SECOND OPTIONAL SERVICE QUEUING MODEL WITH FUZZY PARAMETERS

Transcription

1 International Journal of Applied Engineering and Technolog ISSN: X (Online) 2014 Vol. 4 (1) Januar-March, pp.46-53/mar and Jenitta SECOND OPTIONA SERVICE QEING MODE WITH FZZY PARAMETERS *K. Julia Rose Mar 1 and P. Angel Jenitta 2 1 Department of Mathematics, Nirmala College for Women (Autonomous), Coimbatore-18, Tamil Nadu, India 2 Nirmala College for Women (Autonomous), Coimbatore-18, Tamil Nadu, India *Author for Correspondence ABSTRACT The aim of this paper is to analze Bi-level threshold polic of M X /(G 1, G 2 )/1 with single vacation (SV) queue in a fuzz environment. A mathematical Non linear programming (NP) method is used to construct the membership function of the sstem characteristic in which the batch arrival-rate, expected group size, setup time, vacation time, service time for first essential service(fes), service time for second optional service(sos) and probabilit for opting second channel are all fuzz numbers. The α-cut and zadeh s extension principle are used to transform a fuzz queue into a famil of conventional crisp queue. B means of the membership functions of the sstem characteristic, a set of parametric NP is developed to calculate the lower and upper bound of the sstem characteristics function at α. Moreover a numerical example is also illustrated to check the validit of the proposed approach. Kewords: Bi evel Threshold Polic, First Essential Service, Second Optional Service, Earl Setup, Vacation Polic, α-cut, Membership Function and Zadeh s Extension Principle INTRODCTION The first stud of batch arrival queuing sstem with N polic was carried out b ee and Srinivasan (1989). In their paper the have discussed the mean waiting time of an arbitrar customer and a procedure to find the stationar optimal polic under a linear cost structure. ater, man authors including ee et al., (1994a, 1994b) have analzed the N polic of M X /G/1 queuing models with servers having multiple and single vacation. But these models do not involve the server s setup time. In queuing models, server s setup time corresponds to the preparator work of the server before starting his service. Hur and Park (1999) and Ke (2001) are some of the authors who analzed the N polic of M X /G/1 queuing models with server s setup time. The batch arrival queuing sstem with double threshold polic, setup time and vacation are analzed b ee et al., (2003) is among the most general queuing sstem with threshold policies But in everda life there are queuing situations where all the arriving customers require the first essential service and some ma require the second optional service provided b the same server. Madan (2000) has introduced the concept of second optional service, where the customers ma depart from the sstem either with probabilit (1-r) or ma immediatel opt for second optional service with probabilit r. Choudhur and Paul (2006) have extended the results of Madan. ater Madan and Choudhur (2006) have studied the stead state analsis of the M X /(G 1, G 2 )/1 queue with restricted admissibilit. Recentl Julia Rose Mar et al., (2011) introduced Bi-level threshold polic of M X /(G 1, G 2 )/1/SV queue. In this model the obtained the stationar probabilit generating function of the queue length distribution through supplementar variable techniques and derived the expected length of the ccle (orbit size). The various performance measures were also calculated. In their paper, the inter arrival time, service times, setup time, vacation time are assumed to follow certain probabilit distributions with fixed parameters. In real life in man situations the parameter ma onl be characterized subjectivel (i.e) the sstem parameters are both possibilistic and probabilistic. Thus fuzz analsis would be potentiall much more useful and realistic than the commonl used crisp concepts. i and ee (1989) investigated analtical results for two fuzz queues using a general approach based on Zadeh s extension principle. Nagi and ee (1992) proposed a procedure using α cut and two variable simulations to analze fuzz queues. sing Copright 2014 Centre for Info Bio Technolog (CIBTech) 46

2 International Journal of Applied Engineering and Technolog ISSN: X (Online) 2014 Vol. 4 (1) Januar-March, pp.46-53/mar and Jenitta parametric programming Kao et al., (1999) constructed the membership function of sstem characteristics for fuzz queues. Chauan et al., (2007) have obtained the membership function of sstem characteristics of a retrial queuing model with fuzz arrival, retrial and service rate. Jeeva and Rathnakumari (2012) analzed a batch arrival single server, Bernoulli feedback queue with fuzz vacations. Recentl Ramesh and Kumara (2013) also introduced a batch arrival queue with multiple servers and fuzz parameters. With the help of these available literatures we determine the bi-level threshold polic of bulk arrival queue with second optional service and single vacation in fuzz environment. Model Description Consider bi-level threshold polic of M X /(G 1, G 2 )/1/SV queue. The orbit size and the normal queue size are assumed to be infinite. The customer arrives according to the compound Poisson process with random arrival size X. The server is turned off and leaves the sstem for a vacation of random length V, each time when the sstem becomes empt. After returning from the vacation if the server finds the sstem size is less than m then he remains idle (build up period) in the sstem until the queue length reaches at least m. On the other hand, if the server finds m or more customers in the sstem at the end of the vacation then he immediatel starts the setup operation which takes the random length D (earl setup).after the setup period, if the number of customers in the queue is less than N (N m), then the server remains again idle (dormant) until the queue length reaches at least N. Instead, at the end of the set up period if the server finds more than N customers waiting in the sstem then the server immediatel begins to serve the customers. Here, the idle period of the server is made up of vacation period; build up period, setup period and dormant period. If the queue length reaches N or more either at the end of the setup period or at the end of the dormant period, the server begins a bus period. During bus period the server provides to each unit, two stages of heterogeneous service of which one is optional, that is, the server begins to serve the first phase of essential service(fes) for all the units. After the completion of FES of a unit, the customer ma leave the sstem with probabilit(1-r) or ma opt for a second optional service(sos) in an additional channel b the same server with probabilit r (0 r 1). The server continues this tpe of service until the sstem becomes empt and then turned off the sstem. Thus a ccle is completed. The sstem will be turned on again for setup when at least m customers are present in the sstem. The service times s 1 and s 2 of two channels (FES and SOS) are assumed to be mutuall independent of each other. According to this model the expected length of the ccle (orbit size) and mean sstem size E(N) are obtained as E(T ccle ) = E D +E V + N 1 n =m 1 E X [E s 1 +re s 2 ] (1) E(N) =(E X ) 2 (E(s 1 2 ) + 2rE s 1 E s 2 + re(s 2 2 )) + E (X(X 1))(rE s 2 +E s 1 ) 2(1 (E(x)[E s 1 +re s 2 ]) + 1 E D +E V + N 1 n =m N 1 n =m n ] (2) [ E V (E D 2 +2E D E V +E(V 2 ) 2 + n =0 nψ n + n=0 nψ n E D E X + REST We extend the above queuing sstem in fuzz environment. Suppose the arrival rate, expected group size X, setup time D, vacation time V, service time for FES S 1, service time for SOS S 2 and probabilit for opting second channel r are approximatel known and can be represented as fuzz sets, X, D, V, S 1, S 2 and r. sing α-cuts for the arrival rate, expected group size, setup time, vacation time, service times and probabilit for opting second channel, which are represented b different levels of confidence. et this interval of confidence be represented b [x 1α, x 2α ]. Since the probabilit distribution for the α 1 cuts can be represented b uniform distributions. We have P (x α ) = [x x 2α x 1α x α x 2α ] Then the 1α Copright 2014 Centre for Info Bio Technolog (CIBTech) 47

3 International Journal of Applied Engineering and Technolog ISSN: X (Online) 2014 Vol. 4 (1) Januar-March, pp.46-53/mar and Jenitta mean and the second order moment of the distribution are obtained as 1/2[x 2α + x 1α ] and x 2α 3 x 3 1α. 3(x 2α x 1α ) Further its variance is given b 1/12[x 2α x 1α ] 2 et η (), η X (x), η D (u), η V (v), η S1 (s), η S2 (t), η r (q) denote the membership function of, X, D, V, S 1, S 2, r respectivel. Now we have the following fuzz sets as, = { (, η, ) / Y} (3) X = { ( x, η X x / x X} (4) D = { ( u, η D u / u } (5) V = { ( v, η V v / v V} (6) S 1 = { (s, η S1 s / s S } (7) S 2 = { (t, η S2 t / t T } (8) r = { (q, η r q / q Q} (9) where Y,X,,V,S,T and Q are the crisp universal sets of the batch arrival rate, expected group size, setup time, vacation time, service time for FES, service time for SOS and probabilit for opting second service channel respectivel. et z = f(,x,u,v,s,t,q) denote the sstem characteristics of interest. Since, X, D, V, S 1, S 2, r are fuzz numbers, f(, X, D, V, S 1, S 2, r ) is also a fuzz number. B Zadeh s extension principle the membership function of the sstem characteristics f(, X, D, V, S 1, S 2, r ) is defined as η f,x,d,v,s1,s 2,r (z)= sup Y,x X,u,v V, s S,,t T,q Q min { η (), η X (x), η D (u), η V (v), η S1 (s), η S2 (t), η r (q) / z = f(,x,u,v,s,t,q)} (10) Also assume if the sstem characteristics of interest are expected ccle length E(T ccle ), and mean sstem size E(N). Then we have to find the membership function of E(T ccle ), and E(N). Thus we consider E(T ccle ) as f(,x,u,v,s,t,q) = E u +E v + N 1 n =m and the membership function of expected ccle length E(T ccle ) is given b η (z)= sup min { E Tccle η (), η X (x), η D (u), η V (v), η S1 (s), η S2 (t), η r (q) / z = Y,x X,u,v V, s S,,t T,q Q E u +E v + N 1 n =m } (11) ikewise the membership function of the E(N) is also obtained. The membership function in the above equation is not in the usual forms thus making it ver difficult to imagine its shapes. For this we approach the problem b using the mathematical programming techniques. Parametric NPs are developed to find α cut of f(, X, D, V, S 1, S 2, r) based on the extension principle. To construct the membership function of η E Tccle (z) = α, we have to derive the α cuts of E(T ccle ). The α cuts of, X, D, V, S 1, S 2, r are represented as follows (α) = [ α α ] = [ min{ Y/ η, α }, max{ Y/ η, α }] (12a) X(α) = [x α x α ] = [ min{ x X/ η X x α }, max { x X/ η X x α }] (12b) D(α) = [u α u α ] = [min{ u / η D u α }, max{u / η D u α }] (12c) V(α) = [v α v α ] = [ min{ v V/ η V v α }, max{ v V/ η V v α }] (12d) S 1 (α) = [s α s α ] = [ min { s S / η S1 s α }, max{ s S / η S1 s α }] (12e) S 2 (α) = [t α t α ] = [ min { t T / η S2 t α }, max { t T / η S2 t α }] (12f) r(α) = [q α q α ] = [ min{ q Q/ η r, q α }, max { q Q/ η r, q α }] (12g) Further, the bounds of these intervals can be described as functions of α and can be obtained as Copright 2014 Centre for Info Bio Technolog (CIBTech) 48

4 International Journal of Applied Engineering and Technolog ISSN: X (Online) 2014 Vol. 4 (1) Januar-March, pp.46-53/mar and Jenitta α = min η 1, (α) α = max η 1 (α) x α = minη 1 X (α) x α = max η 1 X (α) u α = min η 1 D (α) u α = max η 1 D (α) v α = min η 1 V (α) v α = max η 1 V (α) s α = min η 1 S1 (α) s α = max η 1 S1 (α) t α = min η 1 S2 (α) t α = max η 1 S2 (α) q α = min η 1 r (α) q α = max η 1 r (α) Thus b making use of the α-cuts for E(T ccle ) we construct the membership functions of (11) which is parameterized b α. Now to derive the membership function of E(T ccle ) it is suffice to find the left and right shape function of η (z). This can be achieved b following the Zadhe s extension principle E Tccle for η (z) which is the minimum of η E Tccle,, η X x, η D u, η V v, η S 1 s, η S2 t, η r q. Then at least one the following cases to be hold which satisfies η E Tccle (z) = α. hence, Case(i) η, = α, η X x α,η D u α, η V v α, η S 1 Case(ii) η, α, η X x = α,η D u α, η V v α, η S 1 Case (iii) η, α, η X x α,η D u = α, η V v α, η S 1 Case(iv) η, α, η X x α,η D u α, η V v = α, η S 1 Case(v) η, α, η X x α,η D u α, η V v α, η S 1 s = α, η S2 Case (vi) η, α, η X x α,η D u α, η V v α, η S 1 t = α, η r (q) α Case (vii) η, α, η X x α,η D u α, η V v α, η S 1 t α, η r (q) = α This can be accomplished b using parametric NP techniques. The NP techniques to find the lower and upper bounds of α cut of η E Tccle (z) for case (i) is [E T ccle ] 1 α = min { E u +E v + N 1 n =m N 1 n =m [E T ccle ] α 1 = max { E u +E v + For case (ii) as 2 [ E T ccle ] α = min{ E u +E v + N 1 n =m N 1 n =m [E T ccle ] α 2 = max { E u +E v + For case (iii) as [ E T ccle ] 3 α = min{ E u +E v + N 1 n =m N 1 n =m [E T ccle ] α 3 = max { E u +E v + For case (iv) as } (13 a) } (13 b) } (13c) } (13d) } (13e) } (13f) Copright 2014 Centre for Info Bio Technolog (CIBTech) 49

5 International Journal of Applied Engineering and Technolog ISSN: X (Online) 2014 Vol. 4 (1) Januar-March, pp.46-53/mar and Jenitta [E T ccle ] 4 α = min { E u +E v + N 1 n =m N 1 n =m [E T ccle ] α 4 = max { E u +E v + For case (v) as 5 [E T ccle ] α = min { E u +E v + N 1 n =m N 1 n =m [E T ccle ] α 5 = max { E u +E v + For case (vi) as 6 [E T ccle ] α = min { E u +E v + N 1 n =m N 1 n =m [E T ccle ] α 6 =max { E u +E v + For case (vii) as 7 [E T ccle ] α = min { E u +E v + N 1 n =m N 1 n =m [E T ccle ] α 7 =max { E u +E v + } (13g) } (13h) } (13i) } (13j) } (13k) } (13l) } (13m) } (13n) As (α),x(α),d(α),v(α),s 1 α, S 2 (α),r(α) are given in equations ( 12 a-g ) α, x X α, u D(α), v V α, s S 1 α, t S 2 (α), q r(α) can be replaced b [ α α ], x x α x α, u [u α u α ], v [v α v α ], s [s α s α ], t [ t α t α ], q [q α q α ] and which are given b the α cuts and in turn the form a nested structure with respect to α which are expressed in (13a-n). Hence for given 0< α 2 < α 1 < 1, we have [ α1 α1 ] [ α2 α2 ], [ x α1 x α1 ] [ x α2 x α2 ], [ u α1 u α1 ] [ u α2 u α2 ], [ v α1 v α1 ] [ v α2 v α2 ], [ s α1 s α1 ] [ s α2 s α2 ], [ t α1 t α1 ] [ t α2 t α2 ], [ q α1 q α1 ] [ q α2 q α2 ]. Thus equations (13a), (13c), (13e), (13g), (13i), (13k) and (13m) have the unique smallest element and equations (13b), (13d), (13f), (13h),(13j), (13l) and (13n) have the unique largest element. Now, to find the membership function of η E Tccle (z) which is equivalent to find the lower bound of [ E T ccle ] α and upper bound of [E T ccle ] α are written as, [E T cc le ] α = min{ E u +E v + N 1 n =m } (14a) where, α α, x α x x α, u α u uα, vα v vα, sα s sα,tα t tα, qα q qα. [E T ccle ] α = max{ E u +E v + N 1 n =m } (14b) where, α α, x α x x α, u α u uα, vα v vα, sα s sα, tα t tα, qα q qα. (i.e) At least an one of, x, u, v, s, t, q must hit the boundaries of their α cut that satisf η E Tccle (z) = α The crisp interval [E T ccle α, E Tccle α ]obtained from (14a) and (14b) represent α cuts of E Tccle. B appling the results of Zimmerman (2001) and convexit propert, we obtain [E T ccle ] α1 [E T ccle ] α2 and [E T ccle ] α1 [E T ccle ] α2, where 0< α 2 < α 1 < 1. In other words E T ccle α increases and E Tccle α decreases as α increases, consequentl the membership function η E Tccle (z) can be found from (14) If both [ E T ccle ] α and [ E T ccle ] α are invertible with respect to α then the left shape function (z) =[ E T ccle α ] 1 and right shape function R(z) =[ E T ccl e α ] 1 can be derived, such that Copright 2014 Centre for Info Bio Technolog (CIBTech) 50

6 International Journal of Applied Engineering and Technolog ISSN: X (Online) 2014 Vol. 4 (1) Januar-March, pp.46-53/mar and Jenitta z [ E T ccle ] α=0 z [E T ccle ] α=1 η (z) = 1 [ E T E Tccle ccle ] α=1 z [ E T ccle ] α=1 R z [ E T ccle ] α=1 z [ E T ccle ] α=0 In man cases, the value of {[ E T ccle ] α [E T ccle ] α / α [0 1]} cannot be solved analticall, consequentl a closed form membership function of E T ccle cannot be obtained. The numerical solutions for [E T ccle ] α and [E T ccle ] α at different levels of α can be collected that approximate the shape of (z) and R(z) (i.e.,) the set of intervals{[e T ccle ] α [E T ccle ] α / [0 1] } will estimate the shapes. Numerical Example Consider a fuzz bi level threshold polic of M X /(G 1, G 2 )/1/SV queuing sstem. The corresponding parameters such as the arrival rate, expected group size X, setup time D, vacation time V, service time for FES S 1, service time for SOS S 2 and probabilit for opting second channel r are fuzz numbers. et = [0.4,0.401,0.402,0.403], X = [1.55,1.65,1.75,1.85] D =[1,1.5,2,2.5], v =[0.5, 0.6,0.7,0.8 ], S 1 = [0.1, 0.15, 0.2, 0.25], S 2 = [ 0.5,0.6,0.7,0.8], r =[ 0.2,0.3,0.4,0.5] Then the expected length of the ccle, and the mean sstem size are given b, E T ccle = E u +E v + N 1 n =m E(N) =(E x ) 2 (E(s 2 ) + 2qE s E t + qe(t 2 )) + 1 E D +E V + N 1 n =m [ E V (E D2 +2E D E V +E(V 2 ) + 2 E (x(x 1))(qE t +E s ) + 2(1 (E (x)[e s +qe t ]) n =0 nψ n + n=0 nψ n E D E X + N 1 n =m n and,x,u,v,s,t,q are the fuzz variable corresponding to, X, D, V, S 1, S 2, r respectivel. Thus, α α = α α, x α x α = α α, [ u α u α ]= [1+0.5α, α], [ v α v α ]= [ α, α], s α s α = [ α α] t α t α = [ α α] [ q α q α ]= [ α, α]. B substituting the above values, the effect of parameters on the expected ccle length E(T ccle ), and mean sstem size E(N) are tabulated and their graphical representations are also shown below ] E(Tccle low) E(Tccleup) Figure 1: The membership function for fuzz E(T ccle ) Copright 2014 Centre for Info Bio Technolog (CIBTech) 51

7 International Journal of Applied Engineering and Technolog ISSN: X (Online) 2014 Vol. 4 (1) Januar-March, pp.46-53/mar and Jenitta Table 1: The α cuts for the performance measure of E(T ccle ) α E(T Ccle ) low E(T Ccle )upp Table 2: The α cuts for the performance r measure of E(N) E(N) low E(N) upp E(N) low E(N) up Figure 2: The membership function for fuzz E(N) Here we perform α cuts of fuzz E(T Ccle ) and E(N) at eleven distinct α levels 0,0.1,0.2, 1.0. Crisp intervals for fuzz E(T Ccle ) with single vacation at different possible α levels are presented in table 1, Copright 2014 Centre for Info Bio Technolog (CIBTech) 52

8 International Journal of Applied Engineering and Technolog ISSN: X (Online) 2014 Vol. 4 (1) Januar-March, pp.46-53/mar and Jenitta similarl the other performance measure such as E(N) are presented in table 2. Figure 1 depicts the rough shape of E(T Ccle ) constructed from α values. The rough shape turns out rather fine and look like a continuous function. The other performance measure E(N) is represented b Figure 2.The α cut depicts that these two performance measures will lie onl in the specified associated range. We find that the above information is ver useful for designing the fuzz queuing sstem. Conclusion The fuzz queuing model has more applicabilit in the real environments than the crisp sstems. This paper applies the concept of α cut and Zadeh s extension principle to fuzz Bi level threshold polic of M X /(G 1, G 2 )/1/SV queuing sstem and thereb deriving the membership function of the expected length of the ccle, for this model. We find that it is more meaningful to express E(T Ccle ) and E(N) as a membership function rather than b crisp values (i.e) as fuzz performance measures. The benefit and significance of such a fuzz performance measure include maintaining the fuzziness of input information completel and the results can be used to represent the fuzz sstem more accuratel. REFERENCES Chaun KJ, Hung Hsim I and Hong C (2007). On retrial queuing model with fuzz parameters. Phsica A: Statistical Mechanics and its Applications 374(1) Choudhur G and Paul M (2006). A batch arrival queue with a second optional service channel under N polic. Stochastic Analsis and Application Hur S and Park J (1999). The effect of different arrival rates on N polic of M/G/1 with server setup. Applied Mathematical Modeling Jeeva M and Rathnakumari E (2012). Bulk arrival single server Bernoulli Feedback queue with fuzz vacations and fuzz parameters. ARPN Journal of Science and Technolog 2(5) Julia Rose Mar K, Afthab Begum MI, and Jamila PM (2011). Bi-level threshold polic of M X / (G 1, G 2 )/1 queue with earl setup and single vacation. International Journal of Operational Research 10(4) Kao C, i C and Chen S (1999). Parametric programming to the analsis of fuzz queues. Fuzz Sets and Sstem Ke JC (2001). The control polic of M X /G/1 Queing sstem with startup and two vacation tpes. Mathematical Methods of Operations Research 54(31) ee HS and Srinivasan MM (1989). Control Policies for the M X /G/1 queuing sstem. Management Sciences 35(6) ee HW, ee SS and Park JO (1994 a). Operating Characteristics of M X /G/1 queue with N Polic. Queuing Sstems ee HW, ee SS and Park JO (1994 b). Analsis of the M X /G/1 queue with N Polic and multiple vacations. Journal of the Applied Probabilit ee HW, Park N and Jeon J (2003). Queue length analsis of a batch arrival queues under bi-level threshold control with earl setup. International Journal of Sstem Science 34(3) i RJ and ee ES (1989).Analsis of fuzz queues. Computers and Mathematics with Application 17(7) Madan KC (2000). An M/G/1 queue with second optional service. Queuing Sstems Madan KC and Choudhur G (2006).Stead state analsis of an M X /(G 1, G 2 )/1 queue with restricted admissibilit and random setup time. Information and Management Science 17(2) Nagi DS and ee ES (1992). Analsis and simulation of fuzz queues. Fuzz Sets and Sstems Ramesh R and Kumara G G (2013). A batch arrival queue with multiple servers and fuzz parameters parametric programming approach. International Journal of Science and Research 2(9) Zimmermann HF (2001). Fuzz Set Theor and its Application, 4th edition (Kluwer Academic, Boston). Copright 2014 Centre for Info Bio Technolog (CIBTech) 53