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Partial balance, insensitivity and weak coupling

Published online by Cambridge University Press:  01 July 2016

P. Whittle
P. Whittle*
Affiliation:
University of Cambridge
*
∗Postal address: Statistical Laboratory, 16 Mill Lane, Cambridge CB2 1SB, UK.
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Abstract

The idea is developed of imbedding a given Markov process in a more general Markov process in a definite sense, the sense of weak coupling. It is shown that this gives a natural theory of insensitivity in that, the stronger the balance conditions satisfied by the imbedded process, the weaker the conditions that are required of the imbedding process. The imbeddings associated with a range of balance conditions are discussed.

Keywords

IMBEDDINGJACKSON NETWORKS
Type
Research Article
Information
Advances in Applied Probability , Volume 18 , Issue 3 , September 1986 , pp. 706 - 723
Copyright
Copyright © Applied Probability Trust 1986 

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References

Barbour, A. D. (1982) Generalized semi-Markov schemes and open queueing networks. J. Appl. Prob. 19, 469–474.CrossRefGoogle Scholar
Gordon, W. J. and Newell, G. F. (1967) Closed queueing systems with exponential servers. Operat. Res. 15, 254–265.Google Scholar
Horduk, A. (1984) Insensitivity for stochastic networks. In Mathematical Computer Performance and Reliability , ed. Iazeolla, G., Courtois, P. J. and Hordijk, A., Elsevier North-Holland, Amsterdam.Google Scholar
Hordijk, A. and Van Dijk, N. (1984) Networks of queues. In Modelling and Performance Evaluation Methodology , Ed. Baccelli, F. and Fayolle, G., Springer-Verlag, Berlin.Google Scholar
Jackson, J. R. (1963) Jobshop-like queueing systems. Management Sci. 10, 131–142.Google Scholar
Jansen, U. and König, D. (1980) Insensitivity and steady-state probabilities in product form for queueing networks. Electron. Inf. Kybernet. 16, 385–397.Google Scholar
Kelly, F. P. (1979) Reversibility and Stochastic Networks. Wiley, New York.Google Scholar
Kelly, F. P. (1986) Blocking probabilities in large circuit-switched networks. Adv. Appl. Prob. 18, 473–505.Google Scholar
Kingman, J. F. C. (1969) Markov population processes. J. Appl. Prob. 6, 1–18.Google Scholar
König, D., Matthes, K. and Nawrotzki, K. (1967) Verallgemeinerungen der Erlangschen und Engsetschen Formeln (Eine Methode der Bedienungstheorie). Akademie-Verlag, Berlin.Google Scholar
Matthes, K. (1962) Zur Theorie der Bedienungsprozesse. Trans. 3rd Prague Conf. Inf. Theory. Google Scholar
Muntz, R. R. (1972) Poisson departure processes and queueing networks. IBM Research Report RC4145, IBM Thomas J. Watson Research Center, Yorktown Heights, New York.Google Scholar
Ross, S. M. (1983) Stochastic Processes. Wiley, New York.Google Scholar
Schassberger, R. (1977) Insensitivity of steady-state distributions of generalised semi-Markov processes. Part I. Ann. Prob. 5, 87–99.Google Scholar
Schassberger, R. (1978) The insensitivity of stationary probabilities in networks of queues. Adv. Appl. Prob. 10, 906–912.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. (1978) Insensitivity of steady-state distributions of generalised semi-Markov processes. Part II. Ann. Prob. 6, 85–93.Google Scholar
Schassberger, R. (1979) A definition of discrete product-form distributions. Z. Operat. Res. 23, 189–195.Google Scholar
Whittle, P. (1967) Nonlinear migration processes. Bull. Internat. Inst. Statist. 42, 642–647.Google Scholar
Whittle, P. (1984) Weak coupling in stochastic systems. Proc. R. Soc. London A 395, 141–151.Google Scholar
Whittle, P. (1985) Partial balance and insensitivity. J. Appl. Prob. 22, 168–176.Google Scholar
Whittle, P. (1986) Systems in Stochastic Equilibrium. Wiley, New York.Google Scholar