Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-24T01:37:51.207Z Has data issue: false hasContentIssue false
  • English
  • Français

Insensitive average residence times in generalized semi-Markov processes

Published online by Cambridge University Press:  01 July 2016

A. D. Barbour  and
R. Schassberger
A. D. Barbour*
Affiliation:
Gonville and Caius College, Cambridge
R. Schassberger*
Affiliation:
Technical University of Berlin
*
∗Postal address: Statistical Laboratory, University of Cambridge, 16 Mill Lane, Cambridge CB2 1SB, U.K.
∗∗Postal address: Fachbereich Mathematik, Technische Universität Berlin, Strasse des 17. Juni 135, D-1000 Berlin 12, Germany.
Get access Rights & Permissions [Opens in a new window]

Abstract

For a broad class of stochastic processes, the generalized semi-Markov processes, conditions are known which imply that the steady state distribution of the process, when it exists, depends only on the means, and not the exact shapes, of certain lifetime distributions entering the definition of the process. It is shown in the present paper that this insensitivity extends to certain average and conditional average residence times. Particularly interesting applications can be found in the field of networks of queues.

Keywords

GENERALIZED SEMI-MARKOV PROCESSESINSENSITIVITYNETWORKS OF QUEUES
Type
Research Article
Information
Advances in Applied Probability , Volume 13 , Issue 4 , December 1981 , pp. 720 - 735
Copyright
Copyright © Applied Probability Trust 1981 

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

Barbour, A. D. (1976) Networks of queues and the method of stages. Adv. Appl. Prob. 8, 584–591.Google Scholar
Cohen, J. W. (1979) The multiple phase service network with generalized processor sharing. Acta Informatica 12, 245–284.Google Scholar
Cooper, R. B. and Marmon, R. I. (1978) A conjecture on a generalization of a theorem concerning response times in a processor-sharing operating system, and its investigation by simulation. Unpublished.Google Scholar
Cramèr, H. and Leadbetter, M. R. (1967) Stationary and Related Stochastic Processes. Wiley, New York.Google Scholar
Doob, J. L. (1948) Asymptotic properties of Markoff transition probabilities. Trans. Amer. Math. Soc. 63, 393–421.Google Scholar
Helm, W. E. and Schassberger, R. (1979) Insensitive generalized semi-Markov schemes with point process input. Technische Hochschule Darmstadt, Preprint 502.Google Scholar
Jain, N. and Jamison, B. (1967) Contributions to Doeblin's theory of Markov processes. Z. Wahrscheinlichkeitsth 8, 19–40.Google Scholar
Jansen, U., König, D. and Nawrotzki, K. (1979) A criterion of insensitivity for a class of queueing systems with random marked point processes. Math. Operationsforsch. Statist. Ser. Optimization 10, 379–403.Google Scholar
Kelly, F. P. (1976) Networks of queues. Adv. Appl. Prob. 8, 416–432.Google Scholar
König, D. and Jansen, U. (1976) Eine Invarianzeigenschaft zufälliger Bedienungsprozesse mit positiven Geschwindigkeiten. Math. Nachr. 70, 321–364.Google Scholar
Matthes, K. (1962) Zur Theorie der Bedienungsprozesse. Trans. 3rd Prague Conf. Information Theory. Google Scholar
Orey, S. (1971) Limit Theorems for Markov Chain Transition Probabilities. Van Nostrand Reinhold, London.Google Scholar
Ryll-Nardzewski, C. (1961) Remarks on processes of calls. Proc. 4th Berkeley Symp. Math. Statist. Prob. 2, 455–465.Google Scholar
Sakata, M., Noguchi, S. and Oizumi, J. (1969) Analysis of a processor-shared queueing model for time sharing systems. Proc. 2nd Hawaii Conf. Systems Science. Google Scholar
Schassberger, R. (1978) Insensitivity of steady-state distributions of generalized semi-Markov processes with speeds. Adv. Appl. Prob. 10, 836–851.Google Scholar