Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-26T20:59:38.430Z Has data issue: false hasContentIssue false
  • English
  • Français

The insensitivity of stationary probabilities in networks of queues

Published online by Cambridge University Press:  01 July 2016

R. Schassberger
R. Schassberger*
Affiliation:
University of Calgary
Get access Rights & Permissions [Opens in a new window]

Abstract

The stationary probabilities for certain networks of queues as defined by Kelly [4] were recently shown by Barbour [1] to depend on the service-time distributions involved only through their means. This type of insensitivity has been studied by König and Jansen [5] for a general class of stochastic processes. Kelly's networks yield special cases of such processes. We point this out in the present paper, thus shedding new light on the insensitivity phenomenon observed in these networks and its connection with the phenomenon of local balance. As a consequence of our recent study [8] we also obtain a new insensitivity result for these networks.

Keywords

GENERALIZED SEMI-MARKOV PROCESSESINSENSITIVITYLOCAL BALANCENETWORKS OF QUEUES
Type
Research Article
Information
Advances in Applied Probability , Volume 10 , Issue 4 , December 1978 , pp. 906 - 912
Copyright
Copyright © Applied Probability Trust 1978 

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

[1] Barbour, A. D. (1976) Networks of queues and the method of stages. Adv. Appl. Prob. 8, 584–591.Google 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.Google Scholar
[3] Jacobi, H. (1965) Eine Unempfindlichkeitseigenschaft für geordnete Bündel ungeordneter Teilbündel. Wiss. Z. Fr.-Schiller Univ. Jena, Math.-Naturwiss. Reihe 14, 251–260.Google Scholar
[4] Kelly, F. P. (1976) Networks of queues. Adv. Appl. Prob. 8, 416–432.CrossRefGoogle Scholar
[5] König, D. and Jansen, U. (1976) Eine Invarianzeigenschaft zufälliger Bedienungsprozesse mit positiven Geschwindigkeiten. Math. Nachr. 70, 321–364.CrossRefGoogle Scholar
[6] König, D., Matthes, K. and Nawrotzki, K. (1974) Unempfindlichkeitseigenschaften von Bedienungsprozesse. Appendix in Introduction to Queueing Theory (German edition) by Gnedenko, B. V. and Kovalenko, I. N.. Akademie-Verlag, Berlin.Google Scholar
[7] Matthes, K. (1962) Zur Theorie der Bedienungsprozessen. Trans. 3rd Prague Conf. Inf. Theory. Google Scholar
[8] Schassberger, R. (1978) Insensitivity of steady-state distributions of generalized semi-Markov processes with speeds. Adv. Appl. Prob. 10, 836–851.Google Scholar
[9] Stoyan, D. (1978) Queuing networks—insensitivity and a heuristic approximation. Elektron. Informationsverarbeit. Kybernetik. 14 (3). Abstract in Adv. Appl. Prob. 10, 318.Google Scholar
[10] Wolff, R. W. and Wrightson, C. W. (1976) An extension of Erlang's loss formula. J. Appl. Prob. 13, 628–632.CrossRefGoogle Scholar