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Insensitivity of steady-state distributions of generalized semi-Markov processes by speeds

Published online by Cambridge University Press:  01 July 2016

R. Schassberger
R. Schassberger*
Affiliation:
University of Calgary
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Abstract

A generalized semi-Markov process with speeds describes the fluctuation, in time, of the state of a certain general system involving, at any given time, one or more living components, whose residual lifetimes are being reduced at state-dependent speeds. Conditions are given for the stationary state distribution, when it exists, to depend only on the means of some of the lifetime distributions, not their exact shapes. This generalizes results of König and Jansen, particularly to the infinite-state case.

Keywords

GENERALIZED SEMI-MARKOV PROCESSESINSENSITIVITYLOCAL BALANCEPRODUCT PROPERTYPOINT PROCESSES
Type
Research Article
Information
Advances in Applied Probability , Volume 10 , Issue 4 , December 1978 , pp. 836 - 851
Copyright
Copyright © Applied Probability Trust 1978 

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References

[1] Barbour, A. D. (1976) Networks of queues and the method of stages. Adv. Appl. Prob. 8, 584–591.CrossRefGoogle Scholar
[2] Baskett, F., Chandy, K. M., Muntz, R. R. and Palacios, F. G. (1975) Open, closed and mixed networks of queues with different classes of customers. J. Assoc. Comput. Mach. 22, 248–260.CrossRefGoogle Scholar
[3] Hermann, U. (1965) Ein Approximationssatz für Verteilungen stationärer zufälliger Punktfolgen. Math. Nachr. 30, 377–381.CrossRefGoogle Scholar
[4] Hordijk, A. and Schassberger, R. (1977) Weak convergence theorems for generalized semi-Markov processes. Technical Report, Department of Mathematics and Statistics, University of Calgary.Google Scholar
[5] Karlin, S. (1966) A First Course in Stochastic Processes. Academic Press, New York.Google Scholar
[6] Kelly, F. P. (1976) Networks of queues. Adv. Appl. Prob. 8, 416–432.CrossRefGoogle Scholar
[7] König, D. and Jansen, U. (1976) Eine Invarianzeigenschaft zufalliger Bedienungsprozesse mit positiven Geschwindigkeiten. Math. Nachr. 70, 321–364.CrossRefGoogle Scholar
[8] König, D., Matthes, K. and Nawrotzki, K. (1967) Verallgemeinerungen der Erlangschen und Engsetschen Formeln. Akademie-Verlag, Berlin.Google Scholar
[9] König, D., Matthes, K. and Nawrotzki, K. (1974) Unempfindlichkeitseigenschaften von Bedienungsprozessen. Appendix in Introduction to Queueing Theory (German edition) by Gnedenko, B. W. and Kovalenko, I. N.. Akademie-Verlag, Berlin.Google Scholar
[10] Matthes, K. (1962) Zur Theorie der Bedienungsprozesse. Trans. 3rd Prague Conf. Inf. Theory. Google Scholar
[11] Schassberger, R. (1973) Warteschlangen. Springer, Vienna.CrossRefGoogle Scholar
[12] Schassberger, R. (1977) Insensitivity of steady-state distributions of generalized semi-Markov processes, Part I. Ann. Prob. 5, 87–99.CrossRefGoogle Scholar
[13] Schassberger, R. (1978) Insensitivity of steady-state distributions of generalized semi-Markov processes, Part II. Ann. Prob. 6, 85–93.CrossRefGoogle Scholar