Study of Impact of Negative Arrivals in Single. Server Fixed Batch Service Queueing System. with Multiple Vacations

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1 Appled Mathematcal Sceces, Vol. 7, 23, o. 4, HIKARI Ltd, Study of Impact of Negatve Arrvals Sgle Server Fxed Batch Servce Queueg System wth Multple Vacatos G. Ayyappa Dept of Mathematcs Podcherry Egeerg College Podcherry, Ida G. Devprya Dept of Mathematcs Sr Gaesh College of Egeerg & Techology Podcherry, Ida A. Muthu Gaapath Subramaa Dept of Mathematcs Kach Mamuvar Cetre for Post Graduate Studes Podcherry, Ida Copyrght 23 G. Ayyappa, G. Devprya ad A. Muthu Gaapath Subramaa. Ths s a ope access artcle dstrbuted uder the Creatve Commos Attrbuto Lcese, whch permts urestrcted use, dstrbuto, ad reproducto ay medum, provded the orgal work s properly cted. Abstract Cosder a sgle server fxed batch servce queueg system uder multple vacato wth a possblty of egatve arrval whch the arrval rate λ follows a Posso process, the servce tme follows a expoetal dstrbuto wth parameter μ. Further we assume a egatve arrval rate occur at the rate of ν whch follows a Posso process ad the legth of tme the server vacato follows a

2 6968 G. Ayyappa, G. Devprya ad A. Muthu Gaapath Subramaa expoetal dstrbuto wth parameter α. Assume that the system tally cotas k customers whe the server eters to the system ad starts the servce mmedately wth a batch of sze k. After completo of a servce, f he fds less tha k customers the queue, the the server goes for a multple vacato of a legth α. If there are more tha k customers the queue the the frst k customers wll be selected from the queue ad servce wll be gve as a batch. We are aalyzg the possblty of egatve arrval ths model. Negatve customers have the effect of deletg some customer the queue. I the smplest verso, a egatve arrval removes a ordary postve customer or a radom batch of postve customers accordg to some strategy. It s oted that the exstece of a flow of egatve arrvals provdes a cotrol mechasm to cotrol excessve cogesto at the queue ad also assume that the egatve customers oly act whe the server s busy. Ths model s completely solved by usg the geeratg fucto techque. We have derved the closed form solutos for probablty of umber of customers the queue durg the server busy ad vacato. Further we are provdg the aalytcal soluto for mea umber of customers ad varace of the system. Varous partcular cases of ths model have bee dscussed. Keywords: Sgle Server, Batch Servce, Negatve arrval, Multple vacato, Steady state dstrbuto.. INTRODUCTION Batch servce queues have umerous applcatos to traffc, trasportato, producto ad maufacturg systems. Baley[2] obtaed the trasform soluto to the fxed-sze batch servce queue wth Posso arrvals. Mller[5] studed the batch arrval batch servce queues ad Jaswal[2] cosdered batch servce queues whch servce sze s radom. Neuts [7] proposed the "geeral bulk servce rule" whch servce tates oly whe a certa umber of customers the queue s avalable. Neuts geeral bulk servce rule was exteded by Borthakur ad Medh [3]. Studes o watg tme a batch servce queue were also redered by Dowto [8], Cohe [6], Medh [4] ad Powell [8]. Fakos [] derved the relato betwee lmtg queue sze dstrbutos at arrval ad departure epochs. Brere ad Chaudhry [4], Grassma ad Chaudhry [], ad Kambo ad Chaudhry [3] used umercal approaches to obta the performace measures. Chaudhry ad Templeto [5] gves more extesve study o batch arrval/servce queues. Gelebe (99) has troduced a ew class of queueg processes whch customers are ether Postve or Negatve. Postve meas a regular customer who s treated the usual way by a server. Negatve customers have the effect of deletg some customer the queue. I the smplest verso, a egatve arrval removes a ordary postve customer or a batch of postve customers accordg to some strategy. Ayyappa et al [] has studed the effect of egatve arrval rate

3 Study of mpact of egatve arrvals 6969 for the retral queueg system. Muthu Gaapath Subramaa et al [6 ] has studed the effect of egatve arrval rate for the prorty retral queueg system. For batch servce queues wth vacatos, there have bee a few related works. Dhas [7] cosdered Markova batch servce systems ad obtaed the queue legth dstrbutos by matrx-geometrc methods. Lee et al. [9] obtaed varous performace measures for M/G B / queue wth sgle vacato. Dshalalow ad Yelle [9] cosdered a o-exhaustve batch servce system wth multple vacatos whch the server starts a multple vacato wheever the queue drops below a level r ad resumes servce at the ed of a vacato segmet whe the queue accumulates to at least r. They called such a system (r, R)-quorum system. Lee et al. [2] showed that for some batch servce queue; mea queue legth may eve decrease systems wth server vacatos. I ths paper we are aalyzg a specal batch servce queue called the fxed sze batch servce queue uder multple vacatos wth egatve arrval. The model s descrbed Secto 2. I Secto 3, we have derved the system steady state equatos ad usg these equatos, the probablty geeratg fuctos for umber of customers the queue whe the server s busy or vacato are derved ad also obtaed steady state probablty dstrbutos. Secto 4 deals wth stablty codto of the system. Closed form solutos of system performace measures are obtaed 6. We are provdg the aalytcal soluto for mea umber of customers ad varace of the system. Also varous partcular cases of ths model have bee dscussed secto DESCRIBITION OF THE MODEL Cosder a sgle server fxed batch servce queueg system uder multple vacato wth a possblty of egatve arrval whch the arrval rate λ follows a Posso process, the servce tme follows a expoetal dstrbuto wth parameter μ. Further we assume egatve arrval occur at the rate of ν whch follows a Posso dstrbuto ad the legth of tme the server vacato follows a expoetal dstrbuto wth parameter α. Assume that the system tally cotas k customers whe the server eters to the system ad starts the servce mmedately wth a batch of sze k. After completo of a servce, f he fds less tha k customers the queue, the the server goes for a multple vacato of a legth α. If there are more tha k customers the queue the the frst k customers wll be selected from the queue ad servce wll be gve as a batch. We are aalyzg the possblty of egatve arrval ths model. Negatve customers have the effect of deletg some customer the queue. I the smplest verso, a egatve arrval removes a ordary postve customer or a radom batch of postve customers accordg to some strategy. It s oted that the exstece of a flow of egatve arrvals provdes a cotrol mechasm to cotrol excessve cogesto at the queue ad also assume that the egatve customers oly act

4 697 G. Ayyappa, G. Devprya ad A. Muthu Gaapath Subramaa whe the server s busy. If there are less tha k customers the queue upo hs retur from the vacato, he mmedately leaves for aother vacato ad so o utl he fally fds k or more customers the queue. Let < N(t),C(t) > be a radom process where N(t) be the radom varable whch represets the umber of customers queue at tme t ad C(t) be the radom varable whch represets the server status (busy/vacato) at tme t. We defe P, (t) - Probablty that there are customers the queue whe the server s busy at tme t. P,2 (t) - Probablty that there are customers the queue whe the server s o vacato at tme t. The Chapma- Kolmogorov equatos are ' P () t = ( λ + μ) P () t + μp () t + ν P () t + αp (t) (),, k,, k,2 P ( t) = ( λ + μ + ν) P ( t) + λp ( t) + νp ( t) + μp ( t) + αp (t) ; =,2,3,... (2) ',,, +, + k, + k,2 ' P,2() t = λp,2() t + μp, () t (3) ' P () t = λp () t + λp () t + μp () t ; =,2,3,...,k- (4),2,2,2, ',2 λ α,2 λ,2 P () t = ( + ) P () t + P () t ; k (5) 3. EVALUATION OF STEADY STATE PROBABILITY: I ths secto, we are fdg the closed form solutos for umber of customers the queue whe the server s busy or the umber of customers the system whe the server s vacato usg geeratg fucto techque. Whe steady state prevals, the equatos () to (5) becomes ( λ + μ) P, = μp k, + νp, + αp k,2 (6) ( λ + μ + ν) P = λp + νp + μp + αp ; =,2,3,...,, +, + k, + k,2 (7) λ P,2 = μ P, (8) λp ; =,2,3,...,k-,2 = λp,2 + μp, (9) ( λ + α) P,2( t) = λp,2; k () Geeratg fuctos for the umber of customers the queue whe the server s busy or the umber of customers the queue whe the server s vacato are defed as,,2 = = Gz ( ) = P z ad H( z) = P z Multply the equato (7) by z o both sdes ad summg over = to ad

5 Study of mpact of egatve arrvals 697 add wth equato (6), we get k ν ( z) z p + λ( z) H( z) Gz ( ) = k+ k k λ z ( λ + μ + ν) z + νz + μ Gz ( ) = G( z) + G2( z) (2) k ν ( zz ) P, λ( zhz ) ( ) where G( z) = ad G ( z) = z+ k k z+ k k λ ( λ + μ + ν) z + νz + μ λ ( λ + μ + ν) z + νz + μ Addg equato (8),(9) ad () after multply wth, z ad z ( =, 2,...) respectvely, we get k k,,2 = = k k P, z + α P,2 z = = H( z)[ α + λ( z)] = μ P z + α P z μ H( z) = () (3) α + λ( z) Equato (4) represets the probablty geeratg fucto for umber of customers the queue whe the server s vacato. Equato (2) represets the probablty geeratg fucto for umber of customers the queue whe the server s busy. The geeratg fucto G (z) has the property that t must coverge sde the ut crcle z <. We otce that the expresso the deomator of G (z), k k λz ( λ μ ν) z μ (4) has k+ zeros. By Rouche's theorem, we otce that k zeros of ths expresso les sde the crcle z < ad must cocde wth k zeros of umerator of G (z) ad oe zero les outsde the crcle z <. Let z be a zero whch les outsde the crcle z <. As G (z) coverges, k zeros of umerator ad deomator wll be cacelled, we get A G ( z) =,whe z =, λ( z z ) A ν P, A ν ( z ) P, G () =, = A = λ( z ) kμ + ν λ λ( z ) kμ + ν λ ν ( z) P, The G ( z) = ( kμ + ν λ)( z z ) ν ( rp ), G ( z) = z r wherer = (5) ( kμ + ν λ) = z The geeratg fucto G 2 (z) has the property that t must coverge sde the ut crcle z <. We otce that the expresso the deomator of G 2 (z), k k λz ( λ μ ν) z μ has k+ zeros. By Rouche's theorem, we otce that k

6 6972 G. Ayyappa, G. Devprya ad A. Muthu Gaapath Subramaa zeros of ths expresso les sde the crcle z < ad must cocde wth k zeros of umerator of G 2 (z) ad oe zero les outsde the crcle z <. Let z be a zero whch les outsde the crcle z <. As G 2 (z) coverges, k zeros of umerator ad deomator wll be cacelled, we get B G2 ( z) =, whe z = λ( z z)( α + λ λz) B λh() B G2 () =, = λ( z ) α kμ+ ν λ αλ( z) To fd H() Put z = equato (), ad usg equato (3) we get ν P, kμ + ν λ H() = kμ + ν λ kμ + ν λαλ( z ) ν P, λαλ( z ) ν P, kμ + ν kμ + ν λ B = the G 2 ( z ) = kμ + ν kμ + ν λ ( z z )( α + λ λz) By applyg partal fractos, we get ν P α, s( r) + + G2 ( z) = r z s z kμ ν kμ ν λ r s (6) + + = = λ where r = ad s = z λ + α Substtute the values of G (z) ad G 2 (z) (2), we get ν( rp ) νp, α, s( r) + + Gz ( ) = zr + zr zs (7) ( kμ + ν λ) = kμ + ν kμ + ν λ r s = = Comparg the coeffcet of z o both sdes of the equato (6), we get ν( rp ), α νp, s( r) + + P, = r + r s for =,2,3,...(8) kμ + ν λ kμ + ν kμ + ν λ r s Usg equato (7) (8), (9) ad (),apply recursve for =,2,3,...,k- ad we get μ P = P ; =,,2...,k- (9),2 t, λ t = k+ λ P,2 = Pk,2 ; k (2) λ + α The ormalzg codto s k (2) P + P + P + P =,,,2,2 = = = k

7 Study of mpact of egatve arrvals 6973 Substtute (8),(9) ad (2) (2), we get, Nr p, = ;k> where (22) Dr k k α( r+ s rs) μαs( r) r μs( r) r Nr = - ( k ) s s ( kμ + ν)( s) λ( kμ + ν) = = s ( kμ + ν) = = s k k νr αν( r + s rs) kμ μ μν( r) μν ( r) Dr = ( k ) r + r kμ + ν λ ( kμ + ν)( kμ + ν λ)( s) λ α λ( kμ + ν λ) = α( kμ+ ν λ) μν ( ) ( ) k k μαν s( r) r s( r) r ( k ) s s λ( μ ν)( μ ν λ) ( μ ν)( μ ν λ) k + k + = = s k + k + = = s Equatos (8),(9),(2) ad (22) represet the steady state probabltes for umber of customers the queue whe the server s busy /vacato. = 4. STABILITY CONDITION The ecessary ad suffcet codto for the system to be stable s ν P, λ + < kμ + ν kμ + ν 5. PARTICULAR CASE If we take ν =, the results cocdes wth the results of the model sgle server batch servce uder multple vacato. 6. SYSTEM PERFORMANCE MEASURES I ths secto, we wll lst some mportat performace measures alog wth ther formulas. These measures are used to brg out the qualtatve behavour of the queueg model uder study. Numercal study has bee dealt very large scale to study the followg measures.. p, Nr = ; k> where Dr k k ( r+ s rs) s( r) r s( r) α μα μ r Nr = - ( k ) s s ( kμ + ν)( s) λ( kμ + ν) = = s ( kμ + ν) = = s

8 6974 G. Ayyappa, G. Devprya ad A. Muthu Gaapath Subramaa k νr αν( r + s rs) kμ μ μν( r) Dr = ( k ) r kμ + ν λ ( kμ + ν)( kμ + ν λ)( s) λ α λ( kμ + ν λ) = μν ( r) α( kμ + ν λ) k = r μν ( ) ( ) μαν s( r) r s( r) r k k ( k ) s s k + k + = = s k + k + = = s λ( μ ν)( μ ν λ) ( μ ν)( μ ν λ) + ν( rp ) α νp s( r) r + ; =,2,3,... kμ + ν λ kμ + ν kμ + ν λ r s μ = P ; =,,2,...,k-,, P, = r s P 3.,2 t, λ t = 4. P λ = λ+ α k+ P,2 k,2 5. Ls = ( + k) P, + P,2 = = 6. Lq = P, + P,2 = ( ) ; k V( x) = ( + k) P, + P,2 ( Ls) = = 7. CONCLUSION: Varous specal cases have bee dscussed, whch are partcular cases of ths research work. Ths research work ca be exteded further by troducg varous cocepts lke breakdow ad repar, secod optoal servce etc. Refereces [] Ayyappa. G, Muthu Gaapath Subramaa. A ad Gopal Sekar, Sgle server Retral queueg system wth egatve arrval uder Pre- emptve prorty servce - Iteratoal Joural of Computatoal ad Cogto (2),Vol 8, No.4,pp 92- [2] Baley, N.T.J., O queueg process wth bulk servce,joural of Royal Statstcal Socety, (954),B6, pp [3] Borthakur, A. ad Medh, J., A queueg system wth arrval ad servce batches of varable sze, Trasportato Scece, (973), 7, pp

9 Study of mpact of egatve arrvals 6975 [4] Brere, G. ad Chaudhry, M.L., Computatoal aalyss of sgle server bulk-servce queues, M/G Y /, Advaced Appled Probablty, (989), 2, pp [5] Chaudhry, M.L. ad Templeto, J.G.C., A Frst Course Bulk Queues, (983),Wley, New York. [6] Cohe, J.W., The Sgle Server Queue, (98), 2d edto, North-Hollad, Amsterdam. [7] Dhas, A.H., Markova Geeral Bulk Servce Queueg Model, (989), Ph.D. thess, Dept. of Math, PSG College of Tech, Ida. [8] Dowto, F., Watg tme bulk servce queues, Joural of Royal Statstcal Socety, (955),B7, [9] Dshalalow, J.H. ad Yelle, J., Bulk put queues wth quorum ad multple vacatos, Mathematcal Problems Egeerg, (996), 2:2, [] Fakos, D., The relato betwee lmtg queue sze dstrbutos at arrval ad departure epochs a bulk queue, Stochastc Processes, (99), 37, [] Grassma, W.K. ad Chaudhry M.L., A ew method to solve steady state queueg equatos, (982) Naval Res. Logst. Quart. 29:3. [2] Jaswal, N.K., A bulk servce queueg problem wth varable capacty, Joural of Royal Statstcal Socety, (964),B26, [3] Kambo, N.S. ad Chaudhry, M.L., A sgle-server bulk-servce queue wth varyg capacty ad Erlag put, INFOR 23:2 (985), [4] Medh, J., Watg tme dstrbuto a Posso queue wth a geeral bulk servce rule, Mgmt. Sc, (975) 2:7, [5] Mller, R.G., A cotrbuto to the theory of bulk queues, Joural of Royal Statstcal Socety, (959), B2, [6] Muthu Gaapath Subramaa, Ayyappa. G, ad Gopal Sekar, Sgle server Retral queueg system wth egatve arrval uder No-Pre emptve prorty servce Malaysa Joural of Fudametal ad Appled Sceces, (29) Volume 5, No.2, pp [7] Neuts, M.F., A geeral class of bulk queues wth Posso put, A.Math.Stat, (967), 38,

10 6976 G. Ayyappa, G. Devprya ad A. Muthu Gaapath Subramaa [8] Powell, W.B., Watg tme dstrbuto for bulk arrval, bulk servce queues wth vehcle holdg ad cacellato strateges, Naval Res. Logst, (987), 34, [9] Lee, S.S., Lee, H.W. ad Chae, K.C., Batch arrval queue wth N-polcy ad sgle vacato, Computer ad Operatos Research, (994), 22:2, [2] Lee, H.W., Lee, S.S., Park, J.O. ad Chae, K.C., Aalyss of M X /G/ queue wth N polcy ad multple vacatos, Joural of Appled Probablty, (994), 3, Receved: October, 23