A BATCH-ARRIVAL QUEUE WITH MULTIPLE SERVERS AND FUZZY PARAMETERS: PARAMETRIC PROGRAMMING APPROACH

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1 Mathematical and Computational Application Vol. 11 No. pp Aociation for Scientific Reearch A BATCH-ARRIVA QEE WITH MTIPE SERVERS AND FZZY PARAMETERS: PARAMETRIC PROGRAMMING APPROACH Jau-Chuan Ke * Hin-I Huang ** Chuen-Horng in ** *Department of Statitic National Taichung Intitute of Technology Taichung 404 Taiwan R.O.C. jauchuan@ntit.edu.tw; **Department of Information Management National Taichung Intitute of Technology Taichung 404 Taiwan R.O.C. yhuang@ntit.edu.tw; linch@ntit.edu.tw Abtract-Thi work contruct the memberhip function of the ytem characteritic of a batch-arrival queuing ytem with multiple erver in which the batch-arrival rate and cutomer ervice rate are all fuzzy number. The -cut approach i ued to tranform a fuzzy queue into a family of conventional crip queue in thi context. By mean of the memberhip function of the ytem characteritic a et of parametric nonlinear program i developed to decribe the family of crip batch-arrival queue with multiple erver. A numerical example i olved uccefully to illutrate the validity of the propoed approach. Becaue the ytem characteritic are expreed and governed by the memberhip function the fuzzy batch-arrival queue with multiple erver are repreented more accurately and the analytic reult are more ueful for ytem deigner and practitioner. Keyword- Fuzzy et Memberhip function Multiple erver Nonlinear programming 1. INTRODCTION Queueing model with multiple erver are effective method for performance analyi of computer and telecommunication ytem manufacturing/production ytem and inventory control (Kleinrock [11] Buzacott and Shanthikumar [1] Gro and Harri [7] Trivedi [19]. In general thee analye conider a queueing ytem where requet for ervice arrive in unit one at a time (ingle-unit arrival. In many practical ituation however requet for ervice uually arrive in batche. For example in manufacturing ytem of the job-hop type each job order often require the manufacture of more than one unit; in computer communication ytem meage which are to be tranmitted could conit of a random number of packet. If the uual crip batch-arrival queue with multiple erver can be extended to fuzzy batch-arrival queue uch queueing model would have wider application. For queueing model with multiple erver under variou conideration the M/M/c vacation ytem with a ingle-unit arrival have attracted much attention from numerou reearcher ince evy and Yechiali [1]. The extenion of thi model can be referred to Vinod [0] Igaki [8] Tian et al. [17] Tian and Xu [18] and Zhang and Tian [ 4]. Zhang and Tian [ 4] tudied the M/M/c vacation ytem with a ingle-unit arrival and a partial erver vacation policy. They proved everal conditional tochatic decompoition reult for the queue length and waiting time. Chao and Zhao [] invetigated the GI/M/c vacation model with a ingle-unit arrival and provided iterative algorithm for computing the tationary probability ditribution. In the literature decribed above cutomer inter-arrival time and cutomer ervice time are required to follow certain probability ditribution with fixed parameter.

2 18 J. C. Ke H. Huang and C. H. in However in many real-world application the parameter ditribution may only be characterized ubjectively; that i the arrival and ervice are typically decribed in everyday language ummarie of central tendency uch a the mean arrival rate i around 5 per day or the mean ervice rate i about 10 per hour rather than with complete probability ditribution. In other word thee ytem parameter are both poibilitic and probabilitic. Thu fuzzy queue are potentially much more ueful and realitic than the commonly ued crip queue (ee i and ee [1] and Zadeh []. By extending the uual crip batch-arrival queue to fuzzy batch-arrival queue in the context of multiple erver thee queuing model become appropriate for a wider range of application. i and ee [1] invetigated the analytical reult for two typical fuzzy queue (denoted M/F/1/ and FM/FM/1/ where F repreent fuzzy time and FM repreent fuzzified exponential ditribution uing a general approach baed on Zadeh extenion principle (ee alo Prade [15] and Yager [1] the poibility concept and fuzzy Markov chain (ee Stanford [16]. A ueful modeling and inferential technique would be applied their approach to general fuzzy queuing problem (ee Stanford [16]. However their approach i complicated and not uitable for computational purpoe; moreover it cannot eaily be ued to derive analytic reult for other complicated queuing ytem (ee Negi and ee [14]. In particular it i very difficult to apply thi approach to fuzzy queue with more fuzzy variable or multiple erver. Negi and ee [14] propoed a procedure uing α-cut and two-variable imulation to analyze fuzzy queue (ee alo Chana and Nowakowki []. nfortunately their approach provide only crip olution; i.e. it doe not fully decribe the memberhip function of the ytem characteritic. ing parametric programming Kao et al. [9] contructed the memberhip function of the ytem characteritic for fuzzy queue and uccefully applied them to four imple fuzzy queue model: M/F/1/ F/M/1/ F/F/1/ and FM/FM/1/. Recently Chen [45] developed FM/FM/1/ and FM/FM [K] /1/ fuzzy ytem uing the ame approach. All previou reearche on fuzzy queuing model are focued on ordinary queue with a ingle erver. In thi paper we develop an approach that provide ytem characteritic for batch-arrival queue with multiple erver and fuzzy parameter: fuzzified exponential batch-arrival and ervice rate. Through -cut and Zadeh extenion principle we tranform the fuzzy queue to a family of crip queue. A varie the family of crip queue i decribed and olved uing parametric nonlinear programming (NP. The NP olution completely and uccefully yield the memberhip function of the ytem characteritic including the expected number of cutomer in the ytem and the expected waiting time in the queue. The remainder of thi paper i organized a follow. Section preent the ytem characteritic of tandard and fuzzy batch-arrival queuing model with multiple erver. In Section a mathematical programming approach i developed to derive the memberhip function of thee ytem characteritic. To demontrate the validity of the propoed approach one realitic numerical example i decribed and olved. Dicuion i provided in Section 4 and concluion are drawn in Section 5. For notational convenience our model in thi paper i hereafter denoted FM [x] /FM/c.

3 A Batch-Arrival Queue with Multiple Server and Fuzzy Parameter 18. FZZY BATCH QEE WITH MTIPE SERVERS We conider a batch-arrival queuing ytem with c erver where the cutomer arrive in batche to occur according to a compound Poion proce with batch-arrival rate. et A k denote the number of cutomer belonging to the k th arrival batch where A k k 1 are with a common ditribution Pr[ Ak n] an n 1 and E [ na n. Cutomer arriving at the ervice facility (erver form a n1 ingle-file queue and are erved in order. The ervice time for each of all c erver i exponentially ditributed with rate and each erver can erve only one cutomer at a time. Cutomer who upon entry the ervice facility find that all erver are buy have to wait in the queue until any one erver i available. et N and W q repreent the expected number of cutomer in the ytem and the expected waiting time in the queue repectively. Through a Markov proce we can eaily obtain N and W q in term of ytem parameter c1 (E[ E[ A 1] n( c n Pn ( n0 N (1 ( c E[ c1 (E[ E[ A 1] n( c n Pn ( n0 1 Wq ( E[ ( c E[ where P n ( repreent the probability that there are n cutomer in the ytem. And the probability depend on and. In teady-tate it i neceary that we have E[ 0 1. c To extend the applicability of the batch-arrival queuing model with multiple erver we allow for fuzzy pecification of ytem parameter. Suppoe the batch-arrival rate for cutomer and ervice rate for each erver are approximately known and can be repreented by the fuzzy et ~ and ~. et ( ~ x and ( denote the memberhip function of ~ and ~. We then have the following fuzzy et: ~ ( x ~ ( x x X (a ~ ( y ~ ( y y Y (b where X and Y are the crip univeral et of the batch-arrival and ervice rate. et f ( x y denote the ytem characteritic of interet. Since ~ and ~ are ~ fuzzy number f ( ~ i alo a fuzzy number. Following Zadeh extenion principle (ee Yager [1] and Zadeh [] the memberhip function of the ytem ~ characteritic f ( ~ i defined a: ~ y ~ ( z up min ~ ( x ~ ( y z f ( x (4 ~ y f ( xx yy 0xE[ / cy1 Aume that the ytem characteritic of interet i the expected number of

4 184 J. C. Ke H. Huang and C. H. in cutomer in the ytem. It follow from (1 that the expected number of cutomer in the ytem i: x(e[ E[ A 1] y n( c n Pn ( x y n0 f ( x y. (5 ( cy xe[ The memberhip function for the expected number of cutomer in the ytem i: c 1 x(e[ E[ A 1] y n( c n P ( n x y n0 ~ ( z up min ~ ( x ~ ( y z. N xx yy 0xE[ / cy1 ( cy xe[ (6 nfortunately the memberhip function i not expreed in the uual form making it very difficult to imagine it hape. In thi paper we approach the repreentation problem uing a mathematical programming technique. Parametric NP are developed ~ to find the -cut of f ( ~ baed on the extenion principle.. PARAMETRIC NONINEAR PROGRAMMING To re-expre the memberhip function ~ ( z N of N ~ in an undertandable and uable form we adopt Zadeh approach which relie on -cut of N ~. Definition for the -cut of ~ and ~ a crip interval are a follow: c1 x x min x ~ ( x max x ( x xx xx ( ~ y y min y ~ ( y max y ( y yy yy ( ~ (7a (7b The contant batch-arrival and ervice rate are hown a interval when the memberhip function are no le than a given poibility level for. A a reult the bound of thee interval can be decribed a function of and can be obtained a: x min ~ ( x max ~ ( y min ~ ( and y max ~ (. Therefore we can ue the -cut of N ~ to contruct it memberhip function ince the memberhip function defined in (6 i parameterized by. ing Zadeh extenion principle ~ ( z N i the minimum of ~ ( x and ~ ( y. To derive the memberhip function ~ ( z we need at leat one of the x(e[ E[ A 1] y n( c n Pn ( x y n0 following cae to hold uch that z = ( cy xe[ atifie ~ ( z : N Cae (i: ( ~ ( x ~ ( y Cae (ii: ( ~ ( x ~ ( y Thi can be accomplihed uing parametric NP technique. The NP to find the lower and upper bound of the -cut of ~ ( z for Cae (i are: N N c1

5 A Batch-Arrival Queue with Multiple Server and Fuzzy Parameter 185 x(e[ E[ A 1] y n( c n P ( x y 1 ( n n0 min ( cy xe[ A ] (8a c1 x(e[ E[ A 1] y n( c n P ( x y 1 ( n n0 max ( cy xe[ A ] (8b and for Cae (ii are: c1 x(e[ E[ A 1] y n( c n P ( x y ( n n0 min ( cy xe[ A ] (8c x(e[ E[ A 1] y n( c n P ( x y n n0 max ( cy xe[ A ]. (8d From the definition of ( and ( in (7 x ( and y ( can be replaced by x [ x x ] and y [ y y ]. The -cut form a neted tructure with repect to (ee Kaufmann [10] and Zimmermann [5]; i.e. given we have [ x x ] [ x x ] 1 1 and [ y ] [ ] y y y 1 1. Therefore (8a and (8c have the ame mallet element and (8b and (8d have the ame larget element. To find the memberhip function ~ ( z it uffice to find the left and right hape function of ~ ( z which i equivalent to finding the lower bound upper bound N N N c1 c1 ( of the -cut of N ~ which can be rewritten a: x(e[ E[ A 1] y n( c n P ( x y n n0 min ( cy xe[ A ].t. x x x and y y y c1 ( and N x(e[ E[ A 1] y n( c n P ( x y n n0 max ( cy xe[ A ] (9b.t. x x x and y y y At leat one of x and y mut hit the boundarie of their -cut to atify ~ ( z =. Thi model i a et of mathematical program with boundary contraint N and lend itelf to the ytematic tudy of how the optimal olution change with x y and c1 (9a x y a varie over (01]. The model i a pecial cae of parametric NP (ee Gal [6]. The crip interval [ ] obtained from (9 repreent the -cut of N ~. Again by applying the reult of Kaufmann [10] and Zimmermann [5] and convexity propertie to N ~ we have 1 and 1 where 1. In other word ( increae and ( decreae a 0 1 N N

6 186 J. C. Ke H. Huang and C. H. in increae. Conequently the memberhip function ~ ( z can be found from (9. If both and in (9 are invertible with repect to then a left 1 1 hape function ( z [ ] and a right hape function R( z [ ] can be derived from which the memberhip function ~ ( z i contructed: ( z α0 z α1 ~ ( z 1 α1 z α1 (10 N R( z α1 z α0. In mot cae the value of and cannot be olved analytically. Conequently a cloed-form memberhip function for ~ ( z cannot be obtained. However the numerical olution for and at different poibility level can be collected to approximate the hape of (z and R (z. That i the et of interval {[ ] [01]} how the hape of ~ ( z N although the exact function i not known explicitly. Note that the memberhip function for the expected waiting time in the queue can be expreed in a imilar manner. 4. NMERICA EXAMPE Thi ection we preent one example motivated by real-life ytem to demontrate the practical ue of the propoed approach which i baed on Example: Conidering one ewerage treatment ytem collect ewage from the urban area and end them to the ewerage treatment plant. The ewerage treatment plant ha three upply pipe (referred to -erver. Each pipe can ettle the larger olid and put the ettled into the chemical proce tank. After the chemical proce the treated water i dicharged to the ea. We aume that the number of arriving ewage olid each time follow a geometric ditribution with parameter p = 0.5 ; i.e. the ize of arriving k 1 ewage olid A i Pr( A k 0.5(1 0.5 k 1. Clearly thi problem can be decribed by FM [x] /FM/ ytem. For efficiency the management want to get the ytem characteritic uch a the expected number of ewage olid in the ytem and the expected waiting time in the queue. Suppoe the batch-arrival rate and ervice rate are trapezoidal fuzzy number ~ repreented by [1 4] and ~ [111114]. Firt it i eay to find that [ x x ] [1 4 ] and [ y y ] [11 14 ]. Next it i obviou that when x x and y y the expected number of ewage olid in the ytem attain it maximum value and when x x and y y the expected number of ewage olid in the ytem attain it minimum value. According to (9 the -cut of N ~ are: N N N

7 A Batch-Arrival Queue with Multiple Server and Fuzzy Parameter With the help of MATAB the memberhip function i: ( z z ~ ( z 1 z N R( z z where: with: P 115z ( 1 ( z 900z i P (85z 4 P 1 (1 (5z P 1 (5z Q 1 1 i(5z Q (5z 88 Q (5z 80z 684 R( z 7060z 155 (50z 0 500z z 80z z 69988z Q 115z 900z 7060z 155 (50z 0 500z 1100z 65z 69988z 9569 a hown in Fig. 1. The overall hape turn out a expected. The memberhip function (z and R (z have complex value with their imaginary part approaching zero when z for (z and z for R (z. Hence the imaginary part of thee two function have no influence on the computational reult and can be diregarded. Next we perform -cut of batch-arrival and ervice rate and fuzzy expected number of ewage olid in the ytem at eleven ditinct value: Crip interval for fuzzy expected number of ewage olid in the ytem at different poibilitic level are preented in Table 1. The fuzzy expected number of ewage olid in the ytem N ~ ha two characteritic to be noted. Firt the upport of N ~ range from 0.74 to 1.444; thi indicate that though the expected number of ewage olid in the ytem i fuzzy it i impoible for it value to fall below 0.74 or exceed Second the -cut at 1 contain the value from 0.59 to which are the mot poible value for the fuzzy expected number of ewage olid in the ytem.

8 188 J. C. Ke H. Huang and C. H. in N ~ N Fig. 1. The memberhip function for fuzzy expected number of ewage olid in the ytem Table 1. -cut of batch-arrival and ervice rate and expected number of ewage olid in the ytem x x y y Similarly the memberhip function for the fuzzy expected waiting time in the queue ( W ~ q i obtained a hown in Fig.. Crip interval for the fuzzy expected waiting time in the queue at different poibilitic level are given in Table. For the fuzzy expected waiting time W ~ q the range of W ~ q at 1 i [ ] indicating that expected waiting time for any ewage olid definitely fall between and Moreover the range of W ~ q at 0 i [ ] indicating that the expected waiting time in the queue will never exceed or fall

9 A Batch-Arrival Queue with Multiple Server and Fuzzy Parameter 189 below W ~ q W q Fig.. The memberhip function for fuzzy expected waiting time in the queue Table. -cut of batch-arrival and ervice rate and expected waiting time x x y y ( W q ( W q CONCSIONS Thi paper applie the concept of -cut and Zadeh extenion principle to a batch-arrival queuing ytem with multiple erver and contruct memberhip function of the expected number of cutomer and the expected waiting time uing paired NP model. Following the propoed approach -cut of the memberhip function are found and their interval limit inverted to attain explicit cloed-form expreion for the ytem characteritic. Even when the memberhip function interval cannot be inverted ytem deigner or manager can pecify the ytem

10 190 J. C. Ke H. Huang and C. H. in characteritic of interet perform numerical experiment to examine the correponding -cut and then ue thi information to develop or improve ytem procee. For example in Example a deigner (manager can et the range of the number of ewage olid to be [ ] to reflect the deired ervice and find that the correponding level i 0.8 with y and y 1.0. In other word the deigner can determine that the ervice rate i between and 1.0. Similarly a deigner can alo et the expected waiting time with rounder number like [ ] to reflect the deired ervice and the correponding level i 0.4 with y and y A thi example demontrate the approach propoed in thi paper provide practical information for ytem deigner and practitioner. REFERENCES [1] J. Buzacott and J. Shanthikumar Stochatic Model of Manufacturing Sytem Englewood Cliff NJ: Prentice-Hall 199. [] S. Chana and M. Nowakowki Single value imulation of fuzzy variable Fuzzy Set and Sytem [] X. Chao and Y. Zhao Analyi of multiple-erver queue with tation and erver vacation European Journal of Operational Reearch [4] S. P. Chen Parametric nonlinear programming for analyzing fuzzy queue with finite capacity European Journal of Operational Reearch [5] S. P. Chen Parametric nonlinear programming approach to fuzzy queue with bulk ervice. European Journal of Operational Reearch [6] T. Gal Potoptimal Analyi Parametric Programming and Related Topic New York: McGraw-Hill [7] D. Gro and C. M. Harri Fundamental of Queueing Theory rd Ed NewYork: JohnWiley [8] N. Igaki Exponential two erver queue with N-policy and multiple vacation Queueing ytem [9] C. Kao C. C. i and S. P. Chen Parametric programming to the analyi of fuzzy queue Fuzzy Set and Sytem [10] A. Kaufmann Introduction to the Theory of Fuzzy Subet Volume 1 New York: Academic Pre [11]. Kleinrock Queueing Sytem Vol. I. Theory New York: Wiley [1] Y. evy and. Yechiali A M/M/c queue with erver vacation INFOR [1] R. J. i and E. S. ee Analyi of fuzzy queue Computer and Mathematic with Application [14] D. S. Negi and E. S. ee Analyi and imulation of fuzzy queue Fuzzy Set and Sytem [15] H. M. Prade An Outline of Fuzzy or Poibilitic Model for Queueing Sytem Fuzzy Set Ed. P. P. Wang and S. K. Chang New York: Plenum Pre (1980 [16] R.E. Stanford The et of limiting ditribution for a Markov chain with fuzzy tranition probabilitie Fuzzy Set and Sytem [17] N. Tian Q. i and J. Cao Conditional tochatic decompoition in the M/M/c queue with erver vacation Stochatic Model 14(

11 A Batch-Arrival Queue with Multiple Server and Fuzzy Parameter 191 [18] N. Tian and X. Xu M/M/c queue with ynchronou multiple vacation of partial erver OR Tranaction 5( [19] K. S. Trivedi Probability and Statitic with Reliability Queueing and Computer Science Application. New York: John Wiley & Son Inc 00. [0] B. Vinod Exponential queue with erver vacation Journal of Operational Reearch Society [1] R. R. Yager A characterization of the extenion principle Fuzzy Set and Sytem []. A. Zadeh Fuzzy et a a bai for a theory of poibility Fuzzy Set and Sytem [] G. Zhang and N. Tian Analyi of queueing ytem with ynchronou ingle vacation for ome erver Queueing ytem [4] G. Zhang and N. Tian Analyi on queueing ytem with ynchronou vacation of partial erver Performance Evaluation [5] H. J. Zimmermann Fuzzy Set Theory and it Application 4th Ed Boton: Kluwer Academic 001.