A MEMBERSHIP FUNCTION SOLUTION APPROACH TO FUZZY QUEUE WITH ERLANG SERVICE MODEL

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1 International Journal of Mathematical Sciences and Applications Vol. No. (May, 0) Copyright Mind Reader Publications A MEMBERSHIP FNCTION SOTION APPROACH TO FZZY QEE WITH ERANG SERVICE MODE V.Ashok Kumar** ** Department of Mathematics, Shinas College of Technology, Oman. vaashok963@rediffmail.com ABSTRACT This paper develops a non-linear programming approach to derive the membership functions of the steady-state performance measures in Erlang service model where the arrival rate and service rate are fuzzy numbers. The basic idea is based on Zadeh s extension principle. Two pairs of mixed integer Non-linear programs with binary variables are formulated to calculate the upper and lower bonds of the system performance at possibility level. sing -cut approach FM/FE K /I fuzzy queue can be reduced to a family of M/E K /I queue with different cuts. Trapezoidal fuzzy numbers are used to demonstrate the validity of the proposal. Numerical examples are solved successfully. Keywords: Erlang service, Fuzzy number, Queue, Mixed Integer non-linear Programming..INTRODCTION Till now all probability queuing models studied have assumed poisson input and exponential service times. In many practical situations, the exponential assumptions may be rather limiting, especially the assumption concerning service times being distributed exponentially. Most of the related studies are based on traditional queuing theory, in that the inter arrival times and service times are assumed to follow certain probability distribution. However, in practice there are cases that these parameters may be obtained subjectively [5]. The fuzzy queues are much more realistic than the commonly used crisp queues [,,5]. Clearly when the arrival rate and service rate are fuzzy the performance measure of the queue also is fuzzy as well. The basic idea is to apply Zadeh s extension principle [7,8,9]. Two pairs of mixed integer non-linear programming models are formulated to calculate the lower and upper bounds of the -cut of the system performance measure. The membership function of the system performance measure is derived analytically.. TRAPEZOIDA FZZY NMBER

2 The trapezoidal fuzzy number is usually defined as A ~ = [a d, a, a 3, a 3 +d ] The membership function for the Trapezoidal fuzzy number A ~ = [a, a, a 3, a 4 ] is defined as μ A ~ (S) (S a)/(a = (a 4 S)/(a 4 a a 3 ) ) a a a 3 S a S a S a 3 4 A (x) Core O Boundary a a a 3 a 4 Support Boundary X 3. FZZY QEE WITH ERANG SERVICE Consider a queuing system in which the customer arrive at a single-server facility with arrival rate ~ and service rate μ ~ Where ~ is fuzzy in nature poisson rate and μ ~ is fuzzy in nature Erlang service rate made up of K exponential phases. Customers are served according to a first-come-firstserved (FCFS) discipline and both the size of calling population and the system capacity are infinite. The parameters of the Erlang are K and and the mean and variance are given by E(x) μ and V(x) Kμ The relation of the Erlang to the exponential distributions allow us to describe the queuing models where the service (or arrivals) may consist of series of identical phases. For example, in performing a laboratory text, the lab technician must perform k steps, each taking the same mean time (sas /k), with the times distributed exponentially. This is represented in figure. The overall service function is Erlang type k with mean /. If the input process is poisson, we would have M/E K / model.

3 883 Step Step Step K /K /K /K Figure : se of Erlang for Phased Service In this model the group arrival rate ~ and service rate μ ~ are approximately known and can be represented by convex fuzzy sets. Note that a fuzzy set A ~ in its universal set Z is convex if [φ z ( φ)z ] min [μ (z ),μ (z )] Where μ A ~ function [0,] and z, z Z. μ A ~ A ~ A ~ is its membership et μ ~ (x) and μ μ ~ (y) are membership functions of arrival rate and service rate respectively. We have ~ = x, μ (x) x S( ~ ~ ) and μ ~ = y, μ (y) y S(μ ~ μ ) ~ ~ where S( ) and S( μ ~ ) are the supports of ~ and μ ~ which denote the universal set of arrival rate and service rate respectively. Clearly when ~ and μ ~ are fuzzy number then the performance measure ~ ~ ρ(,μ ~ ) are also fuzzy as well. On the basis of Zadeh s extension principle [7,8,9], the membership function of the performance measure is defined as sup ~ = minμ (x),μ (y) z ρ(x, y) ~ μ ~ ~ (z),μ ~ ) ρ( x X, y Y Without loss of generality let us assume that the performance measure of interest is the expected number of customers in queue (q). From the knowledge of traditional queuing theory [4], under the study-state conditions = x/y<, the expected number of (K )K (K ) customers in the queue q (μ x) μ and membership function of ~ q is K μ ~ ~ q (z) = sup (K ) (K ) minμ (x),μ (y) z x X, y Y, z Z ~ μ ~ (μ ) μ K...() By using ittle s formula [] in the same manner, the waiting time in queue μ ~ K Wq and the membership function of W ~ q K μ(μ ) is

4 μ q (z) W ~ = sup K minμ (x),μ (y) z x X, yy, x Y ~ μ ~ K μ(μ )...() Similarly W = Wq + μ and = W 4. THE SOTION PROCEDRE: One approach to construct the membership function of μ ~ ~ ρ(,μ ~ ) is on the basis of deriving -cuts of μ ~ ~ ρ(,μ ~ ). Denote -cuts of ~ and μ ~ as min max = x, x = x μ ~ (x), x, μ ~ (x)...(3) x X x X = y, y = min max y μ (y), y μ (y) y Y ~ μ y Y ~...(4) These intervals indicate where the group arrival rate and service rate lie at possibility level. Consequently, the FM/FE K /I queue can be reduced to a family of crisp M/E K /I queens with different -level sets { / 0 < }. By the convexity of a fuzzy number [9], the bands of these intervals are functions of and can be obtained as x = min μ ~ ( ) and x = max μ ( ) ~, y = min μ μ ~ ( ) and y = max μ μ ~ ( ) respectively Consider the membership function of the expected number of customers in the queue q. According to (), μ ~ q ( ) is the minimum of μ ~ ( x) and μ μ ~ ( y). We need either μ ~ ( x) and μ μ ~ ( y) (or) μ ~ ( x) and μ μ ~ ( y) such that (K ) (K ) z to satisfy μ ~ (μ ) μ K q (z). To find the membership function μ ~ q (z) we have to find the lower bound z and the upper bound z of the -cuts of (z). Since μ ~ q the requirement of μ ~ (x) can be represented by x x (or) x x this can be

5 885 formulated as the constraint of x β x ( β )x, where =0 (or). Similarly μ μ ~ ( y) can be formulated as the constraint y ( β )y y β, where =0 (or). Moreover from the definition of (3) and (4) x and y can be respectively replaced by x x, x and y y, y cases above, the membership function d d μ ~ q consequently, considering both of these two can be constructed via finding the lower bound (q) We set where and upper bound (q). (q) = min (q), (q), and (q) =max (q), (q), respectively (q) = min K (K ) x y R (μ ) μ x, y K... (5) such that x = t x ( t)x, y y y and t = 0 or (q) = min K x y R μ x, y (K ) μ K... (6) such that y = y ( t )y t, x x x and t = 0 or (q) = max K (K ) x y R (μ ) μ x, y K... (7) such that x= t 3 x ( t 3 )x, y y y and t 3 = 0 or (q) = max K x y R μ x, y (K ) μ K... (8) such that y = 4 y ( t 4 )y t, x x x and t 4 = 0 or where x y. From the knowledge of calculus, a unique minimum and a unique maximum of the objective function of models (5), (6), (7) and (8) are assumed, which shows that the lower bound (q) these four models. and upper bound (q) of the -cuts of q can be found by solving

6 According to Zadeh s extension principle, ~ q defined in () is a Fuzzy number that possesses convexity [6,9]. Therefore for two values of and such that 0< <, we have (q) (q) and (q) (q). In other words (q) is non-decreasing with respect to and (q) is non-decreasing with respect to. This property assures the convexity of ~ q. Consequently, the membership function solutions of models (5), (6), (7) and (8). μ ~ q (z) can be obtained from the If both (q) function and (q) are invertible with respect to, then a left shape S (z) (q) a right shape function can be obtained. From S (z) and R S (z) the membership function μ ~ q is constructed as μ q (z) ~ = S(z),, R (z), S (q) (q) (q) 0 z (q) z (q) z (q) 0.(9) Since the above performance measures are described by membership functions, they conserve completely all fuzziness of arrival rate and service rate. 5.NMERA EXAMPE Consider a centralized parallel processing system in which the service consists of two phases. Both the arrival rate and service rate are trapezoidal fuzzy numbers represented by ~ = [,3,4,5] and μ ~ = [3,4,5,6] per minute, respectively. The system manager wants to evaluate the performance measures of the system such as the expected number of customers in the queue. The confidence interval at are [+, 5 ] & [3+, 6 ] ( ~ q) = ( ~ q) is invertible (0 a)

7 887 = (9z ) 36 z (4z 3) 6z 0, (9z+)± 36 z 6z z =, z =.07 or 0.75 (9z ) 36 z (4z 3) 6z z = 40 30, 4 30 =.75 or.075 ( ~ q) = (9z ) 36 z (4z 3) 6z.07 z ( ~ q) = )... (0 b) ( ~ q) is invertible = (36z 30) 7 (8z 3) z 3z 0, (36z 30) 7 z 3z 0 0, z =.375 or.80, (36z 30) 7 (8z 3) z 3z, z =.375 or.086

8 ( ~ q) = (36z 36) 7 (8z 3) z 3z.086 z.375 q is defined as follows: From the inverse function of (q) and (q) the membership function of μ ~ q (z) = (9z ) 36 z 6z (4z 3) (36z 30) 7 z 3z (8z 3).07 z z z.375 Wq = Wq = K. K μ(μ ) 3. 4 μ(μ ) z = = (84z 3) 584z 6z 96z 9 0, (84z 3) 584z 96z z = 867 & z = 0 z =.0066 and z = 0, (84z 3) 584z 6z 96z 9 z = 0 and z = 0.05 W ~ q = (84z 3) 584z 6z 96z z 0.05

9 889 W q = = 3 (5 ) 4 (3 )[(3 ) (5 )] 8 (5 3 ) Wq is invertible = (36z 3) 584z 6z 96z 9 0, (36z 3) 584z 96z 9 z = 0 and z = 0.036, (36z 3) 584z 6z 96z 9 μ q (z) W ~ = z = 0 and z = 0.04 (84z 3) 584z 96z z.05 6z.05 z.04 (36z 3) 584z 96z 9.04 z.036 6z cuts of arrival rate, service rate, queue length and waiting time in queue x x y y (q) (q) (Wq) (Wq)

10 Performance measure of q Performance measure of Wq

11 89 6. CONCSION The Erlang family of probability distributions provides much more modeling flexibility than the exponential. In front, the exponential is a special Erlang, namely, type. In many practical situations where observed data might not bear out the exponential distribution assumption, the Erlang can provide greater flexibility by being better able to represent the real world. The other reason why Erlang is useful in queuing analysis is its relation to the exponential distribution with the Markorian property. REFERENCE. J.J. Buckley, Elementary Queuing Theory based on Possibility Theory, Fuzzy Set System 37 (990), J.J. Buckley, T. Feuring, Y. Hasaki, Fuzzy Queuing Theory Resisted, Int, J. ncertain Fuzzy 9 (00), P. Fortemps, M. Roubens, Ranking and Defuzzification Method based on area Compensation, Fuzzy Sets System 8 (996), D. Gross, C.M. Haris, Fundamentals of Queuing Theory, Third Ed., Wiley, New York, J.B. Jo, Y. Tsujimura, M.Gon, G. Yamazaki, Performance Evaluation of Network Models based on Fuzzy Queuing System, JPn. J. Fuzzy Theory System 8 (996), A. Karfmann, Introduction to the Theory of Fuzzy Subsets, Vol. I, Academic Press, New York, R.R. Yovgav, A Characterisation of the Extension Principle, Fuzzy Sets and Systems 8 (986), A. Zardeh, Fuzzy Sets as a Basis for a Theory of Possibility, Fuzzy sets and Systems (978), H.J. Zimmarmann, Fuzzy Set Theory and its Applications, Fourth ed., Kluwar Academic, Boston, 00.