Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-18T02:21:04.832Z Has data issue: false hasContentIssue false
  • English
  • Français

A Quasi-Reversibility Approach to the Insensitivity of Generalized Semi-Markov Processes

Published online by Cambridge University Press:  27 July 2009

Panagiotis Konstantopoulos  and
Jean Walrand
Panagiotis Konstantopoulos
Affiliation:
Department of Electrical Engineering and Computer Sciences and Electronics Research LaboratoryUniversity of California Berkeley, California 94720
Jean Walrand
Affiliation:
Department of Electrical Engineering and Computer Sciences and Electronics Research LaboratoryUniversity of California Berkeley, California 94720
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper is concerned with a certain property of the stationary distribution of a generalized semi-Markov process (GSMP) known as insensitivity. It is well-known that the so-called Matthes' conditions form a necessary and sufficient algebraic criterion for insensitivity. Most proofs of these conditions are basically algebraic. By interpreting a GSMP as a simple queueing network, we are able to show that Matthes' conditions are equivalent to the quasi-reversibility of the network, thus obtaining another simple proof of the sufficiency of these conditions. Furthermore, we apply our method to find a simple criterion for the insensitivity of GSMP's with generalized routing (in a sense that is introduced in the paper).

Type
Articles
Information
Probability in the Engineering and Informational Sciences , Volume 3 , Issue 3 , July 1989 , pp. 405 - 415
Copyright
Copyright © Cambridge University Press 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

Baccelli, F. & Bremaud, P. (1987). Palm probabilities and stationary queues. Berlin, New York: Springer-Verlag.CrossRefGoogle Scholar
Foley, R.D. (1980). The nonhomogeneous M/G/∞ queue. Technical Report VTR 8011. Virginia Polytechnic Institute.Google Scholar
Franken, P., Konig, D., Arndt, U., & Schmidt, V. (1982). Queues and point processes. New York: John Wiley and Sons.Google Scholar
Kelly, F.P. (1979). Reversibility and stochastic networks. New York: John Wiley and Sons.Google Scholar
Matthes, K. (1962). Zur Theorie der Bedienungsprozesse. Transactions of the 3rd Prague Conference on Information Theory.Google Scholar
Rumsewicz, M. P. (1988). Some contributions to the fields of insensitivity and queueing theory. Ph.D. Thesis, University of Adelaide.Google Scholar
Shassberger, R. (1977). Insensitivity of steady-state distributions of generalized semi-Markov processes. Part I. Annals of Probability 5: 87–99.Google Scholar
Schassberger, R. (1978). Insensitivity of steady-state distributions of generalized semi-Markov processes. Part II, Annals of Probability 6: 85–93.CrossRefGoogle Scholar
Schassberger, R. (1978). Insensitivity of steady-state distributions of generalized semi-Markov processes with speeds. Advances in Applied Probability 10: 836–851.CrossRefGoogle Scholar
Schassberger, R. (1978). The insensitivity of stationary probabilities in networks of queues. Advances in Applied Probability 10: 906–912.CrossRefGoogle Scholar
Walrand, J. (1985). Another look at insensitivity, Proceedings of the 23d Annual Allerton Conference, 09 27–29, Allerton, Illinois.Google Scholar
Walrand, J. (1988). An introduction to queueing networks. Englewood Cliffs, NJ: Prentice Hall.Google Scholar
Whitt, W. (1980). Continuity of generalized semi-Markov processes. Mathematics of Operatons Research 4: 494–501.CrossRefGoogle Scholar