Indin Journl of Mthemtics nd Mthemticl Sciences Vol. 7, No., (June ) : 9-38 TANDEM QUEUE WITH THREE MULTISERVER UNITS AND BULK SERVICE WITH ACCESSIBLE AND NON ACCESSBLE BATCH IN UNIT III WITH VACATION G. Ayyppn & S. Sridevi Abstrct This pper nlyses two different queuing model (Model I nd Model II) both consisting of three units connected in series seprted by buffer of finite cpcity with finite number of prllel servers in ech unit. Queuing model differs only on the provision of viling vction to the servers in unit III. In Model I server cn vil exponentil vction, wheres in Model II the server tkes only single vction. Customers of unit I nd Unit II re served singly but in unit III, they re served in groups ccording to bulk service rule. This rule dmits ech btch to be served only if the btch size is not less thn nd not more thn b, customers rriving lte cn lso enter service sttion without ffecting the service time if the size of the btch being served is less thn d ( d d d b). The occurrence time nd service time hve negtive exponentil distributions. The stedy stte probbility vector of the number of customers in the queue is obtined by using modified geometric method. The stbility condition is lso obtined. AMS () Subject Clssifiction Number: 6k5 nd 65k3. Keywords: Bulk service queues, Accessible nd non-ccessible btch service, Mtrix geometric method, Stedy stte solution, Vction.. INTRODUCTION Mny uthors hd lredy discussed bout the queuing models with seril connection of units seprted by n intermedite witing room of finite cpcity. A model in which two units connected in series with n finite witing room in between hs been introduced by Neuts [968]. The concept of blocking in two or more units in service with generl service time distribution without n intermedite buffer hs been considered by Avi-Itzhk nd Ydin (965), Clrke (977). A three stge multi server queuing system with finite queues in ech stge nd blocking hs been nlyzed by Arndt nd Sulnke [985].Tndem Queue with three multi server units connected in series ws nlyzed by Ayyppn.G nd S.Velmurugn (8).
3 G. Ayyppn & S. Sridevi In this pper we del with queuing model consists of three units in series ech hs finite number of prllel servers c i (i =,, 3). These three units re seprted by two witing rooms of cpcities M nd N, which re finite nd independent. The customers in unit I nd unit II re served singly. A queue of infinite length is llowed for unit I nd of finite cpcity M for unit II. The rrivl of customers t Poisson rte λ is independent of service time. The service time in the i th unit, µ i (i =,, 3) re ssumed to be independently distributed rndom vribles. Customers in unit III re served in groups ccording to bulk service rule. This rule dmits ech btch to be served only if the btch size is not less thn nd not more thn b, customers rriving lte cn lso enter service sttion without ffecting the service time if the size of the btch being served is less thn d ( d b). In ddition, Vction is introduced to servers in unit III, in two models. One of multiple exponentil vction (Model I) nd the other of single vction (Model II). MODEL I This model reltes to queues with ccessible nd non ccessible service system with vction. When server completes service nd the btch size nd btch size in unit III is less thn then the server leves for vction. As soon s server returns from the vction nd finds tht the btch size is still less thn then the server immeditely leves for nother vction. Thus the server vils n exponentil vction until the btch size reched.. Stedy Stte Probbility Vector The stedy stte process under considertion cn be formulted s continues time Mrkov chin with sttes S = {(i, j, k, n); i, j M +c, k N, n c 3 } U {(i, j, k, m, n); i, j M + c, k =, m d, n c 3 } Where i denote number of customers in unit I, j denotes number of customers in unit II, k denotes number of customers in buffer of unit III, n denotes number of busy servers in unit III, m denotes number of customers in ccessible service btch (number of customer in the service btch is less thn d) nd c 3 n servers re in vction. The infinitesiml genertor Q of the continues time Mrkov chin
Tndem Queue with Three Multiserver Units nd Bulk Service... 3 c c c + B A M B A M B A Q = c ( c ) Bc A c c M Bc A c + c + M Bc + A The sub mtrices A, B i ( i c ) nd γm ( γ c ), which re squre mtrices of order (M + c + ) ((N + ) + (d ) c 3 ) re defined s below: A = λi where I is unit mtrix of order (M + c + ) ((N + ) + (d ) c 3 ). c M + c c M + c D M M D M M D M c cm cd cm M + c ()()() M + c M M + c D M + c M B = D c D M 3 c D M3 c c D cm 3 cc c ()()() M + c D M + c M3 M + c c M + c Where i c nd β D, β M re given below nd β M + c, β M + c N c c β D = c c N c nd M N E E β = E E E N where E hving only jµ in the min digonl otherwise other elements re zero. c hvingmin digonl elements s (λ + iµ + jµ + kµ 3 ) nd the lower digonl elements re kµ 3 nd c hving min digonl elements s (λ + iµ + jµ + kµ 3 + (c 3 k) α) otherwise other elements re nd k c 3
3 G. Ayyppn & S. Sridevi + d F β M = F d F N nd F c3α µ 3 ( c3 ) α µ ( c ) α 3 3 = c3µ 3 Where β C hve G s sub mtrixon its min digonl nd mtrix G is given below F () λ + iµ + jµ + µ 3 µ 3 ( λ + iµ + jµ + ) µ 3 µ ( λ + iµ + jµ + 3) µ 3 3 = where β M 3 = d + d G + G G d ( c )() µ λ + iµ + jµ + c µ 3 3 3 3 ij + + + c ij jµ ij jµ β M 3 = ijc jµ 3 3 β D = d N G + G G G d Where G hve µ 3 s its min digonl elements otherwise other element in tht mtrix re zero.
Tndem Queue with Three Multiserver Units nd Bulk Service... 33 c M + c c M + c c M + c jm = c M + c Where the mtrix θ, φ re given by the reltions, θ = µ I, where I is unit mtrix of order (N + ) Φ = µ I, where I is unit mtrix of order (d ) c 3 Let us denote by, the vector of stedy stte probbilities ssocited with Q such tht Ech i Q =, e = () Where e = (,, ) T. Let us prtition s. = [,,, c c,,...] c + Where i for i re vectors of order (M + c + ) ((N + ) + (d ) c 3 ). is given by i = [ i, i, i ic3, i, i, i ic3 i, i, i ic3 in, in, in inc3, i i, i ic3, i, i, i ic3 i, i, i ic3 in, in, in inc3, im + c, im + c, im + c im + c c 3, im + c, im + c, im + c im + c c 3 im + c, im + c, im + c im + c c 3 im + c N, im + c N, im + c N im + c Nc 3 i, i ic3, i +, i + i + c3 id, id id c 3, i, i ic3, i +, i + i + c 3 id,, id,... id c3 im + c 3, im + c 3 im + c3 c 3, im + c +,, im + c +,... im + c + c 3 im + c d, im + c d im + c d c 3 ]
34 G. Ayyppn & S. Sridevi The equilibrium eqution Q = cn be expressed in mtrix difference form s k A + k + B c + k + (c M ) = k = c, c, c +,... () With boundry equtions B + M = A + B + ( M ) = A + B + 3 (3 M ) =......... c A + c B c + c (c M ) = (3) In the stble cse there exists the stedy stte probbility vector i i c c, = R + for i c. (4) If mtrix geometric solution exists, the system () becomes i c + c [()] c R A + RB + R c M for i c. (5) The mtrix R is the miniml solution to mtrix nonliner eqution c () A + RB + R c M =, (6) where R Wllce (969) nd it is n irreducible nonnegtive mtrix of spectrl rdius less thn one. Ltouche nd Neuts (98) hve proposed n itertive pproch for finding the mtrix R s follows: R () = (7) R (n + ) = A (B c ) R (n) (c M ) (D c ). For Mrkov process with such genertor, Neuts (978) obtined the stbility condition Π A e < Π A e (8) The corresponding equilibrium condition in this cse is Π A e < Π (c M ) e (9) where the row vector Π is defined s follows. Consider the infinitesiml genertor A = A + B c + c M. A is irreducible. There is unique vector Π, such tht Π A = nd Π e = ()
Tndem Queue with Three Multiserver Units nd Bulk Service... 35 In this cse Π = [Π, Π, Π Π c3, Π, Π, Π Π c3 Π, Π, Π Π c3 Π N, Π N, Π N Π Nc3, Π Π, Π Π c3, Π, Π, Π Π c3 Π, Π, Π Π c3 Π N, Π N, Π N Π Nc3, Π M + c, Π M + c, Π M + c Π M + c c 3, Π M + c, Π M + c, Π M + c Π M + c c 3 Π M + c, Π M + c, Π M + c Π M + c c 3 Π M + c N, Π M + c N, Π M + c N Π M + c Nc 3 Π, Π Π c3, Π +, Π + Π + c3 Π d, Π d Π d c 3, Π, Π Π c3, Π +, Π + Π + c 3 Π d,, Π d,... Π d c3 Π M + c 3, Π M + c 3 Π M + c3 c 3, Π M + c +,, Π M + c +,... Π M + c + c 3 Π M + c d, Π M + c d Π M + c d c 3 ] Since Π A e = λ nd Π (c M ) e = c µ [ (Π M + c + Π M + c + Π M + c +... + Π M + c c 3 + Π M + c + Π M + c + Π M + c +... + Π M + c c 3 +... + Π M + c + Π M + c +... + Π M + cc 3 +... + Π M + cn + Π M + c N +... + Π M + c Nc 3 + Π M + c +... + Π + Π M + cc 3 + Π M + c + c 3 +... + Π M + c d c 3 )]. The stbility condition (9) becomes N c3 c3 d λ < c µ ΠM + c, k, l ΠM + c,, + ik k = l = k = i =. The vectors,, c, c re left to be determined. Let Q * be defined by Q * B A M B A M B A. = A ( c )() M Bc + R cm
36 G. Ayyppn & S. Sridevi To prove tht Q * e = it is enough to prove tht the lst row of Q * e = since the other rows re identicl to tht of Q. (The lst row of Q * ) e = [( c )()] M + Bc + R cm e = = = i c i = [()](()) A + R c M e + R A + RB + R c M e i i+ i+ ()() c i= i= i= A e+ R c M e+ R A e+ R B e+ R c M e i i + i + c + () k = i = i = R A e R B e R c M e = i = i R () A + B + c M e c = since (A + B c + c M ) e = Therefore Q * is n infinitesiml genertor. It is lso irreducible. Let * = (,, c ) be solution of the eqution * Q * =. () The eqution () cn be expressed in mtrix eqution s B + M = A + B + ( M ) = A + B + 3 (3 M ) =......... c A + c B c + c (c M ) = The vectors,, c cn be expressed in terms of c. Using the bove set of equtions nd c my be normlized by c i e + c () I R e =. i = Thus, the below vector re uniquely determined,, c, c. 3. MODEL II In this model slight vrition in model I is considered. As in model I, the underlying structure is queues with server s vction. However the server here tkes only
Tndem Queue with Three Multiserver Units nd Bulk Service... 37 single vction t time. When the server returns to the min system then immeditely strts servicing if the btch size is greter thn else if the btch size is less thn then the server wits until the queue size becomes. The rriving rte is λ, the service rte is µ i (i =,, 3) nd vction for unit III servers is n exponentil distribution rndom vrible /α. 3. Stedy Stte Probbility Vector The stedy stte process under considertion cn be formulted s continues time Mrkov chin with sttes S = {(i, j, k, n, m); i, j M + c, k, n, m c 3, n + m c 3 } U {(i, j, k, l, n, m); i, j M + c, k =, l d, n, m c 3, n + m c 3 } U {(i, j, k, n); i, j M + c, k N, n c 3 } Where i denote number of customers in unit I, j denotes number of customers in unit II, k denotes number of customers in buffer of unit III, n denotes number of busy servers in unit III, l denotes number of customers in ccessible service btch (number of customer in the service btch is less thn d), m denotes number of idel servers in unit III nd c 3 (n + m) servers re in vction. The technique used for the nlysis of model I is successfully pplied for the bove described model II. The detils re not presented here s they re similr to tht of model I. REFERENCES [] Arndt K., nd H. Sulnke, (985), Anlysis of Three-Stge Mny-Server Exponentil Queuing System with Blocking, J. Info. Process. Cyber. EIK, (4/5), 9-. [] Aviltzhk B., nd M. Ydin, (965), A Sequence of Two Queues with no Intermedite Queue, Mgmt. Sct.,, 553-564. [3] Ayyppn G., nd S. Velmurugn, (8), Tndem Queue with Three Multi Server Units with Intermedite Witing Room, Interntionl Journl, 3(), 3-6. [4] Clrke A. B., (977), A Two Server Tndem Queuing System with Storge between Servers, Mth Rept. No.5, Western Michign University, Klmzoo. [5] Ltouche G., nd M. F. Neuts, (98), Efficient Algorithmic Solution to Exponentil Tndem Queues with Blocking, SIAM. J. Algebric Discrete Mth.,, 93-6. [6] Neuts M. F., (967), A Generl Clss of Bulk Queues with Poisson Input, Ann. Mth. Sttist., 38, 759-77. [7] Netus M. F., (968), Two Queues in Series with Finite Intermedite Witing Room, J. Appl. Prob., 5, 3-4.
38 G. Ayyppn & S. Sridevi [8] Neuts M. F., (978), Mrkov Chins with Applictions in Queuing Theory which hve Mtrix-Geometric Invrint Probbility Vector, dv. Appl. Prob.,, 85-. [9] Wllce, (969), The Solution of Qusi Birth nd Deth Processes Arising from Multiple Access Computer Systems, Ph.D. Thesis, Systems Engineering Lbortory University of Michign. G. Ayyppn Deprtment of Mthemtics, Pondicherry Engineering College, Puducherry-654, Indi. S. Sridevi Deprtment of Mthemtics, St.Anne s Engineering nd Technology, Pnruti-676, Indi.