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The basic equations for a supplemented GSMP and its applications to queues

Published online by Cambridge University Press:  14 July 2016

Masakiyo Miyazawa  and
Genji Yamazaki
Masakiyo Miyazawa*
Affiliation:
Science University of Tokyo
Genji Yamazaki*
Affiliation:
Tokyo Metropolitan Institute of Technology
*
∗ Postal address: Department of Information Sciences, Science University of Tokyo, Noda City, Chiba 278, Japan.
∗∗ Postal address: Tokyo Metropolitan Institute of Technology, 6–6 Asahigaoka, Hino-city, Tokyo 191, Japan.
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Abstract

A supplemented GSMP (generalized semi-Markov process) is a useful stochastic process for discussing fairly general queues including queueing networks. Although much work has been done on its insensitivity property, there are only a few papers on its general properties. This paper considers a supplemented GSMP in a general setting. Our main concern is with a system of Laplace–Stieltjes transforms of the steady state equations called the basic equations. The basic equations are derived directly under the stationary condition. It is shown that these basic equations with some other conditions characterize the stationary distribution. We mention how to get a solution to the basic equations when the solution is partially known or inferred. Their applications to queues are discussed.

Keywords

PALM DISTRIBUTIONPOINT PROCESSSTATIONARY DISTRIBUTIONGENERATOR OF MARKOV PROCESSBASIC EQUATION
Type
Research Papers
Information
Journal of Applied Probability , Volume 25 , Issue 3 , September 1988 , pp. 565 - 578
Copyright
Copyright © Applied Probability Trust 1988 

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References

Burman, D. Y. (1981) Insensitivity in queueing systems. Adv. Appl. Prob. 13, 846–859.Google Scholar
Ethier, S. N. and Kurtz, T. G. (1986) Markov Processes: Characterization and Convergence. Wiley, New York.Google Scholar
Feller, W. (1971) Introduction to Probability Theory and Its Applications, Vol. II. Wiley, New York.Google Scholar
Franken, P., König, D., Arndt, U. and Schmidt, V. (1982) Queues and Point Processes. Wiley, Chichester.Google Scholar
Henderson, W. (1983a) Non-standard insensitivity. J. Appl. Prob. 20, 288–296.Google Scholar
Henderson, W. (1983b) Insensitivity and reversed Markov processes. Adv. Appl. Prob. 15, 752–768.Google Scholar
Jansen, U. and König, D. (1980) Insensitivity and steady-state probabilities in product form for queueing networks. Electron. Inf. Kybernet. 16, 385–397Google Scholar
Kelly, F. P. (1979) Reversibility and Stochastic Networks. Wiley, New York.Google Scholar
Matthes, K. (1962) Zur Theorie der Bedienungsprozesse. Trans. 3rd Prague Conf. Information Theor. Statist. Decision Function. Random Processes.Google Scholar
Miyazawa, M. (1979) A formal approach to queueing processes in the steady state and their applications. J. Appl. Prob. 16, 332–346.Google Scholar
Miyazawa, M. (1983) The derivation of invariance relations in complex queueing systems with stationary inputs. Adv. Appl. Prob. 15, 874–885.Google Scholar
Miyazawa, M. (1985) The intensity conservation law for queues with randomly changed service rate. J. Appl. Prob. 22, 408–418.Google Scholar
Miyazawa, M. (1986) Approximation of the queue-length distribution of an M/GI/s queue by the basic equations. J. Appl. Prob. 23, 443–458.Google Scholar
Miyazawa, M. (1987) A generalized Pollaczek-Khinchine formula for the GI/GI/1/k queue and its application to approximation. Stoch. Models 3.Google Scholar
Miyazawa, M. (1988) The characterization of the stationary distributions of the supplemented processing jump process (PJP). In preparation.Google Scholar
Schassberger, R. (1977) Insensitivity of steady state distributions of generalized semi-Markov processes. Part I. Ann. Prob. 5, 87–99.Google Scholar
Schassberger, R. (1978) Insensitivity of steady state distributions of generalized semi-Markov processes. Part II. Ann. Prob. 6, 85–93.Google Scholar
Whitt, W. (1980) Continuity of generalized semi-Markov processes. Math. Operat. Res. 5, 494–501.Google Scholar
Whittle, P. (1986) Partial balance, insensitivity and weak coupling. Adv. Appl. Prob. 18, 706–723.Google Scholar
Wolff, R. W. and Wrightson, C. W. (1976) An extension of Erlang's loss formula. J. Appl. Prob. 13, 628–632.Google Scholar
Yamazaki, G. and Sakasegawa, H. (1987) Limited processor sharing discipline in G/GI/1 models. Management Sci. Submitted.Google Scholar