All of us have experienced the annoyance of waiting in a line. In our. everyday life people are seen waiting at a railway ticket booking office, a

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1 This chapter deals with characteristics of queues, fundamental concepts such as definition of fuzzy set, membership function, fuzzy numbers etc. Description about different queuing models are given in this chapter INTRODUCTION TO QUEUES All of us have experienced the annoyance of waiting in a line. In our everyday life people are seen waiting at a railway ticket booking office, a doctor s clinic, post office, bank, petrol pump and many other places for getting service. Unfortunately this phenomenon continues to be common in congested, urbanized and high tech societies. As customers we do not generally like these waits and managers of establishment at which we wait also do not like us to wait since it may cost them business. Why then there is waiting? The answer is simple : there is more demand for service than there is facility for service available. There may be many reasons : for example there may be a shortage of available servers, it may be infeasible economically for a business to provide the level of service necessary to prevent waiting or there may be a limit to the amount of service that can be provided. To know how much service should be made available one must know answers to such 1

2 questions as how long will a customer wait and how many people will form in the line. Queuing theory attempts to answer these questions through detailed mathematical analysis and in many cases it succeeds. The subject of queuing theory may seem rather esoteric to some readers. However, it must be admitted that it is just as unpleasant to spend time in a queue as in waiting line. Queuing theory is a quantitative technique which consists in constructing mathematical models of various types of queuing systems. These models can be used for making predictions about how the system can adjust with demands. Queuing theory deals with analysis of queues and queuing behavior. Queuing theory, as such, was developed to provide mathematical models to predict the behavior of systems that attempt to provide service for randomly arising demands and can trace its origins back to pioneer investigator, Danish mathematician named A.K Erlang [34], who in 1909 published The Theory of Probabilities and Telephone Conversations based on his work he did for Danish Telephone company Crisp Queues The study on queues is inevitable in today s busy world as waiting in some order to get served is a common phenomenon. Science, engineering, medicine, business, service centers, reservation counters, communication 2

3 network and computer centers are some fields where queues are formed from time to time. Queuing theory is a branch of Applied Probability Theory which attempts to explore and compare various queuing situations for an approximate optimization. As queues are omnipresent a study on the maintenance is highly essential. Operations Research can quite effectively analyze queuing or congestion phenomena. The theory permits performance measures including average waiting time in the system, expected number waiting in the system or queue, and the probability of encountering the system in certain states such as empty, full, having an available server or having to wait before service, etc. A queuing system consisting of customers arriving for service, waits for service (if not immediate) and leaves the system after receiving service. Arriving Customers Served Customers Queue Service Mechanism Waiting Customers Queue Discipline 1.2. CHARACTERISTICS OF A QUEUING MODEL These are : Six Characteristics of queuing processes are explored in this section. (i) Arrival pattern of customers. 3

4 (ii) Service pattern of servers. (iii) Queuing discipline. (iv) System capacity. (v) Number of service channels. (vi) Number of service phases. Among the various characteristics of a queuing system input/arrival pattern of customers, service pattern, number of servers, capacity of the system, queue discipline and the number of phases of service decide the strength of the system. The theory enables mathematical analysis of several related processes. The arrival pattern/input is the number of arrivals per unit time. This is expressed in terms of the mean number or mean inter-arrival time. The arrival may be single, bulk or batches of fixed or varied sizes. Service pattern may also be described by the number of customers served per unit time or the time required to serve a customer. Queue discipline refers to the rule based on which customers are selected for service. There are many queue disciplines and the commonly observed one is First in First out (FIFO). The capacity refers to the limitation on the number of customers allowed for that service. The parallel counters offering the same type of service is also important in the study of queues. A customer may receive his service in one or many stages depending on the requirement. 4

5 1.3. PERFORMANCE MEASURES OF A QUEUING MODEL In optimization, a decision maker always aims at the best choice for which certain measures will be of greater use in deciding the effectiveness of the system. Such measures include average waiting line, average number in the system and average waiting time in the system, queue and probability of encountering the system in certain states such as empty, full, having an available server or having to wait INTRODUCTION TO FUZZY SET THEORY The term fuzzy meaning vagueness is omnipresent in all real world situations. This term is no longer fuzzy to many researchers, scientists and engineers today, as the study on fuzzy sets has become one of the emerging areas in the contemporary technologies. Fuzzy can be realized in many areas of life and is particularly frequent in all areas in which human judgment, evaluation and decisions are important. From historical point of view, the issue of uncertainty was not appreciated by scientific experts. In the traditional sense of science, uncertainty represents an undesirable state which must be avoided at all costs. When dealing with real world problems, we can rarely avoid uncertainty. Leading theory in quantifying uncertainty in scientific models was probability theory which was challenged by Max Black in 1937 with his studies in vagueness. Uncertainty can be manifested in many forms ; can be fuzzy, vague, ambiguous in the form of ignorance or can be due to 5

6 natural variability. Vagueness can be used to describe certain kinds of uncertainty associated with linguistic or intuitive information. In terms of semantics even the terms vague and fuzzy cannot be synonyms as explained by Zadeh in A significant number of direct real world implementations range from home appliances to industrial installations involve fuzzy sets. Fuzzy sets emerged as a new way of representing uncertainty. As such they naturally got involved in fervor of philosophical and methodological disputes with proponents of probability and statistics. It was determined earlier that notions of randomness and fuzziness are mostly orthogonal and could eventually coexist. It would be fair to say that by their nature fuzzy sets are inclined towards multifaceted interaction with certain other methodologies. In operations research, a number of standard methods of linear programming have been augmented by fuzzy sets to handle imprecise objectives and constraints Fuzzy Set A fuzzy set is a set where the members are allowed to have partial membership and hence the degree of membership varies from 0 to 1. It is expressed as, A = {(x, (x) A )/x V) } where X is the universe of discourse and (x) A is the membership function of the universe of discourse and (x) A = 0 or 1, i.e., x is a non-member in A if (x) A = 0 and x is a member in A if (x) A = 1. 6

7 Support of a Fuzzy Set where = 0. The support of a fuzzy set is the set of all members with a strong -cut Height of a Fuzzy Set The height of a fuzzy set h{a(x) x X} is the maximum value of its membership function (x) such that A = {x X A (x) } and Normal Fuzzy Set A fuzzy set where Max {(x)} = 1 is called as a normal fuzzy set, otherwise, it is referred as sub-normal fuzzy set GENERAL FUZZY NUMBERS The description of a problem under fuzzy environment needs a conversion in terms of a quantitative measure. Data which are imprecise in nature can be dealt effectively under fuzzy set theory. For manipulation of such data like very small, low, medium, moderate, high which are linguistic terms fuzzy numbers serve as a tool more suitable for interpretation. These numbers explain the data with reference to a base variable, the values of which are real numbers within a specified range. A base variable is explained by any physical variable like temperature, pressure, speed, voltage or any other numerical variable such as age, rate of interest, salary or the like. The approximate values of the base variables specific to a particular application 7

8 can be mapped to a fuzzy number which is an interval. These fuzzy numbers play a vital role in production centers, machine tools, communication, socioeconomic problems, inventory control, etc. Though variety of fuzzy numbers exist, as far as this thesis is concerned trapezoidal and triangular numbers are used. The following sections define a few types of fuzzy numbers Fuzzy Numbers The notion of fuzzy numbers was introduced by Dubois.D and Prade. H [27]. 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 exist an element x 0 such that A (x 0) = 1 ii. A is fuzzy convex, (i.e.) [ ] x + (1 - )x (x ) (x ), x 1, x 2 R, [0, 1] A 1 2 A 1 A 2 iii. A is upper continuous, and iv. Sup A is bounded, where sup A = { x R : (x) > 0} A Definition - Fuzzy Numbers A real fuzzy number a is a fuzzy subset of the real line R with membership function a conforming to the following conditions. 1. a is a continuous mapping from R to the closed interval [0,1]. 2. a (x) = 0 x (-, a 1 ] 8

9 3. a is strictly increasing and continuous on [a 1, a 2 ] 4. a (x) = 1 x [a 2, a 3 ] 5. a is strictly decreasing and continuous on [a 3, a 4 ] 6. a (x) = 0 x [a 4, ) where a 1, a 2, a 3, a 4 are real numbers and the fuzzy number is denoted by a = (a 1, a 2, a 3, a 4 ) Triangular Fuzzy Number A cover and normal fuzzy set defined on R whose membership function is piecewise continuous, is called a fuzzy number. A triangular fuzzy number a with membership function a (x) is defined on R by (x) = a x - a1 for a x a a 2 - a 1 x - a3 for a x a a 2 - a3 0 otherwise where (a 1, a 2, a 3 ) = a Operations on Triangular Fuzzy Number The function principle was introduced by Chen [20] to treat fuzzy arithmetical operations. This principle is used for the operation of addition, multiplication, subtraction and division of fuzzy numbers. 9

10 Suppose A = (a 1, a 2, a 3 ) and B = (b 1, b 2, b 3 ) are two triangular fuzzy numbers. Then i. The addition of A and B is A + B = (a 1 + b 1, a 2 + b 2, a 3 + b 3 ) where a 1, a 2, a 3, b 1, b 2, b 3 are real numbers. ii. The product of A and B is A x B = (c 1, c 2, c 3 ) where T = {a 1 b 1, a 1 b 3, a 3 b 1, a 3 b 3 } c 1 = min T, c 2 = a 2 b 2, c 3 = max T If a 1, a 2, a 3, b 1, b 2, b 3 are all non zero positive real numbers, then A x B = (a 1 b 1, a 2 b 2, a 3 b 3 ) iii. - B = (-b 3, -b 2, -b 1 ) then the subtraction of B from A is A - B = (a 1 - b 3, a 2 - b 2, a 3 b 1 ) where a 1, a 2, a 3, b 1, b 2, b 3 are real numbers. iv. 1 B = -1 B = (1/b 3, 1/b 2, 1/b 1 ) where b 1, b 2, b 3 are all non zero real numbers then A / B = ( a 1 / b 3, a 2 / b 2, a 3 / b1 ) v. Let R, then A = (a 1, a 2, a 3 ) if Trapezoidal Fuzzy Number = (a 3, a 2, a 1 ) if < 0 The fuzzy number a = (a 1, a 2, a 3, a 4 ) is a trapezoidal number. Its membership function a is illustrated in Fig

11 (x) a 1 0 a 1 a 2 a 3 a 4 x Fig.1.1. Membership function of a fuzzy number a (x) = a (x - a 1) a x a (a 2 - a 1) 1 a x a (a 4 - x) a x a (a 4 - a 3) Operation on Trapezoidal Fuzzy Numbers Let a and b be two trapezoidal fuzzy numbers. The fuzzy arithmetical operations under function principle is given below : 1. Addition of a and b : a b = (a 1 + b 1, a 2 + b 2, a 3 + b 3, a 4 + b 4 ) where a = (a 1, a 2, a 3, a 4 ), b = (b 1, b 2, b 3, b 4 ) a 1, a 2, a 3, a 4, b 1, b 2, b 3 and b 4 are real numbers. 11

12 2. The multiplication of a and b : a b = (c 1, c 2, c 3, c 4 ) where T = {a 1 b 1, a 1 b 4, a 4 b 1, a 4 b 4 } T 1 = {a 2 b 2, a 2 b 3, a 3 b 2, a 3 b 3 } c 1 = min T, c 2 = min T 1, c 3 = max T, c 4 = max T 1 If a 1, a 2, a 3, a 4, b 1, b 2, b 3 and b 4 are all non zero positive real numbers then a b = {a 1 b 1, a 2 b 2, a 3 b 3, a 4 b 4 } 3. -b = (-b 4, -b 3, -b 2, -b 1 ) then a b = (a 1 - b 4, a 2 - b 3, a 3 - b 2, a 4 - b 4 ) where a 1, a 2, a 3, a 4, b 1, b 2, b 3 and b 4 are all real numbers b = b -1 = ,,, b4 b3 b2 b1 where b 1, b 2, b 3 and b 4 are all positive real numbers. a b = a a a a,,, b b b b Let R then i) 0, a = (a 1, a 2, a 3, a 4) 0, a = (a, a, a, a )

13 Cuts The crisp set of elements that belong to the fuzzy set A atleast to the degree [0, 1] is called the level set a = { x X (x) } / a Strong and Weak -cuts If a fuzzy set A is defined on X, for any [0, 1], the -cuts A is represented by the following crisp set, Strong -cuts : Weak -cuts : + A = { x X µ A (x) > ; [0, 1]. A = {x X µ A (x) ; [0, 1] Fuzzy Integrals Let f(x) be a fuzzifying function from [a, b] R to R such that x [a, b], f(x) is a fuzzy number and - f (x) and + f (x) are level curves of a fuzzifying function f(x). The integral of f(x) over [a, b] is then defined to the fuzzy set as follows : f(x) dx f (x) dx f (x) dx b b b - + = + a a a 13

14 Fuzzy Little s Law We introduce fuzzy set theory into Little s Law. The fuzzy number of customers in the system observed upto time t is as follows : L = W s W q = Lq 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 P = P. We ij write P = P = ij P 0 P 1 P 2... P 0 P 1 P P 0 P P Zadeh s Extension Principle The membership function of performance measures of the queuing model is derived by using Zadeh s principle. 14

15 Let P(x, y) denote the system performance measure of interest clearly when arrival rate and service rate are fuzzy numbers, P(, ) will be fuzzy as well. On the basis of Zadeh s extension principle, the membership function of the performance measure P(, ) is defined as, p(, ) x X y Y { } (z) = sup min (x), (y)/z = P(x, y) Deffuzzification Deffuzzification is the transformation of fuzzy into crisp expression (real numbers, symbols, etc.). In decision making, for instance, we want to achieve semantical correctness. The defuzzification strategies use external values of the membership function to define the crisp equivalent value. Defuzzification of triangular fuzzy number using Median Rule a 1 + 2a 2 + a3 is a = 4. Then, a is the associated crisp value of a = (a 1, a 2, a 3 ). Defuzzification of trapezoidal fuzzy number using Graded Mean Integration Representation Method is a = a 1 + 2( a 2 +a 3 ) +a4 6 where a, is the associated crisp value of a = (a 1, a 2, a 3, a 4 ) 15

16 Yager s Ranking Index Method We adopt this method for transforming the fuzzy optimal threshold into a crisp one since the method possess the property of area compensation. The recommended optimum value is calculated by O(N*) = 1 * L * U 0 (N ) + (N ) 2 dx 1.6. FUZZY QUEUES The fuzzy queues are first analysed by R.J. Lie and E.S. Lee [ 52 ] in The analysis of fuzzy queues is based on Zadeh s Extension Principle, the possibility concept and fuzzy Markov Chain. In Poisson arrival queuing system is a fairly reasonable assumption, the arrival rate and service rate are really more possibilistic than probabilistic. Furthermore, in many practical situations the parameters (arrival rate) and (service rate) are frequently fuzzy and cannot be expressed in exact terms. Thus linguistic expressions for these parameters such as the mean arrival rate is approximately 5 and the mean service rate is approximately 10 are much more realistic under these circumstances. They investigated two queuing model in fuzzy environment as M/F/1 and FM/FM/1 where the first queuing model denotes fuzzy service rate and the second queuing model, the fuzzified exponential arrivals and service rates. 16

17 In 1992, D.S. Negi and E.S. Lee [61] combined the ability of fuzzy sets to represent a practical situation with the well known established traditional queuing approach which are rigorous. But the assumptions are frequently far from real. Both cut sets and the use of random variable to represent fuzzy numbers are investigated. C.Kao, C.C.Li, S.P. Chen [41] introduced a parametric programming approach to analyze fuzzy queues in He investigated four simple 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. Fuzzy queues are much more realistic than the commonly used crisp queues. Within the context of traditional queuing theory, the arrival time and service time are required to follow certain distribution. However in many practical situations, the statistical data may be obtained subjectively. They may be more suitably described in linguistic terms like fast, slow or moderate rather than by probability distribution. Fuzzy set theory has been applied to some queuing systems to provide wider applications in instrumentation technology, information technology and communication technology Machine Repair Problem Machine repair models have wide applications in many practical situations, such as production line systems, client server computing 17

18 maintenance operations, and manufacturing systems consider a manufacturing system consisting of K machines subject to breakdowns from time to time. When a machine breaks down, it is repaired by one of a crew of R repair persons, thus this repair person cannot repair other broken machines for a period of time. Thus, during this busy period of time, it is possible that there are broken machines have to wait and are interfered with by the machine being repaired. Many studies have been published on the machine interference problem and its cost/profit analysis recent studies include, and so on. Efficient methods have been developed for analyzing the machine interference problem when its parameters, such as the breakdown rate and service rate, are known exactly. One commonly used type of solution methods is the queuing theory approach in that the machine interference problem is modeled as a finite calling population queuing system. The machine breakdowns are treated as the customers and the repair persons are servers in the system. The interest performance measures FM/FM/2/GD/4/4 are given as L(x, y) = 4 x j j! 4 x j + y j i j-r j y 2! 2 j 4 4 x j j! 4 x j = 3 j + y j-r j y 2! j = 0 j = 3 2 j = 0 j 18

19 L q = W = W q = 4 ( ) j = 3 x j - 2 4! y j 2 4 x x 4! 4! y y + j! (4 - j)! (4 - j)! 2! 2 j = 0 j = 3 L x(4 - L) L q x(4 - L ) q j j j Erlang s Loss Model Suppose that in a queuing system which has m service channels, the demand for service arises in accordance with a Poisson process with parameter a. Suppose that the servicing time in each channel is exponential with parameter b. Further suppose there is no storage capacity in the sense that a demand which arrives at the moment when any channel is free is received and is processed, whereas a demand which is received when all the m channels are busy is rejected and leaves the system. This is called a loss system. This model envisages that a unit who finds, on arrival that all the channels are busy leaves the system without waiting for service. This model was first investigated by Erlang. The performance measures used are Expected number of Busy Channels x 1 B(C, x/y) y E(B) = [ ] 19

20 Busy probability P(X = 1) = x 1 B(C, x/y) y C [ ] Tandem Queues Queues in series (or tandem), which is a simple one with 2 service stations such that a unit arrives from outside to the first station, receives service there and proceeds to second one and after receiving service there departs from the system. In open networks, the response time or sojourn time of a customer is defined as the time from its entry into the network until it exits from the network. Consider the two stage tandem network. The system consists of two nodes with respective service rates µ 0 and µ 1 which are fuzzy numbers. The external arrival rate is which is also a fuzzy number. After an exponentially distributed service time departures are routed to node 2 with probability 1 q and fed back to node 1 with probability q, where q is considered as a fuzzy number. In this model we try to fuzzify the mean and variance of sojourn time distribution in queuing network with overtaking. The performance measures are The variance sojourn time of a queuing network for a fuzzy queuing system is N 1 1 = + 2 y 1(q - 1) - xq ( y (1 - q) - x) y 1(q - 1) + xq ( y - x) 1 2 The mean sojourn time of a queuing network 20

21 L 1 1 = + y - x ( y (1 - q) - x) 1 2 Number of customers in the system L s x x = + y - x ( y - x) 1 2 Expected waiting time in the system W q 1 x x = + R ( y 1 - x) y 2 - x where R is the response time N Policy Queues with Infinite Capacity One of the well known control operating policies of queuing model is the N-policy. In N-policy the server is turned on when N 1 or more units are in presence, and off when the system is empty. In other words each time the system becomes empty, the server waits or gets busy with other works until the queue length becomes N. Then commences servicing customers one by one and continues until the system becomes completely empty. The period the server waits until the arrival of the N th customer is called server vacation. Here the performance measures are W q = N S = N N ( ) ( ) 21

22 M x /M(a,b)/1 Queue with Random Break Downs Server vacations or breakdowns are common in many queuing situations. During the server vacation periods or the repair time of the service facility the units or the customers have to wait until the server returns to the system or the system becomes operable again. In our M x /M(a,b)/1 model we assume that the customers arrive in batches of variable size with a minimum batch size a and a maximum batch size b. Such models may find applications in transportation and communication system and much other similar queuing system. Here the steady state solution of PGF is calculated as follows. If the server renders service to the customers in batches of fixed size b then with a = b we have W(Z) = F(Z) = P(Z) = b-i i 0 WiZ (Z -1) i = 0 - (Z) + b Z - (Z) + + a-1 n b WnZ (Z -1) - (Z) + i = 0 - (Z) + b Z - (Z) + + a-1 n b 1+ WnZ (Z -1) - (Z) + i = 0 - (Z) + b Z - (Z) + + These are the PGFs of FM x / FM(a,b) / 1 queue with batch arrivals, service in batches of fixed size b, random breakdowns and exponential repairs. 22

23 Queuing System with Removable and Reliable Server Consider an M/G/1 queuing system in which a removable and reliable server operates with an N policy. The term removable server stands for the system turning on and off the server depending upon the number of customers in the system. A non reliable server means that the server is typically subject to unpredictable random breakdowns. The performance measures used are E(N S ) = + ( ) where ρ = S. 2 ( R ) ( S S ) S ( R R ) N q + + q q R ( 1 + qr ) N Policy M/E K /1 Queuing Model with Removable Service Station A.K. Erlang [34] pioneered queuing systems theory for its application to congestion in telephone networks, considers an M/E K /1 queuing system with a removable service station. The term removable server states the operation policy of the system allows one to turn on and turn off the server, depending on the number of customers in the system. The server is removable and applies the N policy. That is the server operation starts only when N (N 1) customer have accumulated, and is shut down (turned off) when no customer is present. After the server is turned off, the server may not operate until N customers are present in the system. The management policy that the decision maker can turn a single service station on at the customer s arrival 23

24 epochs or off at service completion epochs is investigated. A pioneer work in this field is Yadin and Naor[79] who first introduced the concept of an N policy which turns the server on when the number of customers in the system reaches a certain number, N (N 1), and turns the server off when there is no customer in the system. The performance measured is The total expected cost function per unit time is given by G(N) = N (V + u) x (y - x) W + 2 Ny where x, y, w, v, u are the fuzzy variables. Corresponding to arrival rate, service rate, C h holding cost, C S start-up cost and C r rem ovable cost. N is the optimal management policy given by N* = 2(V + U) x (y - x) Wy 1/ M/D/n Queue This model with constant service time can be considered as a limiting case of M/E k /n. As k the whole mass of the distribution E k is concentrated at the mean 1/µ, so E k can be considered deterministic ( constant = 1/µ). The performance measures used are The expected number of passengers in queue is L q = 2( - ) and expected number of passengers in system is L S = L + q. 24

25 1.7. NOTATIONS USED : arrival rate : service rate L : number of customers in system Lq : number of customers in queue W : number of customers in system Wq : number of customers in queue : fuzzy rate of arrival. : fuzzy rate of service P n : probability that there are n customers P 0 : probability that the system is empty ρ : traffic intensity + A : strong -cut A : weak cut N : capacity of the system IBF : instantaneous Bernoulli s feed back MINLP : mixed integer non- linear programming NLP : non linear programming (x ) : membership function of x 0 M 0 P(M) : graded mean integration representation P k(m) : K-preference integration representation OTT : on task time 25

26 : fuzzy mean service time S : fuzzy mean repair time R 2 R : fuzzy variance of the repair time iid : identically and independently distributed O(N*) : optimum threshold PGF : probability generating function 1.8. LITERATURE REVIEW Parametric non linear programming, mixed integer non linear programming, operations of fuzzy numbers and Monte Carlo method are also most relevant from the practical point of view in pursuit of fuzzy queues. We furnish below the element which could be fuzzy in nature in general queuing model. i) The fuzzy numbers can be assigned as the arrival rate and service rate in queuing model. ii) Cost Analysis of M / E k / 1 queuing system with removable service station in which holding cost, service cost are also fuzzy numbers. iii) Erlang s loss model, the input parameters arrival rate and the service rate are fuzzy numbers. iv) The machine interference problem, where breakdown rate and service rate are trapezoidal fuzzy numbers. 26

27 v) Membership grades are compared by employing Yager s ranking index method [80] Thus, we have the following types of fuzzy queues. i) Optimizing the Problems using ranking methods with fuzzy input. ii) iii) Problems using non-linear parametric programming with fuzzy input. Problems using mixed integer non-linear parametric programming with fuzzy input. iv) Problems with fuzzy input using Function Principle. v) problems using Monte Carlo method [1] with fuzzy input. There exists a large mass of literature review on the traditional queuing models. The finite capacity queuing systems have been extensively studied by D. Gross and C.M. Harris [32] in The steady state equation of different queuing models is discussed and the closed forms of performance measures are obtained. D.Gross and C.M.Harris 1998 [32] Hillier and Lieberman, 2001 [35] and Taha 2003 [73], suggest that the queuing decision problems are often encountered in many practical systems such as flexible manufacturing system and in telecommunication system. In 1998, Papadopolus [62] published many papers on queuing decision model. Many types of queuing models with a single removal server have been investigated so far. In 1972 Bell [9] discussed about the optimal operation of 27

28 an M/G/1 queue with removable server. Kimura in 1981 [46] analysed an M/G/1 queuing system with removable server via diffusion approximation. Yadin and Naor [79] discussed queuing system removable service state and introduced the concept of an N policy which turns the server on when the number of customers in the system reaches a certain number (N 1) and turns the server off when there are no customers in the system. Pearn and Chang [66] considered a sensitivity investigation of the N-policy M/E K /1 queuing system with removable service station. Baburaj and Manoharan [7] analyzed a bulk service queuing system with removable severs and obtained steady state probabilities of the system. Gross and Harris [33] discussed about an ordinary M/E K /1 queuing system. In this series of investigations, Wang and Huang [76]have made an attempt to study the optimal operation of an M/E K /1 queuing system with a removable service station. Avi-itzhak and Naor (1963) [4] studied the M/M/1 queuing system for a non-reliable server where the service rule does not depend on the number of customers in queue. In 1965, the concept of fuzzy set theory gained currency with Lotfi A. Zadeh [ 82]. H.M. Prade [63 ] discussed an outline of fuzzy or possibilistic models for queuing system in In 1989 R.J. Lie and E.S. Lee [52] made a general approach for queuing system in a fuzzy environment based on Zadeh s extension principle. In 1992, D.S. Negi and E.S. Lee [61] combined the ability of fuzzy sets to represent the queuing system. Both -cut sets and 28

29 the use of random variables to represent fuzzy numbers were investigated. Buckley [13] investigated multiple channel queuing system with finite or infinite waiting capacity and calling population. Negi and Lee [61] formulated the -cut and two variable simulation approach for analyzing fuzzy queues on the basis of Zadeh s extension principle. Li and Lee [51] investigated the analytical results for M/F/1/ and FM/FM/1/ (where F represents fuzzy time and FM represents fuzzified exponential distribution) using Markov Chain. Unfortunately their approach provided only crisp solutions. In other words the membership functions of the performance measures are not completely described. Kao et al applied (1999) [41] parametric programming to construct the membership functions of the performance measures for four simple fuzzy queues with one or two fuzzy variable 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. Schmucker (1984) [67] presented a method for fuzzy risk analysis based on fuzzy number arithmetic operations. Hsieh and Chen (1999) [36] presented a similarity measure between fuzzy numbers using Graded Mean Integration Representation method. In 1985 Chen S.H [17] had given operations on Fuzzy Numbers with Function Principle. In 2004, Shin-pin Chen [69] propounded a non linear programming approach for a bulk queuing model. In 2005, a bulk arrival 29

30 queuing model with fuzzy parameters and varying batch sizes was developed by Ship-pin Chen [70] In 2007 Shin- Pin Chen [71] used a mixed integer programming approach. In 2006, Fuzzy analysis of queuing systems with an unreliable server was suggested by J.C. Ke, C.H. Lin [44]. In 2007, Pardo and Fuente [64 ] optimized two fuzzy queuing models with priority-discipline in which one is a non-preemptive priority system and another is a preemptive priority system. Aydin and Apaydin [5] considered the multi-channel fuzzy queuing systems and computed fuzzy queuing characteristics via different membership functions. In 2008, Lin, C.H., Huang, H.I and Ke, J.C. [54] developed batch arrival queue with setup and uncertain parameter patterns. In 2009, María José Pardo [65] optimized the functions of fuzzy profit of queuing models with publicity and renouncement. In 2010, R. Kalayanaraman, N. Thillaigovindan and G. Kannadasan [40] suggest a single server fuzzy queue with unreliable Server. In 2011 Ebrahim Teimoury [28] Optimized Multi Supplier Systems with Fuzzy Queuing Approach : Case Study of SAPCO 1.9. MOTIVATION AND SCOPE OF THE THESIS Motivated and developed by the works of S.P. Chen et al [20] [21] this thesis documents a wider application from the traditional conventional crisp queues to fuzzy queues. Trapezoidal fuzzy numbers and triangular fuzzy numbers have been chosen to serve as the parameter to describe the fuzzy 30

31 queues with optimization techniques. In this thesis, to demonstrate the validity of the proposed approach, numerical examples are solved successfully through software package MATLAB 6.0 [56]. The software version has been adopted for alleviating the strain in computation and the shape for constructing the membership function of the system performance measures ORGANISATION OF THE THESIS The thesis analyzed and documented the application of queuing model in a fuzzy environment. This thesis consists of six chapters as follows : Chapter I deals with characteristics of queues, fundamental concepts such as definition of fuzzy set, membership function, fuzzy numbers etc. Description about different queuing models is given in this chapter. Taking into cognizance, the existing results on crisp queuing model, the researchers focus on the application of queuing model to varied fuzzy environment situation. A brief historical note on fuzzy queuing model is also dealt with the motivation and scope of the thesis has been mentioned. Chapter II discusses the Machine Repair Problem under uncertain environment. Using Zadeh s Extension principle a parametric programming is developed to describe the performance measure of membership function. 31

32 The content of Chapter II has been published in the International Journal of Mathematical Sciences and Engineering Applications Vol.2 No : 3, Oct. 2010, PP Section 1, of Chapter III discusses a procedure for constructing the membership function of the performance measures of Erlang s Loss Model using non linear programming approach. Section 2, of Chapter III Suggest the performance measures of Sojourn Time Distribution of Tandem queues. The content (i) of chapter III has been published in the International Journal of Fuzzy Mathematics and Systems Vol.1 No : 1(2011) PP 1-10 and the content (ii) has been presented in the International Conference ICMCM 11 and published in the proceedings organized by PSG College of Technology, Coimbatore ( to ). Section 1 of Chapter IV proposes a procedure to construct the membership function of the system characteristic of N-policy queue with infinite capacity under imprecise data. Triangular membership function has been used. Section 2 of Chapter IV uses Zadeh s extension principle is used to develop parametric non linear programming to describe the membership function of performance measures and Yager s index method is used to find the optimum threshold. 32

33 The content (i) of Chapter IV has been published in the Journal of Physical Sciences Vol.15 (Dec. 2011) The content (ii) has been published in the International Journal of Advanced and Innovative Research Vol.1, No.1. Section 1 of Chapter V presents fuzzy analysis of a queuing system with removable and non reliable server using mixed integer non-linear programming. Section 2 of Chapter V is to find the fuzzy steady state values of M x /M(a,b)/1. Function principle is used as an arithmetic operator and for defuzzification Graded Mean Integration method is used. The content (i) of Chapter V has been published in the Journal of Computing Technologies Vol.2, No.1. The contents (ii) of Chapter V has been accepted for publication in the International Journal of Computing. Chapter VI gives applications of queuing theory in real life situation under fuzzy environment. Section 1 of Chapter VI discusses the waiting time of passengers in an escalator at the railway station using Monte Carlo method in fuzzy environment. Section 2 of Chapter VI presents a method for fuzzy risk analysis based on child staff ratio, group size and care giver ability. 33

34 The content (i) of chapter VI has been published in Arya Bhatta Journal of Mathematics & Informatics. The content (ii) of chapter VI has been published in Applied Mathematics, Elixir Publication. MATLAB 6.0 coding have been given in Annexure. 34