Analysis of Fuzzy Erlang s Loss Queuing Model: Non Linear Programming Approach

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1 International Journal of Fuzzy Mathematic and Sytem. Volume 1, Number 1 (2011), pp Reearch India Publication Analyi of Fuzzy Erlang o Queuing Model: Non inear Programming Approach W. Ritha 1 and Sreelekha Menon B. 2 1 Dept. of Mathematic, Holy Cro College, Tiruchirappalli, India 2 Dept. of Mathematic, SCMS School of Engg. and Technology, Kerala, India *Correponding Author ritha_prakah@yahoo.co.in Abtract Thi paper propoe a procedure for contructing the memberhip function of the performance meaure in a finite capacity lo queuing ytem with arrival rate and ervice rate being fuzzy number. To invetigate the performance meaure of the finite capacity lo queuing ytem a pair of mathematical non linear programme are formulated to calculate the upper and lower bound of ytem characteritic. By extending finite capacity lo queue in fuzzy environment would have wider application. Keyword: Fuzzy et, fuzzy trapezoidal number queuing model, memberhip function, non linear programming Introduction In teletraffic engineering the term turn i ued to decribe any entity that will carry one call. The number of trunk, to be provided obviouly depend on the traffic to be carried. On a group of trunk, the average number of call in progre depend on both the number of call which arrive and their duration. In order to obtain analytical olution to teletraffic problem it i neceary to have a mathematical model of the traffic offered to telecommunication ytem. Erlang determind the grade of ervice i, the lo probability of a lot call ytem having N trunk, when offered traffic i A. A ytem in which cutomer have to leave when the pace i full becaue of limited waiting pace i called lo ytem the formula hold irrepective of the form of ervice time ditribution. Thi i known a robutne property. Propertie of Erlang lo formula have been dicued by everal reearche Viz Vaulot (1951), Fortet (1948),Jagerman (1974), Jage and Van Doorn (1986), Takac (1969), Harel(1987), Berezner et al (1998), J Medhi (2002), Newell (1984) dicue

2 2 W. Ritha and Sreelekha Menon B. aymptotic approximation for Erlang lo formula when the number of erver a well a offered load are large. We think of waiting line a the circumtance we encounter at the grocery tore but in the telecommunication world line can alo form for packet waiting for trunk to become available. Queuing theory define, a et of formula that decribe waiting line behavior and can be applied to thee and imilar telecommunication ituation. Telecommunication baed ervice activitie can be independent and random a well. For example the length of a telephone cell or ize of a packet of data will impact the ervice time of telephone witche and route. Efficient method have been developed for analyzing the queuing ytem when it parameter uch a arrival rate and ervice rate are known exactly. However, there are cae that thee parameter may not be preented preciely due to uncontrollable factor. Specifically in many practical application, the tatitical data may be obtained ubjectively ie. arrival rate and ervice rate are more uitably decribed by linguitic term uch a fat, moderate, or low rather by probability ditribution baed on tatitical theory. Imprecie information of thi kind will determine the ytem performance meaure accurately. To deal with imprecie information, Zadeh introduced the concept of fuzzine. Fuzzy et theory i a well known concept for modeling impreciion or uncertainty ariing from mental phenomenon. Specifically fuzzy queue have been dicued by everal reearcher. Buckley invetigated multiple channel queueing ytem with finite or infinite waiting capacity and calling population. Negi and ee formulated the cut and two variable imulation approach for analyzing fuzzy queue on the bai of Zadeh extenion principle. i and ee [15] invetigated the analytical reult for M/F/1/ and FM/FM/1/ (where F repreent fuzzy time and FM repreent fuzzified exponential ditribution) uing Markov Chain. nfortunately their approach provided only crip olution. In other word the memberhip function of the performance meaure are not completely decribed. Kao et al applied parametric programming to contruct the memberhip function of the performance meaure for four imple fuzzy queue with one or two fuzzy variable namely M/F/1, F/M/1, F/F/1 and FM/FM/1 where F denote fuzzy time and FM denote fuzzified exponential time. Clearly when the arrival rate or ervice rate are fuzzy number, the performance meaure of tate dependent ervice queue will be fuzzy a well. In thi paper, we develop a olution procedure that i capable of evaluating the fuzzy performance meaure for tate dependent ervice queue with fuzzified exponential arrival rate and ervice rate. The memberhip function of performance meaure are derived by applying cut and Zadeh extenion principle. A pair of mathematical program i formulated to calculate the lower and upper bound of the -cut of the performance meaure. Conequently the memberhip function of the performance meaure i derived analytically or numerically by enumerating different value of. M/M/S/S: o Model Sytem Thi model enviage that a unit who find, on arrival that all S channel are buy

3 Analyi of Fuzzy Erlang o Queuing Model 3 leave the ytem without waiting for ervice. Thi i called S channel lo ytem and wa invetigated by Erlang. For thi birth death proce we have λ n l, n n, n 0, 1, 2, λ n 0, n n, n. We get the teady tate probability a ( λ/) n P 0 n 1, 2,.... P n n! ( λ/ ) K 1 + P 0 1 K 1 K! ( ) 1 K P 0 λ/ K 0 K! n P n ( λ/ ) n! n 0, 1, 2,.... K ( λ/ ) K 0 K! Thi i known a Erlang Formula or Erlang Firt Formula The probability that an arriving unit i lo to the ytem (which the ame a that all the channel are buy) i given by P ( λ/ )! K ( λ/ ) K 0 K! Thi i known a Erlang o Formula or blocking formula and i denoted by B(S, λ/). Sytem with imited Waiting Time Apart from limitation of pace, a econd kind limitation which arie in problem dealing with practical ituation need conideration. For eg. a long ditance telephone call may be booked to be put through within a limited time (ay, during office hour). While a λ/ i the offered load. a a[1 B(c, a)] i the carried load. The overflow rate of blocked (lot) cutomer i ab(c, a) ; the um of the carried load and the overflow rate equal the offered load a. The throughput, defined a the rate at which cutomer (unit) depart from the ytem after being erved, i given by a l[1

4 4 W. Ritha and Sreelekha Menon B. B(c, a)] ; thi i the rate at which cutomer are accepted for ervice. A ytem in which cutomer have to leave when the pace i full becaue of limited waiting pace i called a lo ytem, wherea a ytem where all the arriving cutomer can wait i called delay ytem. Expected no. of buy channel et B be the random variable denoting the number of buy channel, we have n ( λ/) n E[B] np n P 0 n 1 n 1 n! ( ) n n λ/ λ P ( λ/) n-1 0 P 0 (n - 1)! (n-1)! n 1 n 1 n ( λ/) ( λ/ ) λ P 0 n 1 n!! λ [ 1 - Pc ] λ λ 1 - B C, FM/FM/S/S: o Model Sytem Conider a queueing model in which cutomer arrive at the ytem according to Poion proce with fuzzy arrival rate λ %. All cutomer are erved according to exponential time fuzzy ervice rate % 1. A ytem in which cutomer have to leave when the pace i full becaue of limited waiting pace i called a lo ytem. Thi model i denoted by FM/FM/S/S. In thi model the arrival rate λ % and ervice rate % are approximately known and are repreented by the following convex fuzzy et. λ % (x, (x) x X (1) { } { % Y} % (y, (y) y where X and Y are the crip univeral et of the arrival rate and ervice rate (x) and are the correponding memberhip function. et P(x, y) denote the ytem % (y) performance meaure of interet. When λ % and % are fuzzy number P( λ %, % ) i alo a fuzzy number. According to Zadeh extenion principle, the memberhip function of the performance meaure P( λ %, % ) i defined.

5 Analyi of Fuzzy Erlang o Queuing Model 5 λ λ Sup min (x), (2) % (y) Z 1 - B x X y Y P(λ, % ) % ( ) Without lo of generality, aume that the ytem performance meaure of interet are, q, w and w q. From the knowledge of traditional queueing theory the expected ytem length for a crip tate dependent ervice queueing ytem i Solution procedure To re-expre the memberhip function yatem characteritic in an undertandable and uual form we adopt Zadeh approach, which relie on cut of The cut or level et of λ % and % are defined a { } { % } λ() x X (x) () y Y (y) P(λ, % ) % (3) can be expreed in another form λ() min { x (x) }, m ax λ { x (x) λ } % % x X x X x (4a), x () min{ y (y) }, max{ y (y) } % % y Y y Y y (4b), y By the convexity of a fuzzy number, the bound of thee interval are function of and can be obtained a -1 x min () -1 x max () -1 y min % () -1 y max % () (5) Clearly the memberhip function of P(λ, % ) % i defined in (2) i alo parametered by. Conequently we can ue it cut to contruct the memberhip function from the memberhip function tated in (4). (z) i the minimum of,. To λ % λ % (x) % (y) tackle from the memberhip value, we need either (x) and (y) or % (3)

6 6 W. Ritha and Sreelekha Menon B. l l λ λ and. Such that z to atify (x) % (y) 1 B C, (z). Thi can be accomplihed via parametric NP technique. For the former cae the parametric non-linear program for finding the parametric non-linear program for finding the lower and upper bound of cut of are [ E(B) ] 1 [ E(B) ] 1 and for the latter cae [ E(B) ] 2 [ E(B) ] 2 x x min 1 - B C, ; x x x y () + y y x,y R x x max 1 - B C, ; x x x y () + y y x,y R x x min 1 - B C, ; y y y x λ() + y y x,y R x x max 1 - B C, ; y y y x λ() + y y x,y R ing (5(a) and 5(b)) we can have [ E(B) ] x x min 1 - B C, + x,y R y y x x x y y y [ ] x E(B) max 1 - B C, + x,y R y x x x y y y [ E(B) ] [ ] x y If both and E(B) are invertible with repect to then a left hape function (z) [ E(B) ] and a right hape function R(Z) [ E(B) ] can be obtained from which the memberhip function E(B) i contructed. λ %

7 Analyi of Fuzzy Erlang o Queuing Model 7 µ E[B] (z) z 1 z z 2 1 z z z R(z) z z z Numerical example In an automatic telegraph witching ytem with 3 equipment incoming meage are tored in a queue until the retranmitting equipment of an outgoing trunk can end them. Meage arrive at the rate that can be repreented by a trapezoidal fuzzy number λ % [1, 2, 3, 4] per hr and the time taken to retranmit meage may be aumed to have an exponential ditribution with a mean time that can be repreented by a trapezoidal fuzzy number. % [6, 7, 8, 9]. The ytem want to know about the carried load (expected no. of buy channel), expected no. of idle channel, buy probability. Expected number of Buy Channel x E(B) [1 - B(3, x/y)] y Buy probability P(X 1) x [1 - B(3, x/y)] y 3 Expected number of buy channel xy + 6x y + 3x E(B) 6y + 6xy + 3x y + x et λ % [ ] % [ ] x, x [ + 1, 4 - ] y, y [6 +, 9 - ] [ ] (1 + )(6 + ) + 6(1 + ) (6 + ) + 3(1 + ) E(B) (6 + ) + 6(1 + )(6 + ) + 3(1 + ) (6 + ) + (1 + ) [ ] (4 - )(9 - ) + 6(4 - ) (9- ) + 3(4 - ) E(B) (9 - ) + 6(4 - )(9 - ) + 3(4 - ) (9- ) + Buy probability

8 8 W. Ritha and Sreelekha Menon B. P[X 1] {P[X 1]} {P[X 1]} E(B) 3 [ ] E(B) 3 [ ] E(B) 3 with the help of MATAB 6.0, following table give the ytem characteritic value with different poibility level from 0 to 1. [E(B)] [E(B)] P[X1] P[X1] Figure 1: indicate Expected number of Buy Channel E ( B ) at different value of. Figure 1 E(B)

9 Analyi of Fuzzy Erlang o Queuing Model 9 Fig 2 depict the memberhip function of Buy probability at poible value of Figure 2 P (X) Concluion Thi paper applie a non linear programming approach, for contructing performance meaure of fuzzy o ytem queuing model. In practical, deigning fuzzy queue are more ueful than the crip queue. The input parameter like arrival rate and ervice rate are fuzzy trapezoidal number the output like Expected number of buy channel, buy probability are parameterized by obtained lower and upper bound of thee cut are adopted to contruct memberhip function. Thi how that the propoed approach preerve the efficiency of the ytem. Reference [1] Buckley, J.J., Elementary queuing theory baed on poibilitic theory, Fuzzy Set and Sytem, 37 (1990) [2] Buckley, J.J., Feuring, T., Hayahi, Y., Fuzzy queuing theory revied, International Journal of ncertainty fuzzine & knowledge baed ytem. 9 (2001) [3] J Chana, S., Nowakowki, M., Single value imulation of fuzzy variable, Fuzzy Set and Sytem, 21 (1988) [4] Chuen-Horng in Jau-Chuan, Hin I Huang,on retrial queuing model with fuzzy parameter Phyica A 374 (2007) [5] S.P. Chen, Parametric nonlinear programming for analyzing fuzzy queue with finite capacity, Eur. J. Oper. Re. 157 (2004) [6] S.P. Chen, Parametric nonlinear programming approach to fuzzy queue with bulk ervice, Eur. J. Oper. Re. 163 (2005)

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