Hostname: page-component-7479d7b7d-t6hkb Total loading time: 0 Render date: 2024-07-14T07:41:49.275Z Has data issue: false hasContentIssue false
  • English
  • Français

Insensitivity of processes with interruptions

Published online by Cambridge University Press:  14 July 2016

W. Henderson  and
P. Taylor
W. Henderson*
Affiliation:
University of Adelaide
P. Taylor*
Affiliation:
University of Western Australia
*
∗ Postal address: Applied Mathematics Department, The University of Adelaide, P.O. Box 498, Adelaide, SA 5001, Australia.
∗∗ Postal address: Mathematics Department, The University of Western Australia, Nedlands, WA 6009, Australia.
Get access Rights & Permissions [Opens in a new window]

Abstract

The theory of insensitivity within generalised semi-Markov processes is extended to cover classes of models in which the generally distributed lifetimes can be terminated prematurely by the deaths of negative exponentially distributed lifetimes. As a consequence of this approach it is shown that there exist classes of processes which are insensitive with respect to characteristics of the general distributions other than the mean. Two examples are given. The first is an analysis of networks of queues in which the generally distributed service times can be interrupted with resulting changes in routing probabilities. The second is a model for the effect of disturbances on the evolution of a vegetation community.

Keywords

GENERALISED SEMI-MARKOV PROCESSESPARTIAL BALANCEPRODUCT FORM
Type
Research Papers
Information
Journal of Applied Probability , Volume 26 , Issue 2 , June 1989 , pp. 242 - 258
Copyright
Copyright © Applied Probability Trust 1989 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Burman, D. Y. (1981) Insensitivity in Queueing Systems, Adv. Appl. Prob. 13, 846–859.Google Scholar
Chandy, K. M. and Martin, A. J. (1983) A characterisation of product form queueing networks. J. Assoc. Comput Mach. 30, 286–299.Google Scholar
Cinlar, E. (1975) Introduction to Stochastic Processes. Prentice-Hall, Englewood Cliffs, N.J. Google Scholar
Haas, P. J. and Shedler, G. S. (1987) Stochastic Petri nets with simultaneous transition firings. Proc. International Workshop on Petri Nets and Performance Models, Madison, Wisconsin, 24–32.Google Scholar
Henderson, W. (1983) Insensitivity and reversed Markov processes. Adv. Appl. Prob. 15, 752–768.Google Scholar
Henderson, W. and Lucic, D. (1988) Applications of generalised semi-Markov processes to stochastic Petri nets. Proc. Internat. Sem. Performance of Distributed and Parallel Systems, Kyoto.Google Scholar
Henderson, W. and Wilkins, C. W. (1975) The interaction of bushfires and vegetation. Search 6, 130–133.Google Scholar
Jacobi, H. (1965) Eine Unempfindlichkeitseigenschaft für geordnete Bündel ungeordnete Teilbündel. Wiss. Z. Friedrich-Schiller-Universität Jena. Math. Nat. 14, 251–260.Google Scholar
Kelly, F. P. (1979) Reversibility and Stochastic Networks. Wiley, London.Google Scholar
König, D. (1965) Verallgemeinerungen der Engsetschen Formeln. Math. Nachr. 28, 145–155.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 Conference Inf. Theory Statist Dec. Fns and Random Procs. 335–343.Google Scholar
Matthes, K. (1962) Zur Theorie der Bedienungsprozesse. Trans 3rd Prague Conference Inf. Theory Statist Dec. Fns and Random Procs.Google Scholar
Miyazawa, M. (1988) The characterization of stationary distributions of the supplemented self-clocking jump process. Math. Operat. Res. To appear.Google Scholar
Nobel, I. R. and Slatyer, R. O. (1980) The use of vital attributes to predict successional changes in plant communities subject to recurrent disturbances. Vegetatio 43, 5–21.Google Scholar
Rumsewicz, M. P. and Henderson, W. (1989) Insensitivity with age dependent routing. Adv. Appl. Prob. 21(2).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 processes. Adv. Appl. Prob. 18, 791–814.Google Scholar
Taylor, P. G. (1987) Aspects of Insensitivity in Stochastic Processes. Ph. D. Thesis, University of Adelaide.Google Scholar
Wilkins, C. W. (1976) A study of the action of wildfires on remote forests. Stoch. Proc. Appl. 4, 187–202.Google Scholar
Whittle, P. (1985) Partial balance and insensitivity. J. Appl. Prob. 22, 168–176.Google Scholar
Wolff, R. and Wrightson, C. (1976) An extension of Erlang's loss formula. J. Appl. Prob. 13, 628–632.Google Scholar