Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-10T14:48:46.911Z Has data issue: false hasContentIssue false
  • English
  • Français

Generalized semi-Markov schemes and open queueing networks

Published online by Cambridge University Press:  14 July 2016

A. D. Barbour
A. D. Barbour*
Affiliation:
Gonville and Caius College, Cambridge
Get access Rights & Permissions [Opens in a new window]

Abstract

Generalized semi-Markov schemes were devised to give a versatile general model embracing queueing networks and similar systems of practical importance, and they have proved particularly successful in uniting many disparate results on insensitivity. However, it turns out that, although closed queueing networks are expressible as GSMS, open networks are not, and that the insensitivity results for such networks are not therefore strictly within their scope. In this paper, it is shown that, as one might hope, open networks can be realized as limits of a suitable sequence of closed networks in such a way that the insensitivity properties of the GSMS are transferred to the open network in the limit, and thus that open networks too can, in a sense, be considered to be GSMS. However, it appears from the technical nature of the arguments involved that, despite this close relationship between GSMS and open networks, it may nonetheless be simpler to treat them separately when constructing the proofs of theorems.

Keywords

GENERALIZED SEMI-MARKOV SCHEMEOPEN QUEUEING NETWORKINSENSITIVITYCONTINUITY OF QUEUES
Type
Short Communications
Information
Journal of Applied Probability , Volume 19 , Issue 2 , June 1982 , pp. 469 - 474
Copyright
Copyright © Applied Probability Trust 1982 

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.CrossRefGoogle Scholar
[2] Barbour, A. D. and Schassberger, R. (1981) Insensitive average residence times in generalized semi-Markov processes. Adv. Appl. Prob. 13, 846–859.CrossRefGoogle Scholar
[3] 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
[4] Helm, W. E. and Schassberger, R. (1979) Insensitive generalized semi-Markov schemes with point process input. Math. Operat. Res. Google Scholar
[5] Karr, A. F. (1975) Weak convergence of a sequence of Markov chains. Z. Wahrscheinlichkeitsth. 33, 41–48.Google Scholar
[6] Kelly, F. P. (1976) Networks of queues. Adv. Appl. Prob. 8, 416–432.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.CrossRefGoogle Scholar
[9] Schassberger, R. (1978) The insensitivity of stationary probabilities in networks of queues. Adv. Appl. Prob. 10, 906–912.Google Scholar
[10] Whitt, W. (1980) Continuity of generalized semi-Markov processes. Math. Operat. Res. 5, 494–501.Google Scholar