Hostname: page-component-848d4c4894-ndmmz Total loading time: 0 Render date: 2024-06-12T16:39:37.894Z Has data issue: false hasContentIssue false

Supplementary variable method applied to the MAP/G/1 queueing system

Published online by Cambridge University Press:  17 February 2009

Bong Dae Choi
Affiliation:
Center for Applied Mathematics and Department of Mathematics, Korea Advanced Institute of Science and Technology, 373–1 Kusong-Dong, Yusong-Gu, Taejon, 305–701, Korea
Gang Uk Hwang
Affiliation:
Center for Applied Mathematics and Department of Mathematics, Korea Advanced Institute of Science and Technology, 373–1 Kusong-Dong, Yusong-Gu, Taejon, 305–701, Korea
Dong Hwan Han
Affiliation:
Department of Mathematics, Sun Moon University, Asan-Kun, Chungnam, 337–840, Korea
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we consider the MAP/G/1 queueing system with infinite capacity. In analysis, we use the supplementary variable method to derive the double transform of the queue length and the remaining service time of the customer in service (if any) in the steady state. As will be shown in this paper, our method is very simple and elegant. As a one-dimensional marginal transform of the double transform, we obtain the generating function of the queue length in the system for the MAP/G/1 queue, which is consistent with the known result.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1998

References

[1]Bellman, R., Introduction to matrix analysis (McGraw-Hill, New York, 1970).Google Scholar
[2]Hokstad, B. T., “A supplementary variable technique applied to the M/G/l queue”, Scand. J. Stat. 2 (1976) 95103.Google Scholar
[3]Karlin, S. and Taylor, H. M., A first course in stochastic processes (Academic Press, Inc., New York, 1975).Google Scholar
[4]Lucantoni, D. M., Meier-Hellstem, K. S. and Neuts, M. F., “A single-server queue with server vacations and a class of non-renewal arrival processes”, Adv. Appl. Prob. 22 (1990) 676705.CrossRefGoogle Scholar
[5]Lucantoni, D. M., “New results on the single server queue with a batch Markovian arrival process”, Stock. Mod. 7 (1991) 146.Google Scholar
[6]Lucantoni, D. M., Choudhury, G. L. and Whitt, W., “The transient BMAP queue”, Stoch. Mod. 10 (1994) 145182.Google Scholar
[7]Lucantoni, D. M. and Neuts, M. F., “Simpler proofs of some properties of the fundamental period of the MAP/G/1 queue”, J. Appl. Prob. 31 (1994) 235243.CrossRefGoogle Scholar
[8]Mine, H., Nonnegative Matrices (John Wiley & Sons, Inc., New York, 1988).Google Scholar
[9]Neuts, M. F., “A versatile Markovian point process”, J. Appl. Prob. 16 (1979) 764779.CrossRefGoogle Scholar
[10]Neuts, M. F., Matrix-Geometric Solutions in Stochastic Models (The John Hopkins University Press, Baltimore, MD, 1981).Google Scholar
[11]Neuts, M. F., Structured Stochastic Matrices of M/G/l Type and Their Applications (Marcel Dekker, New York, 1989).Google Scholar
[12]Ramaswami, V., “The N/G/l queue and its detailed analysis”, Adv. Appl. Prob. 12 (1980) 222261.CrossRefGoogle Scholar
[13]Ramaswami, V., “From the matrix-geometric to the matrix-exponential”, Queueing Systems 6 (1990) 229260.CrossRefGoogle Scholar
[14]Seneta, E., Nonnegative Matrices: An Introduction to Theory and Applications (John Wiley & Sons, New York, 1980).Google Scholar
[15]Sengupta, B., “Markov processes whose steady state distribution is Matrix-exponential with an application to the GI/PH/1 queue”, Adv. Appl. Prob. 21 (1989) 159180.CrossRefGoogle Scholar
[16]Takagi, H., Queueing Analysis: A Foundation of Performance Evaluation Volume I: Vacation and Priority Systems, Part I (Elsevier Science Publisher B. B., (North Holland), Amsterdam, 1991).Google Scholar
[17]Takine, T. and Hasegawa, T., “The workload in the MAP/G/1 queue with state-dependent services: Its application to a queue with preemptive resume priority”, Stock Mod. 10 (1994) 183204.Google Scholar