Abstract
We consider a finite buffer batch service queueing system with multiple vacations wherein the input process is Markovian arrival process (MAP). The server leaves for a vacation as soon as the system empties and is allowed to take repeated (multiple) vacations. The service- and vacation- times are arbitrarily distributed. We obtain the queue length distributions at service completion, vacation termination, departure, arbitrary and pre-arrival epochs. Finally, some performance measures such as loss probability, average queue lengths are discussed. Computational procedure has been given when the service- and vacation- time distributions are of phase type (PH-distribution).
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References
Blondia C. (1991). Finite Capacity Vacation Model with Non-Renewal Input.Journal of Applied Probability 28, 174–197.
Chakravarthy S. (1993). Analysis of a FiniteMAP/G/1 Queue with Group Services.Queueing Systems 13, 385–407.
Chaudhry M.L. and Templeton J.G.C. (1983).A First Course in Bulk Queues. John Wiley.
Choi B.D., Hwang G.U. and Han D.H. (1998). Supplementary Variable Method Applied to theMAP/G/1 Queueing Systems.Journal of the Australian Mathematical Society 40, 86–96.
Doshi B.T. (1986). Queueing Systems with Vacations: Survey.Queueing Systems 1, 29–66.
Doshi B.T. (1990). Single Server Queues with Vacations. In: Takagi H. (ed.),Stochastic Analysis of Computer and Communication Systems. North-Holland, 217–265.
Ferrandiz J.M. (1993). TheBMAP/G/1 Queue with Server Set-up Times and Server Vacations.Advances in Applied Probability 25, 235–254.
Gupta U.C. and Laxmi P.V. (2001a). The Relations Among the Queue Size Distributions at Departure, Arbitrary and Pre-Arrival Epochs in the MAP/G/1 Queue with Finite/Infinite Buffer — an Alternative Approach.Opsearch 38, 520–530.
Gupta U.C. and Laxmi P.V. (2001b). Analysis of theMAP/G a,b/1/N Queue.Queueing Systems 38, 109–124.
Gupta U.C. and Sikdar K. (2005). On the Finite Buffer Bulk ServiceM/G/1 Queue with Multiple Vacations.Journal of Probability and Statistical Science (to appear).
Kasahara S., Takine T., Takahashi Y., and Hasegawa T. (1996).MAP/G/1 Queues UnderN-Policy With and Without Vacations.Journal of the Operations Research Society of Japan 39, 188–212.
Latouche G. and Ramaswami V. (1999).Introduction to Matrix Analytic Methods in Stochastic Modelling. SIAM & ASA.
Lee H.W., Ahn B.Y. and Park N.I. (2001). Decomposition of the Queue Length Distributions in theMAP/G/1 Queue Under Multiple and Single Vacations withN-Policy.Stochastic Models 17, 157–190.
Lucantoni D.M., Meier-Hellstern K.S. and Neuts M.F. (1990). A Single-Server Queue with Server Vacations and a Class of Non-Renewal Process.Advances in Applied Probability 22, 676–705.
Lucantoni D.M. and Ramaswami V. (1985). Efficient Algorithms for Solving Non-Linear Matrix Equations Arising in Phase Type Queues.Stochastic Models 1, 29–52.
Matendo S.K. (1993). A Single-Server Queue with Server Vacations and a Batch Markovian Arrival Process.Cahiers du CERO 35, 87–114.
Matendo S.K. (1994). Some Performance Measures for Vacation Models with a Batch Markovian Arrival Process.Journal of Applied Mathematics and Stochastic Analysis 7, 111–124.
Neuts M.F. (1981).Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. Johns Hopkins University Press.
Neuts M.F. (1989).Structured Stochastic Matrices of M/G/1 Type and Their Applications. Marcel Dekker.
Niu Z. and Takahashi Y. (1999). A Finite-Capacity Queue with Exhaustive Vacation/Close-Down/Setup Times and Markovian Arrival Processes.Queueing Systems 31, 1–23.
Niu Z., Shu T. and Takahashi Y. (2003). A Vacation Queue with Setup and Close-Down Times and Batch Markovian Arrival Processes.Performance Evaluation 54, 225–248.
Schellhaas H. (1994). Single Server Queues with a Batch Markovian Arrival Process and Server Vacations.OR Spektrum 15, 189–196.
Takagi H. (1991).Queueing Analysis — A Foundation of Performance Evaluation, Vacation and Priority Systems. North-Holland.
Takagi, H. (1993).Queueing Analysis — A Foundation of Performance Evaluation, Finite Systems (vol. 2). North-Holland.
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Sikdar, K., Gupta, U.C. The queue length distributions in the finite buffer bulk-service MAP/G/1 queue with multiple vacations. Top 13, 75–103 (2005). https://doi.org/10.1007/BF02578989
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DOI: https://doi.org/10.1007/BF02578989