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Insensitivity in processes with zero speeds

Published online by Cambridge University Press:  01 July 2016

P. Taylor
P. Taylor*
Affiliation:
University of Adelaide
*
∗Present address: Department of Mathematics, University of Western Australia, Nedlands, WA 6009, Australia.
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Abstract

Many authors have discussed the equivalence of partial balance and insensitivity in in stochastic processes. When speeds are introduced into a stochastic process there arises a difficulty in proving the necessity of partial balance for insensitivity. Previous authors have overcome this difficulty by assuming that a process has the property of instantaneous attention. This property enforces the requirement that no lifetime can be created in a state in which that lifetime has zero speed.

In this paper it is shown that for processes with a finite state space it is unnecessary to make this assumption provided the notion of partial balance is slightly changed. Thus we give a criterion, analogous to partial balance, which is necessary and sufficient for insensitivity even in processes which do not possess the property of instantaneous attention. When a process does have instantaneous attention this criterion is equivalent to partial balance.

Keywords

PARTIAL BALANCEPRODUCT FORMINSTANTANEOUS ATTENTION
Type
Research Article
Information
Advances in Applied Probability , Volume 21 , Issue 3 , September 1989 , pp. 612 - 628
Copyright
Copyright © Applied Probability Trust 1989 

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References

Burman, D. (1981) Insensitivity in queueing systems. Adv. Appl. Prob. 13, 846–859.CrossRefGoogle Scholar
Franken, P., König, D., Arndt, U. and Schmidt, V. (1982) Queues and Point Processes. Wiley, New York.Google Scholar
Henderson, W. (1983) Insensitivity and reversed Markov processes. Adv. Appl. Prob. 15, 752–768.Google Scholar
Hordijk, A. (1984) Insensitivity for stochastic networks. In Mathematical Computer Performance and Reliability, ed. Iazeolla, G., Courtois, P. J. and Hordijk, A., North-Holland, Amsterdam, 77–94.Google Scholar
Hordijk, A. and Van Dijk, N. (1983) Adjoint processes, job local balance and insensitivity for stochastic networks. Proc. 44th Session Internat. Statist. Inst., 776–788.Google Scholar
König, D. and Jansen, U. (1974) Stochastic processes and properties of invariance for queueing systems with speeds and temporary interruptions. Trans. 7th Prague Conf. Information Theory, etc., 335–343.Google Scholar
Matthes, K. (1962) Zur Theorie der Bedienungsprozesse. Trans 3rd Prague Conf. Information Theory, etc., 513–528.Google Scholar
Rumsewicz, M. (1988) Some Contributions to the Fields of Insensitivity and Queueing Theory. , University of Adelaide.Google Scholar
Schassberger, R. (1978) Insensitivity of steady-state distributions of generalised semi-Markov processes with speeds. Adv. Appl. Prob. 10, 836–851.Google Scholar
Schassberger, R. (1986) Two remarks on insensitive stochastic models. Adv. Appl. Prob. 18, 791–814.CrossRefGoogle Scholar
Taylor, P. (1987) Aspects of Insensitivity in Stochastic Processes. , University of Adelaide.Google Scholar
Whittle, P. (1985) Partial balance and insensitivity. J. Appl. Prob. 22, 168–176.CrossRefGoogle Scholar