn), for given value of x(t1), x(t2), - - x(tn-1) depends on X (tn –1 )

The most recent known value i.e.

P[X (tn) ≤ Xn/ X (t1)] = X1 - - -, X (tn-1) = Xn1]

= P(X (tn) ≤ Xn / X(tn-1) = xn-1) 1.4.3 - NON- MARKOVIAN PROCESS A process is non-Monrovian if what happens in the (t, t + dt) depends not only

on dt and the state of the system at time t, but also on one item of the

previous history of the process. For instance consider some object which may

for instance be an electric bulb, a machine on a living organism with a general

distribution of life time, when the life times these objects are over, these will

be replaced by new ones, we may then ask about the probability distribution

of the total number of replacement that have taken place in any specific time.

25

The stochastic process in continuous time is clearly non-markovian process in

general it is perhaps nothing that non markovian process can be made

markov over continues time by adopting a more comprehensive. Definition of

system state infect from a physical point of view it is eminently reasonable to

visualize a non-markovian process as the projection of a markovian process.

1.4.4 - MARKOV CHAIN A Markov process with discrete state space is said to Markov chain.

Mathematically, a stochastic process {xn, n = 0, 1, 2 - -} is called a markov

chain if, for, j.k.

j1, j2- - - - - - jn-1 ∈ N

P [xn = K / xn-1 = j, xn-2 = j1, - - - x0 = jn –1 ]

= P [xn = K / Xn-1 = j] = Pjk

If it is independent on the chain is said to be homogenous and if it is

dependent on the chain is said to be non homogeneous.

A fundamental paper of Imbedded Markov Chain Analysis of a Waiting Line Process

in continuous time is given Gaver, D. P.(1959).

1.5 - QUEUE DISCIPLINE This will provide technical discussion of active queue memory management.

The discipline is the rule determining the formation of the queue, the manner

of the queue of the customer behavior behaves while waiting and the manner

in which they are chosen for service e.g.

I. First Come First Serve (FCFS) or First In First Out :

This is the Simplest, discipline according to this discipline the

customer is served in the order of their arrival e.g. cinema ticket

window, railway ticket window etc.

26

5

5

2

2

4

46

6

3

3

1

1

Flow 1

Flow 2

Flow 3

Flow 4

Flow 5

Flow 6

Flow 7

Flow 8

Port

Fig 1.4

II. Last Come, First Serve (LCFS) or Last In First Out (LIFO) :

This Discipline may be seen in big god owns where the item which

come last are taken out first.

III. Service In Random Order (SIRO) :

An extremely difficult queue Discipline to handle might be “might be

right”. These are useful in allocation of an item whose demand is

high and supply is low e.g. allotting the share to applicant by a

company.

IV. Priority queue (PQ)

Priority queue is the basis for a class of queue scheduling algorithm that

is designed to provide a relatively simple method of supporting

differentiable service class.

Flow 1

Flow 2

Flow 3

Flow 4

Flow 5

Classifire

Flow 6

Flow 7

Flow 8

PortScheduler

Highest Priority

Middle Priority

Lowest Priority

Fig 1.5

V. Fair queueing (FQ)

FQ is the foundation for a class of queue scheduling disciplines that are

designed to ensure that each flow has fair accesses to network resources

and to prevent a bursty flow from consuming more than its fair share of

27

output port bandwidth.

Flow 1

Flow 2

Flow 3

Flow 4

Flow 5

Classifire

Flow 6

Flow 7

Flow 8

Port

Scheduler

Fig 1.6

VI. Weighted fair queueing (WFQ)

1.6 - SYSTEM CAPACITY There are some physical limitations to the amount of waiting room in queueing

process so that when line reaches a certain length, no further customers are

allowed to enter until space becomes available by a service completion.

These are referred to as finite queueing process, i.e. there is a finite queueing

process i.e. finite limit to the maximum queue size. A queue with limited

waiting room can be viewed as one with forced to balking where a customer is

forced to balk if he arrives at a time when queue size is at its limit. This is a

simple case of balking. Since it is known exactly under what circumstances

arriving customers must balk.

1.7 - NUMBER OF SERVICE CHANNELS The number of service channels gives the number of parallel service station

which can service customers simultaneously.

28

Fig 1.7 depicts the variation of multiple channel system

000 0 0

000 000 0 000 0 Fig 1.7 - Multi channel queueing system The two multi channel system differ in the respect that the first has a single

queue, while the second allows a queue for each channel. A cafe shop with

many chair- table is an example of the first type of multi channel system

(assuming no customer is waiting for any particular tables), while a super

market fits the second description. It is in general, assumed that the service

mechanism of parallel channels operate independently of each other. 1.8 - NUMBER OF SERVICE STAGES A queueing system may have only a single stage of service such as the cafe

shop and super market example, or it may have several stages. An example

of a multistage queueing system would be a physical examination procedure,

where each patient must proceed through several stages, such as medical

history, ear, nose and throat examination, blood tests, electrocardiogram, eye

examination, and so on. In some multistage queueing processes where

quality control inspection are performed after certain stages and parts do not

meet quality standards are sent back for reprocessing. A multistage queueing

system with some recycling is depicted in fig 1.8, 1.9, 2.0.

29

0 0 0 00 00 0 0 0 0

Fig 1.8- Multistage queueing system

Fig. 1.9 - Multi Channel queue with single phase multi channel,

multi phase queue discipline

ServiceIV

output ServiceV

Fig. 2.0 - Multi channel multi phase queue discipline

These characteristics of queueing system detailed in this section are generally

sufficient to describe completely a process under study. One can see from the

discussion thus for that there exists a wide variety of queueing system that

can be uncounted. Before performing any mathematical analysis, how ever, it

is absolutely necessary to describe adequately the process being modeled

knowledge of these characteristics is essential in this task. It is extremely

Input

Service I

Departure

Input Queue II

QueueIV

Queue III

Queue I

ServiceIII

ServiceII

ServiceI

Queue I

Queue II

Queue III

Service II

Service III

QueueIV

Output

30

important to use the correct model or at least the model that best describes

the real situation being studied. A good deal of thought is often required in this

model selection procedure. For example, let us reconsider the supermarket

mentioned. Suppose there are checkout counters if customers were to choose

a checkout counter on a purely random basis (Without regard to the queue

length in front of each counter) and never switch lines (no jockeying then we

truly have independent single channel models, if on the other hand, there is a

single waiting line and when a checker becomes idle, the customer at the

head of the line (or with the lowest number if numbers are given out) enters

service. we have a c-channel model. Neither of course, is generally the case

in most super market. What very often happens is that queues from in front of

each counters but new customers enter the queue which is the shortest (or

has shopping costs which are lightly loaded). Also, there is a good deal of

jockeying between lines. Now the question becomes which choice of models

(C independent single channels or a single C – Channel) is more appropriate.

if there are complete jockeying, the single c-channel model would be quite

appropriate since even though in reality there are c lines, there is little

difference, when jockeying is present between there two cases. This is so

because no servers will be idle as long as customers are waiting for service,

which would not be the case with c truly independent single channels. As

jockeying is relatively easy to accomplish in supermarket, the c channel model

would be the more appropriate and realistic model rather than c single

channels model, which one might have been tempted to select initially prior to

giving much thought to the process. Thus is important not to jump to hasty

conclusions but to choose carefully the most suitable model.

31

SECTION B SCHEDULING THEORY

The present thesis deals with queueing and scheduling problems. Scheduling

theory involves the allocation of limited recourses over time. There are two

types of scheduling/sequencing problems. One type is called deterministic

and other is called stochastic. When the processing for job to be operated on

a given set of machines is prescribed in hours or other time units, the

problems are called deterministic. The scheduling problems are called

stochastic when processing time for job follow certain probability distribution.

In a totally deterministic environment the scheduling problems were originally

discussed by Johnson’s (1954). Later on his work was generalized by

several authors and researches in different context since the work of Johnson

a good enough work has done work generally new concept in deterministic

theory. The idea of equivalent job for job-block transportation time, arbitrary

lags times, the weighted job etc, have been introduced to consider more

practical flow- shop scheduling problems.

The scheduling problems usually arise in the production concerns, where the

production of some items is made into distinct but successive stages. At each

stage there is a machine to perform the required set of jobs; scheduling

problems involve placing jobs in a certain order (sequence). For example in a

job shop there are n jobs and different machines and each job must be

processed on m machines in the same order which no passing between

machines. The time between the start of first job on the first machines and

completion of the last job on the machine is minimum. The flow shop

scheduling problems is known as the determination of the sequence in which

two or more jobs should be processed on two or more machines in order to

optimize some measure of effeteness. S.M. Johnson (1954) and Bellman (1956) consider a problem involving flow shop scheduling of n jobs on two

machines. Mitten (1959) and Johnson (1959) treated a scheduling problem

with arbitrary time lags. Maggu and Das (1980) considered 2xn flow shop

problem where in transportation time of job from one machine to another are

32

assumed to occur. Other authors like I.Nabeshima (1963), P. Gilmare and

R.E. Gomory (1964), E.M. Havner (1969) P.C.Bagga (1970), P.L. Maggu &

Das (1970) K.G.Ramurthy (1971), Maggu, Das & Kumar (1981).

I. Nabeshima & S. Maruyama (1984) T.Blacewicz, G.Finike, R.Haupt & G.

Schmidt (1988) S.S. Panwalkar (1991) T.P.Singh (1993) T.P.Singh &

A.B.Chandarmouli (1994) etc. discussed the two machine n jobs problem

According to Johnson, consider a set of j = (1, 2, 3, - - - -n) of n jobs where A

and B are two machines and i be the processing time where Ai ≥ 0 and Bi ≥

0. Each job i must start processing on B after completing to process on A.

then the problem is to find an optimal schedule minimizing the maximum job

completion time.

The assumptions made about sequencing / scheduling problem up to the

recent time studies are as follows:-

1- A machine is supposed to be continuously available for the assignment

of jobs.

2- No significant division of time scale in shift or days for the machines is

considered.

3- No temporary availability of machines is assumed to meet certain

causes of non functioning of machines due to their breakdown or

maintenance except study due to Maggu (1981)

4- No partition of jobs is supposed to be allowable and this is only one

machine in the shop for each type of job except study due to Ikram and Maggu (1976).

5- Each machine can handle at most one operation at a time.

6- Pre-emtion is not allowed that is once a job is started it is performed to

completion.

7- The item interval for processing is independent of the order in which

jobs are done.

8- Each item must be processed through a given set of ordered machines

that is the technological ordering of machines for operation for the job

is fixed.

9- The transportation time of jobs from one machine to other is supposed

to be negligible except the recent study due to Maggu and Das (1980).

33

10- All jobs are supposed to be given and are supposed to start processing

before the period under consideration begins.

11- A job is assumed to be held for the time till the next machine takes it

up, that is, in process inventory for jobs is allowed except the study

due to Ikram (1977).

12- There is no priority job. Each job is assumed to be completed at any

place of the optimal sequence except the recent study by Das (1978).

13- A job is an entity, that is, even through the item, has a lot of individual

parts, no individual part of the lot may be processed by more than one

machine at a time.

14- The set up times are assumed to be included in the processing time for

different jobs.

15- Jobs are supposed to be completed individually and not in blocks

except the recent studies made by Maggu and Das (1977).

1.9 - VARIABLES THAT DEFINE A SEQUEUING

SCHEDULING PROBLEM

The relevant attributes of job i that are given as part of problem description

are denoting by the following variables

ri is the ready time, release time or arrival time. This is the at

which the job is released to the shop by the external job

generation Process.

di is the due date. This is the time at which some external agency

would like to have the job delivered back after completion.

ai is the difference di - ri and is called the total allowance time for

the job in the shop.

Pi is the total processing time for the job.

34

Wij is the waiting time preceding the jth operation of the ith job’

the time that the ith job must wait after completion of (j-i) th

operation and before beginning the jth operation.

Wi is the total waiting time for all operation of the job i.

Ci is the completion time of job I, the time at which processing of the

Last operation of the job is completed, it is seen that

Ci = ri+ pii + wi Fi is the flow time of job i, the total time that the job i has been

through all the operation

Fi = ci – ri = Pi + Wi

Li is the lateness of the job i. it can be seen that Li = c i – d i

Ti Tardiness’ of job i.

Ei is the earliness of jobs i

Ei = Max (0, - Li)

1.10 - Nome Cloture

N = Set of 1, 2, 3, - n jobs

M = Set of M Machines 1, 2, 3, - m or ABC

Tij = Processing time of ith job on jth machine.

Ai = Processing time of i th job on machine A

Dn(s) = ∑ x = Total idle time of last machine for n jobs in a

sequences

β = job block ( i, j,)

35

T = Total elapsed time.

1.11 - METHODS FOR SOLUTION The fundamental paper on n – job, 2- machine flow shop problem was given

by S.M.Johnson (1954). Other author like L.G.Mitten (1959), A.S.Manne

(1960), P.P. Talwar (1967) etc. discussed two-machine, n-jobs problem. n–

Job, 2- machine scheduling flow shop problem the most fundamental paper in

the field of scheduling is given by Johnson (1954). He gives an algorithm for

sequencing n job, all simultaneously available, on 2- machine flow shop so as

to minimize the maximum flow time. Its significant not only for its contents but

for the general acceptance of minimizing the maximum flow time as a criterion

for general job shop problem. Johnson’s accomplishment is not so much the

proposition of an algorithm as it is the offering of a proof of that obvious

algorithm which is fact is optimal.

Johnson’s assumptions are given as follows:-

Ai = is the time required by job i on machine A

Bi = is the time required by job i on machine B

Ti = is the total elapsed time for job 1,2,3,4, ----n

The problem is to determine a sequence (i1, i2, - - - in) where (i1, i2,- - - in) is a

permutation of integers through n which will minimize T.

Johnson investigated that on optimal sequence is obtained by the following

rule:-

Job j proceed job j+1 in an optimal sequence with regard to minimum total

elapsed time

Min (Aj, Bj+1) ≤ Min (A j+1, Bj)

Johnson showed that the result is transitive and its importune lies in the fact

that, it indicates the starting with any sequence. So the optimal sequence can

be obtained by successive interchange of consecutive jobs applying the

above rule.

Johnson’s procedure for finding an optimal sequence may also describe as;

36

Select the smallest processing time occurring in the list A1, A2, - - - - - An, B1,

B2 - - - -Bn. If there is a tie select the processing time arbitrary among

Ai’s and Bi’s.

(i) If the minimum processing time is Ai, do rth job first, if it is Bi, do the sth

job last. Note that this decision would apply on both machines A and B.

(ii) There are n-1 jobs left. Apply step (l) and (ii) separately deleting the

selected jobs and continue in this manner till all jobs have been

ordered.

The resulting sequence of jobs will be optimal with minimum make span.

Ikram (1977) finds that the interchange of jobs i0 by j0 in the Johnson’s

optimal schedule in processing of n-jobs on two machines A and B in the

order AB does not alternate the optimality if the following cancellations hold.

(i) Ai0 < Bi0

(ii) Aj0 < Bj0

(iii) Aj0 > Ai0

(iv) Aj0 - Ai0 = Bj0 – Bi0

(v) Aj0 ≤ Ig

Where Ai and Bi have their usual meaning and IB is the idle time for machine B

according to the schedule S.

Bhatnagar, Das and Mehta (1979) in their paper described method of

scheduling n-jobs on 2- machines which yields a sequence to give minimum

total waiting time for all jobs. They summarize the following theorem. n jobs

1,2,3,4,..n be processed through two machines A,B in order AB with no

passing allowed the processing times satisfy the structural relationship max

tiA < max tiB where tix is the processing tine for job i on machine x (x=A,B i =

1,2,3,...n) then for any n-job sequence S=(α1α2, α3,- - αn) the total waiting

time Tw (say) is given by

nTw = ntα1A + ∑ (n-r)Xαr = ∑tiA

37

r=1 with Xαr = tαr β - tαrA and αr = 1,2,3,4 .n

They give an algorithm laying the rules to find out sequence with minimum

total waiting time.

1.12- EQUIVALENT JOB CONCEPT It is suppose that in a general “n-job 2-machine” flow shop problem every

possible job schedule is feasible; so that which ever best served a given

objective can be selected. This concept consider a more practical scheduling

situation in which certain ordering of jobs are prescribed either by

technological constrains or by externally imposed policy.

For example, consider the case of a flow job-shop in which there are two

machines say printing and binding. The items to be processed are

manuscripts of different languages and categories. Suppose manuscripts of

same category in two different languages (say Hindi and English in this order)

are to be stitched in to a single binding in the form of a book at the binding

machine. This gives an idea of introducing an equivalent job for a job- block of

district but finite jobs. In the above physical striation j1 denotes manuscript of

the same in Hindi and j2 denotes manuscript in English of the same subject

(say statistics). Then the ordered pair ( j1, j2) called job-block has been

designated by a single job β in the sense that β can be considered as

one equivalent job in the book of statistics containing both type of version in

the prescribed order on the other hand one can also say that job j2 can not be

processed before j1 due to technological conditions as a mater of policy that

Hindi version should appear before English version in the same book . It is

further assumed that no more items are to be processed in between j1 and j2

on all the machines.

The concept of equivalent job β for a job block (j1,j2) has been recently

inducted in to the scheduling theory by Maggu and Das (1977). The optimal

algorithm has been described as consequence of e1uivlent job for a job block

theorem due to mugger and Das (1999), which is, describe as follows:

38

In processing a schedule S = (α 1, α 2, α 3,- -α k-1, α k, α k+1,- - α n ) of n -

jobs on two machines A and B in the order AB with no passing allowed, the

job block (α k, α k+1) having processing time (Aα k, Bα k, Aα k+1 Bα k+1) is

equivalent to the single job β (called equivalent job β )

Now the processing time of job β on the machines A and B denoted

respectively by A β and B β are given by

A β = Aα k – Aα k + 1 – Min (Bα k, Aα k+1)

B β = Bα k + Bα k + 1 – min (Bα k, Aα k +1)

Maggu and Dass (1977) described two types of job block first is called fixed –

order and other one is called arbitrary order job block. However in each case

the jobs belongs to same group and there is no job to be operated in between

the two jobs or more jobs in a job block.

1.12.1 - Equivalent Job-Block Theorem

In the processing of a sequence S=(α 1, α 2, α 3,- - α k-1, α k, α k+1,-

- - - α n ) of n jobs on two machines A and B in the order AB, the ordered job

pair, (α k, α k+1) with processing times (tα kA, tα kB, tα k+1A, tα k+1B) have the

equivalent job with processing times (tβA, t βB) defined by t βA = t α kA+ + t α k+1A – min (t α kB t α k+1A)

t βB = t α kB + t α k+1B - min (t α kB, t α k+1A

Proof: we can consider for sequences the following relations.

Tα kB = max (T α kA, Tα k-1B) + t α kB

39

Tα k +1B = max (Tα k A + tα K+1B, Tα k + t α kB, Tα k +1 B+ t α kB)+ t

α K+1B

Tα k +2B = max (Tα k A + tα K+1A+ tα K+2A, Tα k A + t α K+1A+

t α K+2B, Tα k A + tα KB+ t α K+1B, Tα k-1B+ tα kB+ t α k+1B)+ tα K+2B

Since

Max (T α k A + t α K+1A+ tα K+2B, Tα k A + t α KB+ t α K+1B )

= Max [ (Tα k A + max (t α K+1A, t α kB ) + t α K+1B ]

We have

Tα k +2B = max [ T α k A + t α K+1A+ t α K+2A, T α k A + max(tα K+1A, tα kB)

+ tα K+1B, T α k-1B. t α KB. t α kB ) + t α K+1B ]+ t α K+2B (1)

Tα k +2A = T α k-1A+ T α k A + t α K+1A+ t α K+2A (2)

Now define the sequence

S’= {α 1, α 2, α 3,- - α k-1, β, α k+1,- - - - α n } Where, tβA = tα KA + tα K+1 A –C (3)

tβB = tα kB + tα K+1B – C (4) where C is a constant.1

Let T’pq denote the completion time of job p on machine q in sequence S’,

So that

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T’βB = max [ T’βA T’α k+1B ] + t βB

T’α K+2B = (T’βA + tα K+2A. T’βA + tβB + tα K+2B ) (5)

Now T’βA = tα K-1A + t βA

= T’α k-1A + tα KA + tα K+1A – C (from (3))

= Tα k-1A + tα KA + tα K+1A – C ( As T’α k-1A = Tα k-1A )

= Tα kA + tα K+1A – C (6)

Hence , from (3),(4) and (6) we have T’α K+2B = max (Tα kA + tα K+1A-C + tα K+2A, Tα kA +

tα K+1A- C + tα KB + tα K+1B - C, T’α k-1B + tα K B + tα K+1 B – C ) + tα K+2B (7) Let C = min (tα K+1A, tα KB) (8) Then tα K+1A –C + tα KB = max (tα K+1A. tα KB) (9) Also T’α K-1B = Tα K-1B (10) Hence from (7),(8),(9) and (10) we have T’α K+2B = max (Tα kA + tα K+1A + t α K+2A- C. T’α kA + (t α K+1A+ t α KB ) + t α K+1B- C ,Tα K-1B + t α K B + t α K+1 B - C )+ t α K+2B

= max [ Tα kA + t α K+1A + t α K+2A. Tα kA+ (t α K+1A+ t

α KB ) + t α K+1 B. Tα K-1B + t α K B + t α K+1 B ] +tα K+2B

- (11) Hence from (1) and (11), we have

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T’α K+2B = Tα K+2B – C (12) It is obvious that

T’α K+2A. = Tα K+2A+ t’ βA+ t α K+2A From (3) substituting the value of t βA, we have Tα K+2A= T’α K-1A. + (t α KA + t α K+1A - C) + t α K+2A Tα K+2A= Tα K-1A + (t α KA + t α K+1A- C) + t α K+2A (13) Since T’α K-1A =Tα K-1A From (2) and (13) we have T’α K+2A=Tα K+2A-C (14) Here relations (11) and (14) are similar to (9) and (12), therefore, we can

Define job β in S’ as equivalent job for job-block (α k, α k+1) of order job

Pair α k, α k+1 of S. One can easily extend the concept of k= 2 job block to a

job block consisting of any number k of jobs. It can be verified that the

equivalent of job blocks is associative, but not commutative i.e., the equivalent

job for block (α 1, α 2, α 3,- - α k+1) is unique, but the equivalent jobs for

block (α k, α k+1) , (α k-1, α k) are not always same. Further Bellman (1956)

and Johnson (1954) considered a problem involving the scheduling of n – job

on 2- machines. Mitten (1959) and Johnson (1959)

Maggu and Das (1980) considered 2× n flow shop problem where in

transportation of jobs from one machine to another are assumed to occur.

Now the concept of ‘Transportation times ‘as introduced by Maggu and Das

(1980) is given n the following theorem.

THEOREM:-

Consider a problem involving the scheduling of n- job on 2- machines. In their

formulation, there are given two machines A, B and a set of n jobs. Also given

are processing time ti , from machine A to machine B, an optimal ordering of

jobs to minimize total elapsed time is given by the rule.

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Job i proceeds job i+1 if

Min (tiA + ti , ti+1B + ti+1) < min ( ti+1A + ti+1, tiB + ti)

1.12.2 - DECOMPOSITION ALGORITHM:-

The utility of the above theorem can be summarized In to Decomposition

Algorithm to provide us a numerical method to obtain optimal schedule of jobs

in 2× n sequencing problem wherein transportation times of jobs are taken

into account. The optimal schedule conforms to minimizing of total elapsed

time.

Let the given problem be stated in the form as follows:

Job Machine A ti Machine B (i) (Ai) (Bi) 1 A1 t1 B1 2 A2 t2 B2

3 A3 t3 B3

- - - -

- - - -

N An tn Bn

where Ai, Bi denote the processing times of job I on machine A and B and ti

denote the transportation times of job I from machine A to machine B.

1.12 - MULTISTAGE FLOW SHOP SCHEDULING WITH TRANSPORATATION TIME There is one of the deterministic scheduling theories is that the moving time

for a job from one machine to another machine in the processing course of

jobs is ignored. In this concept this assumption is relaxed, since there are

practical scheduling situations when certain times are required by jobs for

their transportation from one machine to another machine.

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In this situation we can be seen that the machines on which jobs are to be

processed are planted at different places and there jobs require additional

times in their transportation from one machine to another machine. In the form

of loading time of jobs, moving time of jobs and then unloading time of jobs.

The transportation time in which involving sum of these three time has been

firstly introduced by Maggu and Dass (1980). Here transportation time for job

i is given as the time elapsed between completion time point of job i on

machine I and its start time point on the subsequent machine 2.

Maggu and Das (1980) originally established a theorem as follows to provide

a decomposition algorithm for determining an optimal scheduler for the “2-

machine, n jobs’ flow-shop scheduling problem involving transportation time of

jobs. A heuristic approach for two machine existing transportation time and

separated setup time has given by A.B. Chandramouli (1995).

Theorem: - consider a problem involving the scheduling of n-jobs on two

machines in formulation we are given two machines A and B, and a set of n

jobs. Also given are the processing time, tix for each job i on machines x = A B

further it is given that each machine can handle only one job at a time and

each job has transportation time ti, from machine A to machine B. An optimal

ordering of jobs to minimize total elapsed time is given by the following rule.

Job i precedes job i+1 if;

Min ( tiA + ti, ti+ 1B + ti+1) < Min ( t i+1A + ti+1, tiB +ti)

Johnson’s (1954) decomposition algorithm follows as a particular case when

we assume ti = 0 Maggu, Das and Pal (1982) established a theorem, as a

consequence of which solution algorithm gives optimal sequence of m-

machine, n-jobs flow shop problem involving transportation time of jobs and

equivalent job for job- block. Theorem; - jobs 1,2,3, -- n are processed on m machines mj (j = 1,2, - m)

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in order of M1,M2,M3- - - - Mn with no passing allowed is used to denote

processing time of job i on machine M.ij let tiα → α +1 be transportation

time of ji from machine Mα to be subsequent machine Mα +1 (α = 1, 2 , - - -

m-1) let the following structural relationship hold.

Min (tis + tis → s + 1) ≥ max ( tis → s +1 + tis+ 1)

(s = 1,2,3, - - m-2)

if β denote the equivalent job for the ordered job block (α k, α k+1) of jobs

(α k, α k+1) in this order.

tβ1 = (tα k1 + Uα k ) + ( tα k+1 + Uα k+1)

min tα k+ Uα k, tα k+1 + Uα k+1 )

tβα → α +1 = 0 (α = 1,2,3, - - - - m-1)

tβr = 0 (r= 1,2,3, - - - - m-1)

tβm = (tα km + Uα k ) +( tα k+1 + Uα k+1)

min (tα km + Uα k, tα k+1 + Uα k+1 )

Ui = ti1 → 2 + ti2 + t i2 → 3 + t i → m-1

1.14 - ARBITRARY LAGS IN JOB SCHEDULING- Some arbitrary legs are adding in which the processing times of job of flow-

shop scheduling problems. These additional lags are known as “start lag” and

“stop lag”. Mitten and Johnson (1959) have been discussed about arbitrary

lags start-log and stop-log for i are defined as follows –

45

Start-lags- The “start lag (Di ≥ 0)” is the minimum time which must elapse

between starting job i on the first machine and starting it on the second

machine.

Stop lags- “stop lag (E i ≥ 0)” for the job i is the minimum time which must

elapse between completion jobs i on the first machine and completing it on

second machine. It may be start lag and stop lag times are smaller as

comparatively processing times of job.

This concept confirm to the introduction of transportation time “ ti” for the job i

in the study of n- job, 2 machine flow shop scheduling problems due to

mitten ( 1959) and Johnson ( 1959). Maggu and Das (1980) have discuss

about Arbitrary lag time which is called transportation time (ti) and it can be

defined as by Maggu and Das (1980) is the minimum time for job i which

must elapse after completion of job i on the first machine and starting it on

the second machine.

There are some following steps related to flow shop model-

(i) The manufacturing system i.e., flow shop consists of two different

machines A and B installed at different places which are ordered as AB

according to the order of production stage.

(ii) Every job is completed through the same production stage that AB

(iii) The processing time is denoted by Ai and Bi of job i onmachine

A and B.

(iv) No passing is allowed in the flow shop i.e. the same job sequence

occurs on each machine.

(v) Let (Di ≥ 0)” be the start - lag for job i.

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(vi) Let (Ei ≥ 0) be the stop- lag for job i.

(vii) Le ti ≥ 0 be the transportation time for job i.

Our problem is stated that To minimizing the total elapse time for n-job ,2

machine flow shop scheduling problem.

Following are the essential steps of optimal algorithm:

Step1 - Let t i denotes the effective transportation times as

ti = max ( Di - Ai, Ei - Bi, ti)

Step2- Let G and H be two imaginary Machine with processing

As Gi and Hi where Gi and Hi are defined by

Gi = Ai + ti

Hi = Bi + ti

Step3 -To obtain optimal sequence by Johnson’s (1954) procedure for n job,

2 machine problem on the reduced problem in step 2.

Step4 -The optimal sequence obtained in step 3 gives the optimal sequence

for the optimal sequences for the original problem particulars studies---

(i) For every job i, Di = Ai and Ei = Bi then the algorithm reduces to Maggu

and Das (1980) problem algorithm.

(ii) If either ti = 0 , or Di ≥ Ai + ti, Ei = Bi+ ti then the algorithm reduces to

the Mitten -Johnson’s problem.

(iii) If ti = 0 , Di = Ai, Ei = Bi then the algorithm reduces to Bellman’s (1956)

nd Johnson’s (1954).

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1.15 - CONCEPT OF BREAK DOWN OF MACHINE

It has been assumed that no machine fails and no disturbance in the

processing of the job. Some times it is practically not possible that machine

may work:

(i) Due to failure of electric supply from power house

(ii) Machines stop working due to failure of one or more components

suddenly

(iii) Machines are required to stop for certain interval of time due to in

component of labor or machine or some other external cause.

These factors into the account of break down interval of machine on the

completion time of the set of jobs.

1.16 - MULTISTAGE STOCHASTIC JOB SCHEDULING A flow-shop or job shop is considered stochastic if either the job processing

time or due dates are uncertain. This is case other the just because of

machine breakdowns, differences in operators abilities, operators fatigue, jobs

and flexibility in setting due dates.

Some assumption are usually made in the stochastic job sequencing is given

below –

(i) There are n jobs to be sequenced which are simultaneously available

at time zero.

(ii) M operations of each job proceed on m machines if m≥ 1 and shop is

a flow shop then the first operation proceeds on first machine and

second on second machine etc.

(iii) Operation cannot be spitted.

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(iv) The transportation time include in the processing time of an operation.

(v) A machine cannot more than one operation at a time.

(vi) Each job, once started, must be performed to completion.

(vii) Labor and machine capacity is fixed.

(viii) Unlimited in process inventory is allowed.

(ix) No operation or job may be processed on more than one machine at a

time.

Maggu, sighed, N. Mohd. And Yadav (1982) gave solution algorithm of

obtaining an optional sequence giving minimum elapsed time in n job, 2 -

machine flow shop scheduling problem in which job involve start lag and stop

lag and transportation time and breakdown time of machine. The optimal

algorithms pertaining to both the studies by Miyazaki (1980) and Maggu in (1982) conform to the objective criterion as the minimization of weighted

mean flow time of job. The present thesis introduced the concept of time of

return in the defined problem.

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THESIS AT A GLANCE

In the present thesis two types of problem are dealt. One type of

problem concerns with queueing theory with special reference to queues

in series and queues in bi-series. The other types of problems pertain

to scheduling theory this thesis is comprised in to five chapters.

Chapter I deals with introductory concepts in the queueing and

scheduling theory. The basic concepts which are involving directly and

in –directly in the study and methods of solution have been used.

Chapter II discusses a network of queues in Bi-series each linked in

series with a common queue channel under the Poisson assumption of

arrival and departures and the service times are distributed exponentially

at each channel. The study state behavior of this queue model has

been discussed.

Chapter III represent a method of n job m machine flow shop

scheduling problem involving transportation time of jobs for minimizing

the total waiting time. A heuristic method is given to find optimal or

near optimal sequence.

Chapter IV Deals with method of 2xn flow shop scheduling problem in

which transportation time and break down interval with arbitrary lags

are given. A heuristic method is given to find optimal or near optimal

sequence.

Chapter IV Concern multistage flow-shop scheduling in which weighted

jobs are involved. A heuristic method is given to find optimal or near

optimal sequence.

All chapters give solution to practical problems arising in business and

50

industries where the production process of each item is divided in to

different but successive parts for completion. Economic measures have

been devised to minimize either ideal time of machine/servers so that

total cost of productions of an item is minimum.

The thesis is conducted by attaching a bibliography at the end which

consists references of the studies and their authors that have been

used in writing thesis.