Chapter – 1

Introduction

CHAPTER – I

INTRODUCTION

This introductory chapter deals with the notion of queuing theory,

fundamental concept of fuzzy set theory, definition of fuzzy numbers,

operations on fuzzy numbers, survey of literature and the summary of the

thesis.

1.1. Fuzzy Logic

Fuzzy logic is best defined as a form of mathematical logic in which

truth can assume a continuum of values between 0 and 11. The notion that

every proposition must be either true or false is known as bivalent logic. The

central idea of fuzzy logic is that every proposition, in addition of being true or

false, can also be partially true or partially false. Furthermore, fuzzy logic

allows a given proposition to be partially true and partially false at the same

time. It is a system of expressing partial truth mathematically.

Like many theories, Fuzzy logic is understood through examples. Take

the statement, “Georgia Tech is a large school”. Bivalent logic allows this

statement to be either true or false. Fuzzy logic, on the other hand, contends

that this statement is 100% true if student enrollment is 10,000 or more, but

only 50% true if enrollment is 3,000 and 0% true if enrollment is 500.

1 In logic, the principle of bivalence is that for any proposition P, either P is true or false.

Lotfi Zadeh, a professor at the University of California at Berkeley, first

introduced the theory of fuzzy logic in a 1965 paper entitled, “Fuzzy sets”,

[78]. The philosophical foundation of Fuzzy logic can be divided into four

major contributions. The first contribution belongs to Aristotle, who in 200

B.C. proposed the “Law of the Excluded Middle”, which held that every

proposition must be either true or false. Plato was among the first to suggest

that this dichotomy did not fully describe reality. He theorized that there was a

state between true and false. In 1920, the logician Jan Lukaisiewicz proposed

the mathematics for a tri-valued logic which included the concept of fractional

truth [54]. He referred to this third logic value (beyond true and false) as

“Possible”. Lukaisiewicz’s ideas formed the basis for a wide body of research

into what eventually became known as many-valued logic.

1.2. Fuzzy Set Theory

Most of our traditional tools for formal modelling, reasoning and

computing are crisp, deterministic and precise in character. By crisp we mean

dichotomous, that is, yes-or-no type rather then more-or-less type.

In conventional dual logic, for instance, a statement can be true or false

and nothing in between. In a set theory, an element can either belong to a set or

not; and in queuing a characteristic is favorable or not. Precision assumes that

the parameters of a model represent exactly either our perception of the

phenomenon modelled or the features of the real system that has been

modelled. Generally precision also implies that the model is unequivocal, that

is, it contains no ambiguities.

Classic set theory holds that an object is either a member or a non-

member of a given set (but never both). Full membership is indicated by the

value 1, while non-membership is indicated by the value 0. Sets of this type are

known as Classic sets or Crisp sets. Fuzzy logic allows an object to have any

value on the continuum between 0 and 1 (including 0 and 1), depending on the

degree of membership of said object in a given set. This idea of partial set

membership is the cornerstone of fuzzy set theory.

As the complexity of a system increases, our ability to make precise and

significant statements about its behavior diminishes until a threshold is reached

beyond which precision and significance become utmost mutually exclusive

characteristics. Moreover in constructing a model, we always attempt to

minimize its usefulness. This aim is closely connected with the relationship

among three key characteristics of every system model, namely complexity,

credibility and uncertainty. Uncertainty has a pivotal role in any effort to

maximize the usefulness of system models.

One of the meanings attributed to the term uncertainty is vagueness.

That is, the difficulty of making sharp or precise decision. This applies to many

terms used in our day to day life such as the set of all brilliant boys, expensive

apartments, highly reputed institutions, numbers much greater than one, etc.

This imprecision or vagueness that is characteristic of natural language does

not necessarily imply less accuracy or meaningfulness.

An important point in the evolution of the modern concept of

uncertainty is the publication of the seminal paper by Lotfi A. Zadeh [78].

Fuzzy set theory is based on Fuzzy sets. A fuzzy set is a class with no sharp

boundary between membership and non-membership. Mathematically, a fuzzy

set is a set whose grade of membership falls within the real incursive interval

[0,1].

Fuzzy set theory is a marvellous tool for modelling the kind of

uncertainty associated with vagueness, with imprecision, and with a lack of

information regarding a particular element of the problem at hand. In fact, the

fuzzy principle is that “everything is a matter of degree”. Thus the membership

in a fuzzy set is not a matter of affirmation or denial, but rather a matter of

degree. Consequently, the underlying logic is the fuzzy logic.

A fuzzy set [78] is a class of objects with continuum of grades of

membership. Such a set is characterized by a membership (characteristic)

function, which assigns to each object a grade of membership ranging between

zero and one. Formally, let x be a non-empty set, a fuzzy set A in x is

characterized by a membership function A(x) for all x X, which associates

with each point in X a real number in the interval [0,1], with the value of A(x)

at x representing the “grade of membership” of x in A.

Zadeh’s ideas have found applications in computer science, artificial

intelligence, decision analysis, information science, system science, control

engineering, expert systems, pattern recognition, management science,

operations research, and robotics. The ideas of fuzzy set theory have been

introduced in topology, abstract algebra, geometry, graph theory and analysis.

Fuzzy set theory provides us not only with a meaningful and powerful

representation of measurement of uncertainties, but also with a meaningful

representation of measurement of vague concepts expressed in natural

languages. Because every crisp set is fuzzy set but not conversely, the

mathematical embedding of conventional set theory into fuzzy sets is as natural

as the idea of embedding the real number, into complex plane. Thus the idea of

fuzziness is one of enrichment, not of replacement.

1.2.1. Definition: Fuzzy Set

If X is a collection of objects denoted generically by x then a fuzzy set A

in X which is a set of ordered pairs;

A = {(x, A(x) / x X}, A(x) is called the membership function or

grade of membership (also degree of compatibility or degree of truth) of x in A

which maps X to be membership space M.

1.2.2. Definition: -cut

An -cut of a fuzzy set A~

is a crisp set A that contains all the elements

of the universal set x that have a membership grade in A greater than or equal

to the specified value of . Thus, 1αα},0(x)μ:X{xAA~α .

1

a bc Theta0

0.8

0.6

0.4

0.2

alpha=0.75

alpha=0.50

alpha=0.25

Mem

ber

ship

(T

het

a)

Fig.1.4. Triangular fuzzy parameters and its alpha-cuts

1.2.3. Definition: Strong -cut

The Strong -cut of a fuzzy set A~

is a crisp set A that contains all the

elements of the universal set x that have a membership grade in A greater than

the specified value of . Thus, 1αα},0(x)μ:X{xAA~α .

1.3. Fuzzy Numbers

Many fuzzy sets representing linguistic concepts such as ‘low’,

‘medium’, ‘high’ and so on are employed to define states of a variable. The

relevance of fuzzy variable is that they facilitate gradual transitions between

states and consequently possesses a neutral capability to express and deal with

observation and measurement of uncertainties. In the traditional sense

computing involves manipulation of numbers and symbols. But in contrast

humans employ mostly words in computing, reasoning and arriving at natural

language or having the form of mental perceptions. A key aspect of computing

with words is that it involves a fusion of natural languages and computation

with fuzzy variables. The notion of a granule plays a vital role in computing

with words. According to Zadeh [78], ‘graduation plays a key role in human

cognition’. For humans it serves as way of achieving data comparison.

Fuzzy sets, which are defined on the set R of real numbers, endow a

special importance. Membership functions : R [0,1] clearly possess a

quantitative meaning and may be viewed as fuzzy numbers provided they

satisfy certain conditions. Initiative conceptions of approximate numbers or

intervals such as numbers that are close to 5’ or numbers that are around the

given real numbers’. Such notions are essential for characterizing states of

fuzzy variables.

These fuzzy numbers play an important role in many applications

including fuzzy control, decision-making, approximate reasoning and

optimization. A fuzzy number is the fuzzy subset of the real line where the

highest membership values are clustered around a given real number. For a

fuzzy number the membership function is monotonic on both sides of the

central value. The following thesis provides the definition of a fuzzy number,

which is commonly accepted in literature.

1.3.1. Definition: Fuzzy Number

A fuzzy subset A of the real line R with membership function

A : R [0, 1] is called a fuzzy number if

(i) A is normal, i.e., there exists an element x0 A such that A(Xo) = 1.

(ii) A is fuzzy convex, i.e., A [x1 + (1-)x2] {A(x1)A(x2)} x1, x2

R and [0, 1].

(iii) A is upper semi continuous and

(iv) Supp A is bounded where supp A = {x R : A (x) > 0}

1.3.2. Types of Fuzzy Numbers

1.3.2.1. Triangular Fuzzy Number

A triangular fuzzy number A~

is a fuzzy number specified by 3-tuples

(a1, a2, a3) such that a1 a2 a3, with membership function defined as

3

32323

21121

1

A~

axif0

axaif)a)/(aa(x

axaif)a)/(aa(x

axif0

(x)μ

This is represented diagrammatically as

1

)(~ xA

0 a1 a2a3

x

Fig.1.1. Membership Function of Triangular Fuzzy Number A~

1.3.2.2. Trapezoidal Fuzzy Number

A trapezoidal fuzzy number A~

is a fuzzy number fully specified by

4-tuples (a1, a2, a3, a4) such that a1 a2 a3 a4, with membership function

defined as

1 2 1 1 2

2 3

A

4 3 4 3 4

(x a ) / (a a ) if a x a

1 if a x aμ (x)

(x a ) / (a a ) if a x a

0 otherwise

This is represented diagrammatically as

1

)(~ xA

0 a1 a2 a3

xa4

Fig.1.2. Membership function of a Trapezoidal fuzzy number A~

1.3.2.3. Piecewise Quadratic Fuzzy Number

A piecewise – quadratic fuzzy number (PQFN) )a,a,a,a,(aA~

54321 is

defined by the membership function as

otherwise0

axafor)a2(a

)a(x

axafor)a2(a

1)a(x

axafor)a2(a

1)a(x

axafor)a2(a

1)a(x

(x)μ

542

45

2

5

432

34

2

3

322

23

2

3

212

12

2

1

A~

The PQFN is a bell shaped function and symmetric about the line

x = a3, has a supporting interval ]a,[aa~ 51 . Moreover, )a(a2

1a 513 and

a3 – a2 = a4 – a3. -cut at level = 2

1 between the points (a2, a4) called cross

over points. Also the interval of confidence at level is a = {a1 + 2(a2 – a1),

a5 – 2(a5 – a4) }.

This is diagrammatically

1

)(~ xA

0 a1 a2 a3

xa4

a5

(a4,1/2)(a2,1/2)

(a3,1)

1/2

Fig.1.3. Membership Function of a Piecewise Quadratic Fuzzy Number

1.3.3. Function Principle

Function principle was introduced by Chen [12] to treat the fuzzy

diametrical operations with triangular, trapezoidal and piecewise quadratic

fuzzy number.

1.3.3.1. Operations on Triangular Fuzzy Numbers

Consider two triangular fuzzy numbers ),,(~

321 andaaaA ),,(bB~

321 bb

i addition of A~

and B~

.

B~

A~ = (a1, a2, a3) + (b1, b2, b3)

= (a1 + b1, a2 + b2, a3 + b3)

Where a1, a2, a3, b1, b2, and b3 and real numbers.

ii. Multiplication of A~

and B~

.

B~

*A~

= (a1, a2, a3) * (b1, b2, b3)

= (a1b1, a2b2, a3b3)

where a1, a2, a3, b1, b2, and b3 and real numbers.

iii. B~

= (-b3, -b2, -b1)

where b1, b2 and b3 are real numbers.

iv. Subtraction of A~

and B~

.

B~

A~ = (a1, a2, a3) - (b1, b2, b3)

= (a1 – b3, a2 – b2, a3 – b1)

where a1, a2, a3, b1, b2, and b3 and real numbers.

v. B~1

= -1B~ =

123 b

1,

b

1,

b

1

where b1, b2 and b3 are all non zero positive real numbers.

vi. Division of A~

and B~

B~A~

= (a1, a2, a3) / (b1, b2, b3)

= (a1/b3, a2 /b2, a3/b1)

where A and B are non-zero positive real numbers.

vii. For any real number K

K A~

= K(a1, a2, a3) = (Ka1, Ka2, Ka3) if K 0

K A~

= K(a1, a2, a3) = (Ka3, Ka2, Ka1) if K < 0

1.3.3.2. Operations on Trapezoidal Fuzzy Numbers

Consider two trapezoidal fuzzy numbers A~

= (a1, a2, a3, a4) and

B~

= (b1, b2, b3, b4)

i. Addition of A~

and B

A~

+ B~

= (a1, a2, a3, a4) + (b1, b2, b3, b4)

= (a1+b1, a2+b2, a3+b3, a4+b4)

where a1, a2, a3, a4, b1, b2, b3 and b4 are real numbers.

ii. Multiplication of A and B

A~

* B~

= (a1, a2, a3, a4) * (b1, b2, b3, b4)

= (a1b1, a2b2, a3b3, a4b4)

where a1, a2, a3, a4, b1, b2, b3 and b4 are real numbers.

iii. B~

= (-b4, -b3, -b2, -b1)

where b1, b2,b3 and b4 are real numbers.

iv. Subtraction of A~

and B~

B~

A~ = (a1, a2, a3, a4) - (b1, b2, b3, b4)

= (a1-b4, a2-b3, a3-b2, a4-b1)

where a1, a2, a3, a4, b1, b2, b3 and b4 are real numbers.

v. B~1

= 1B~ =

4 3 2 1

1 1 1 1, , ,

b b b b

where b1, b2, b3 and b4 are all positive real numbers.

vi. Division of A~

and B~

B~A~

= (a1, a2, a3, a4) / (b1, b2, b3, b4)

=

1

4

2

3

3

2

4

1

b

a,

b

a,

b

a,

b

a

where A~

and B~

are non-zero positive real numbers.

vii. For any real number K:

K A~

= K(a1, a2, a3, a4) = (Ka1, Ka2, Ka3, Ka4) if K 0

K A~

= K(a1, a2, a3, a4) = (Ka4, Ka3, Ka2, Ka1) if K < 0

1.3.3.3. Operations on Piecewise Quadratic Fuzzy Numbers

Consider two piecewise quadratic fuzzy numbers A~

= (a1, a2, a3, a4, a5)

and B~

= (b1, b2, b3, b4, b5)

i. Addition of A~

and B~

B~

A~ = (a1, a2, a3, a4, a5) + (b1, b2, b3, b4, b5)

= (a1+b1, a2+b2, a3+b3, a4+b4, a5+b5)

where a1, a2, a3, a4, a5, b1, b2, b3,b4 and b5 are real numbers.

ii. Multiplication of A~

and B~

By condition,

2

bbb

2

bb;

2

aaa

2

aa 423

51423

51

a3b3 = 4

)bab(a)bab(a

4

)b).(ba(a 511555115151

)bab(a

2

1),bab(a

2

1,ba),bab(a

2

1),bab(a

2

1B~

.A~

551144223342245115

where a1, a2, a3, a4,a5 b1, b2, b3,b4 and b5 are real numbers.

iii. Subtraction of A~

and B~

B~

A~ = (a1, a2, a3, a4, a5) - (b1, b2, b3, b4,b5)

= (a1-b5, a2-b4, a3-b3, a4-b2, a5-b1)

where a1, a2, a3, a4,a5 b1, b2, b3,b4 and b5 are real numbers.

iv. B~

= (-b5, -b4, -b3, -b2, -b1)

where b1, b2,b3,b4 and b5 are real numbers.

v. Division of A~

and B~

By condition,

2

bbb

2

bb;

2

aaa

2

aa 423

51423

51

3

3

42

42

3

3

51

51

b

a

bb

aa;

b

a

bb

aa

2

bb

2a

bb

2a

b

a

2

bb

2a

bb

2a

42

4

42

2

3

351

5

51

1

Then

51

5

42

4

3

3

42

2

51

1

bb

2a,

bb

2a,

b

a,

bb

2a,

bb

2a

B~A~

,

if all bi’s are non-zero.

Also,

51

1

42

2

3

3

42

4

51

5

bb

2a,

bb

2a,

b

a,

bb

2a,

bb

2a

B~A~

If all bi’s are non-zero and B~

is negative.

1.3.4. Fuzzy Mapping

Let X and Y be universes and (Y)P~

be fuzzy set Y. (Y)P~

X:f~

is a

fuzzified mapping iff y)(x,μ(y)μR~

(x)f~ , (x, y) X Y where y)(x,μ

R~ is the

membership function of a fuzzy relation.

1.3.5. Fuzzy Integrals

Let (x)f~

be a fuzzifying function from [a,b] to R R such that x[a,b],

(x)f~

is a fuzzy number and (x)f and (x)f~

αα are level curves of a fuzzifying

function (x)f~

. The integral of (x)f~

over [a,b] is than defined to be fuzzy set as

follows,

b

a

b

aα

b

aα

(x)dxf(x)dxf(x)dxf~

.

1.3.6. Fuzzy Markov Chain

Fuzzy Markov Chain can be viewed as a perception of usual Markov

Chain which is called the original of the Fuzzy Markov Chain. The transition

probability matrix of the embedded fuzzy Markov Chain by ijP~

P~ . We write

...P~

00

...P~

P~

0

...P~

P~

P~

...P~

P~

P~

P~

P~

0

10

210

210

ij

1.3.7. Zadeh’s Extension Principle

The membership function of system characteristics of the queuing model

is derived by using Zadeh’s Principle.

Let P(x,y) denote the system characteristic of Interest. Clearly when

arrival rate λ~

and service rate μ~ are fuzzy numbers, then )μ~,λ~

P( will be fuzzy

as well. On the basis of Zadeh’s extension principle, the membership function

of the performance measure )μ~,λ~

P( is defined as

y)P(x,(y)/zμ(x),μminSup(z)μ μ~λ~

YyXx

)μ~,λ~

(P~

.

1.3.8. Notations used

: Arrival rate

: Service rate

K : Limiting capacity of the queuing system.

P(n N) = PN : Probability that the system has N customers.

E[B] : Expected time the server is busy.

E[I] : Expected time the server is idle.

ρ~ : Fuzzy time the server is busy

λ~

: Fuzzy arrival rate

μ~ : Fuzzy service rate

ω~ : Fuzzy batch size

θ~

: Fuzzy vacation rate

sL = sN : Fuzzy expected number of customers in the system.

qL : Fuzzy expected number of customers in queue.

sW : Fuzzy expected waiting time in the system.

qW~

: Fuzzy expected waiting time in queue.

1.4. Defuzzification

The aggregation defined by a triangular, trapezoidal or piecewise

quadratic fuzzy number has to be expressed by a crisp value which represents

best the corresponding average. This operation is called defuzzification.

There is no unique way to perform the operation defuzzification. There

are several existing methods for defuzzification taken into consideration the

shape of the clipped fuzzy numbers, namely the length of supporting interval,

the height of the clipped triangular, closeness to central triangular fuzzy

numbers. The most popular defuzzification methods are centre of area method,

Mean of maximum method, Graded mean integration method, Height

defuzzification method and Yager Index method.

In the thesis we use Yager ranking index method to defuzzify triangular

or trapezoidal fuzzy numbers.

1.4.1. Defuzzification for Triangular Fuzzy Number

Suppose )a,a,(aA~

321 is a given triangular fuzzy number. Then the

defuzzification of the fuzzy number by graded mean integration method is

6

)a4a(a)A

~P( 321

.

1.4.2. Defuzzification for Trapezoidal Fuzzy Number

Suppose )a,a,a,(aA~

4321 is a given trapezoidal fuzzy number. Then the

defuzzification of the fuzzy number by graded mean integration method is

6

)a2a2a(a)A

~P( 4321

.

1.4.3. Defuzzification for Piecewise Quadratic Fuzzy Number

Suppose )a,a,a,a,(aA~

54321 is a given piecewise quadratic fuzzy

number. Then the defuzzification of the fuzzy number by graded mean

integration method is 6

)aa2aa(a)A

~P( 54321

.

1.4.4. Yager Ranking Index Method

The method of ranking fuzzy numbers has been proposed firstly by Jain

[35]. Since then, a large variety of methods have been developed ranging from

the trivial to complex, including one fuzzy number attribute to many fuzzy

number attributes. In a study conducted by, Chen and Hwang [17], the ranking

methods are classified into four major classes which are preference relation,

fuzzy mean and spread, fuzzy scoring and linguistic expression.

The involvement of centroid concept in ranking fuzzy number only

started in 1980 by Yager [75]. Other than Yager [75], a number of researchers

like Murakani et al. [61], Cheng [15], Chu and Tsao [23], Chen and Chen [13],

Chen and Chen [14], Liang et al. [59] and Wang and Lee [73] have also used

the centroid concept in developing their ranking index. Some of the ranking

indices are based on the value of x alone while some are based on the

contribution of both x and y values.

Yager [75] proposed a procedure for ordering fuzzy sets based on the

concept of area compensation. Area compensation possesses the properties of

linearity. A ranking index I( )P~

is calculated for the convex fuzzy number

P~

from its -cut U

α

L

αP~ P,Pα according to the following formula:

1

0

U

α

L

α PP2

1)P

~I( d which is the centre of the mean value of P

~.

Consider two fuzzy number 21 P~ and P

~, the equation )P

~I()P

~I( 21 implies

that 21 P~

P~ [75, 27].

1.5. Queuing Systems

1.5.1. Crisp Queues

Queuing theory is concerned with developing and investigating

mathematical models of the systems where “customers” wait for “service”.

The terms “customers” and “servers” are generic. Queuing theory started with

the work of Danish Mathematician A.K. Erlang in 1905, which was motivated

by the problem of designing telephone exchanges. The field has grown to

include the application of a variety of mathematical methods to the study of

waiting lines in different contexts. The mathematical methods include Markov

process, linear algebra, transform theory and asymptotic methods. The area of

application includes computer, communication, production and manufacturing

and health care systems.

1.5.2. Characteristic of queuing model

Any queuing system can be completely described with the following

characteristics.

1. Arrival (or inter-arrival) pattern of customers

2. Service pattern of customers

3. Queue discipline

4. Behavior of customers

5. System capacity

Arrival pattern represents the way in which customers arrive and join the

queuing system. It is usually measured in terms of the average number of

customers per unit time called mean arrival rate. The customer may be human

beings, machines or computer documents. Equivalently, it can also be

measured by inter-arrival time between successive customers, called mean

inter-arrival time. If the arrival pattern does not change with time, it is called

stationary arrival pattern and if it is time-dependent, it is referred as non-

stationary.

The service pattern of the servers can be measured in terms of number of

customers served per unit time, called the service rate or in terms of time

required to serve a customer, called the inter-service time.

Queue discipline represents the order in which customers are admitted

for service from a queue. It is also called service discipline. The commonly

employed disciplines are,

1. FCFS (First come, first served)

2. LCFS (Last come, first served)

3. SIRO (Service in Random order)

4. Priority – Customer is served in preference over the other

Queuing behavior of customers plays a role in waiting-line analysis.

This may be one of the following:

Jockeying - When there are parallel queues, human customers may

leave one queue and join another queue to reduce waiting

time.

Balking - Customers may not enter the queue at all because the

queue is too long or they have no time to wait.

Reneging - Owing to impatience in waiting, customers may leave the

queue.

Priorities - Some customers are served before others regardless of

their order of arrival.

System capacity represents the number of customers in the system for

getting service. It may be finite or infinite.

1.5.3. Representation of queuing models

A queuing model can be represented in the form (a/b/c) : (d/e/f)

where a Arrival distribution

b Service distribution

c Number of servers

d Queue discipline

e Capacity of the system (maximum number of customers allowed in

the system)

f Population (size of input source, may be finite or infinite)

The first three elements were designed by Kendall in 1953. The

elements d and e were included by Lee in 1966 and Taha added the element f in

1968.

1.5.4. Fuzzy Little’s Formula

This formula states that

ss W~

λ~

L~

qq W~

λ~

L~

μ~λ~

L~

L~

qs

μ~1

W~

W~

qs

1.5.5. Fuzzy Queues

In most of the real world situations, the experts often, only imprecisely

or ambiguously know the possible value of parameters of mathematical

models. Hence it would be certainly fitting to interpret the expert’s

understanding of the parameters of fuzzy numerical data, which can be

represented by means of fuzzy sets of the real line known as fuzzy numbers.

The resulting mathematical programming problem involving fuzzy parameters

would be viewed as a more realistic version than the conventional one. In the

uncertainty of the real-world problems, the fuzzy numbers play an important

role in many applications including fuzzy queue, fuzzy control, decision-

making, approximate reasoning and optimization.

The fuzzy queues are used to represent the practical situations with the

well-known established traditional queuing approaches which are rigorous but

the assumptions are frequently too far from reality. Within the context of

traditional queuing theory, the inter arrival times and service times are required

to follow certain distributions.

However in many practical applications the arrival pattern and service

pattern are more suitably described by linguistic term such as fast, slow (or)

moderate, rather than probability distributions. Restated, the inter arrival times

and service times are more possibilitistic than probabilistic. If the usual crisp

queues were extended to fuzzy queues [64] queuing models would have even

wider applications.

1.5.6. Finite Capacity Queues

A queue in which the capacity of the system is limited to K, with single

server and service takes place on First Come First served basis. The

performance measure of Ws is given by

Ws = ρ))(1ρλ(1

]kρ1)ρ(kρ[11k

1kk

1.5.7. Queues with Multiple Servers

A queue is one in which service takes place more than in a single server

simultaneously with fuzzified exponential batch-arrival and service rates. The

performance measure of Ws is given by:

Ws = xE[A])-2xE[A](cy

y)(x,n)pn(c2y1)])E[A(Ax(2E[A]1c

0n

n

1.5.8. Bulk arrival queue with fuzzy batch sizes

A queue in which customers arrive in batches with fuzzified Markovian

arrival and service with single server and the service takes place with First

Come First Service basis. The various system characteristics are as follows.

Lq = xE[K]}2y{y

yE[K]}]2x{E[K]}x{yE[K 22

Ls = xE[K]}2{y

}(E[K]x{E[K] 2

Ws = xE[K]}2y{y

E[K]E[K] 2

Wq = xE[K]}2y{y

yE[K])2x(E[K]]yE[K 22

1.5.9. Batch arrival queue with setup time

A queue in which the customers arrive in batches with time taken to start

the service (setup time) with arrival rate, service rate and setup rate are all

fuzzy numbers. The performance measures are given by

μθ

E(A) θ)(μ λ

E(A)] λ[μ 2

1)]-[A(A E λ

E(A)] λ-[μμ

)]λE(A[L

2

s

θ

1

E(A)] λ[μ 2E(A)

1)]E[A(A

E(A)] λ[μμ

E(A) λ Wq

Lq = Ns –

Ws = Wq +

1

E[B] = E(A)] -[

E(A)

1.5.10. Batch arrival queue with vacation policies

A batch arrival queue in which the server leaves for a vacation (single or

multiple vacation) when there are no customers in queue with arrival rate,

service rate, arrival batch size and vacation rate are fuzzy numbers. The

performance measures are given by

Policy I (server takes multiple vacation)

Lq = λθ

1

ρ)μE(A)(1 2

1)E[A(A

ρ)μ(1

ρ

Ws =

μ

1

θ

1λE(A)

ρ)μ(1 2

1)λE[A(A

ρ1

θ

λ

1 2

Policy II (server takes single vacation)

Lq =

)λλθθ(θ

θ)λ(λ

ρ)μE[A](1 2

1)]E[A(A

ρ)μ(1

ρλ

22

Ws =

μ

λE(A)

)λλθθ(θ

θ)E[A](λλ

ρ)μ(1 2

1)]λE[A(A

ρ)(1

ρ

λ

122

22

where = μ

λE(A)

1.6. Literature survey

Queuing models have wider applications in service organizations as well

as manufacturing firms, in that various types of customers are serviced by

various types of servers according to specific queue discipline [31]. A general

approach for queuing systems in a fuzzy environment is proposed based on

Zadeh’s extension principle, the possibility concept and fuzzy Markov Chains.

Developments on fuzzy simulation and on the representation of fuzzy

numbers by random variables can be used to analyze the queuing system.

Stanford [66] has introduced the concept of fuzzy probabilities and the

properties of fuzzy probability Makov chains were discussed. The work of

Wenstop [74], Leung [55], Gericke and stranbe [30], Jumarie [39], Moral [60],

and Delgardo and Moral [24] can all be applied indirectly to the analysis of

fuzzy queuing systems [62]. Li and Lee [58] have derived analytical results for

two fuzzy queuing systems based on Zadeh’s extension principle [78,79], the

possibility concept and fuzzy Markov Chains [66]. However, Negi and Lee

[63] commented, that their approach is complicated and is generally unsuitable

for computational purposes, also it is hardly possible to derive analytical results

for complicated queuing systems. Therefore Negi and Lee [63] proposes two

approaches, the -cut and two-variable simulation to analyze fuzzy queues.

Unfortunately, their approach does not provide the membership functions of

the performance measure completely. Hence C. Kao et al. [11] adopts -cut

approach to decompose a fuzzy queue to a family of crisp queues. As the

-varies, the parametric programming technique is applied to describe the

family of crisp queues and the concept is successfully applied to four typical

fuzzy queues, namely M/F/1, F/M/1, F/F/1 and FM/FM/1 where F denotes

fuzzy time and FM denotes fuzzified exponential time.

The finite-capacity queuing system has been extensively studied by

many researchers like Shi [65], Gouweleeuw and Tijms [32], Bretthaner and

Cote [5], Wagner [72], Kavusturneu and Gupta [43], Laxmi and Gupta [49],

and Kerbache and Smith [46] where in the inter arrival times and service times

are required to follow certain distributions. However, in many practical

situations, the statistical information may be obtained subjectively, i.e., the

arrival pattern and service pattern terms such as fast, slow or moderate, rather

than probability distributions. By using -cuts and Zadeh’s extension principle

[78, 79] Shih-Pin Chen [18] described a pair of parametric nonlinear programs

through which the membership functions of the performance measures are

derived.

M/M/C vacation systems with a single-unit arrival have attracted much

attention from numerous researchers since Levy and Yechiali [57]. The

extensions of this model can be referred to Vinod [71], Igaki [34], Tian et al.

[68], Tian and Xu [69], and Zhang and Tian [80,81] studied the M/M/C

vacation systems with a single-unit arrival and a “partial server vacation

policy”. Chao and Zhao [10] investigated the GI/M/C vacation models with a

single-unit arrival and provided iterative algorithms for computing the

stationary probabilities distributions. Jau-Chuan Ke, Hsin-I Huang, Chuen-

Houng Lin [36] developed a pair of parametric nonlinear programs, using

-cuts and Zadeh’s extension principle to derive the membership functions of

various system characteristic of FM[X]

/FM/C queuing models.

In practical situations the server setup may be required for the

preparatory work before starting the service. During this setup period the

lubricant work can be increasingly extended and so the unsteady conditions of

machine are reduced. Queuing models with a server setup are extensively

studied by many researchers Borthakur and Medhi [4], Li et al.[33] Lee and

Park [51], Krishna Reddy et al. [48] Choudhury [19, 20] and Ke [44]. Li et al.

[33] proposed an effective iterative algorithm to compute the stationary queue

length distributions for M/G/1/N1 queues with setup time and arbitrary state

dependent arrival rate. Lee and Park [51] examined M/G/1 production system

with early setup and developed a procedure to find the joint optimal threshold

that minimizes a linear average cost. Krishna Reddy et al. [48] studied a

N policy M[X]

/G/1 queuing system with multiple vacations and setup times for

a two stage flow line production system.

Vacation queuing systems in which the server is unavailable during non-

deterministic intervals of time have received considerable attention in

literature. Levy and Yechiali [56] first proposed an M/G/1 queuing system

under a single vacation policy when the system is empty. A study of the

variations and extensions of this single vacation model was presented by Doshi

[25] and Takagi [67]. For queuing systems with batch inputs, Choudhury [21]

successfully modelled a M[X]

/G/1 queuing system (where [X] represents

random batch arrival size) with a single vacation policy, extending the results

of Levy and Yechiali [56] and Doshi [25]. The M[X]

/G/1 queuing system with

multiple vacation, where the server goes on vacations repeatedly until it finds

at least one waiting customer at the end of a vacation, was first studied by

Baba [1]. Lee and Srinivasan [50] examined the control operating policy of

Baba’s [1] model using a general approach and presented applications in

production / inventory systems and other areas. Lee et al. [53, 52] analyzed in

detail Lee and Srinivasan’s [50] system with single and multiple vacation

policies. Jau-Chuan Ke, Hsin-I Huang, Chuen-Houng Lin [37] analyzed batch

arrival queues under single and multiple vacation using fuzzy parameters.

1.7. Motivation and scope of the thesis

C. Kao, C.C. Li, S.P. Chen were motivated and prompted by their

works. This thesis is significantly traditional way of dealing with fuzzy

queuing problems. In most cases, the system is considerably complex; we have

chosen a triangular and trapezoidal fuzzy numbers to serve as a variable for

fuzzy queuing problems. In this thesis, an effort has been made to convert

queuing problem into fuzzy queuing problem and the solution of various

system characteristics of queuing models were discussed.

1.8. Organization of the Thesis

The results embodied in this thesis are deadly concerned with the study

of queuing problems in fuzzy environment. The present thesis consists of six

chapters, and the organization of the thesis is as follows.

Chapter I is introductory in nature. In this chapter a notion of triangular,

trapezoidal and piecewise quadratic fuzzy numbers [26] which serve as a

parameter for fuzzy queuing problem is introduced. A Chronological literature

survey is presented about the on going research in this line. The motivation and

scope of the thesis have also been mentioned.

Chapter II has two parts. The first part of Chapter II forms the material

of a research paper published in Bulletin of Pure and applied sciences. Vol.26E

(No.1) 2007: P.87-97.

This part gives the membership function of various system

characteristics of FM/FM/1 queuing system with finite capacity and calling

population is infinite. The second part of Chapter II (part) forms the material of

a research paper published in international review of fuzzy mathematics,

Vol.II, No.1 (June 2008), P.47-59. This part constructs the membership

function of system characteristics of waiting time in the system for analyzing

fuzzy queues with finite capacity. This concept is discussed in two fuzzy

queues often encountered in real life.

Chapter III (part) forms the material of a research paper published in

international Journal of Algorithms computing and mathematics, Volume 2,

Number 4, November 2009, 59-64. This part constructs the membership

function of the waiting time in the system for Batch arrival queue with multiple

servers.

Chapter IV (part) forms the material of a research paper published in

special issue on IJACM, Fuzzy Math, Vol. 2 (No.3), AUG 2009: 1-8. This part

constructs the membership function of various system characteristics of Bulk

arrival queuing model with fuzzy varying batch size using mixed integer non

linear programming (MINLP) method.

Chapter V (part) forms the material of a research paper published in

Bulletin of Pure and Applied Sciences, Vol. 28E (No.2), 2009: 313-331. This

part constructs the membership functions of various system characteristics of

fuzzy queue with setup time using parametric nonlinear programs.

Chapter VI (part) forms the material of a research paper is

communicated to Reflections des ERA, Journal of Mathematical sciences. This

part constructs the membership functions of system characteristics of queuing

model with two different vacation policies, Policy I for multiple vacation and

Policy II for single vacation.