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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.
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