Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-27T14:02:29.699Z Has data issue: false hasContentIssue false
  • English
  • Français

Tight bounds on the sensitivity of generalised semi-Markov processes with a single generally distributed lifetime

Part of: Operations research and management science Special processes

Published online by Cambridge University Press:  14 July 2016

A. J. Coyle  and
P. G. Taylor
A. J. Coyle
Affiliation:
University of Adelaide
P. G. Taylor*
Affiliation:
University of Adelaide
*
∗Postal address: Applied Mathematics Department, The University of Adelaide, GPO Box 498, Adelaide, SA 5001, Australia.
Get access Rights & Permissions [Opens in a new window]

Abstract

There are some generalised semi-Markov processes (GSMP) which are insensitive, that is the value of some performance measures for the system depend only on the mean value of lifetimes and not on their actual distribution. In most cases this is not true and a performance measure can take on a number of values depending on the lifetime distributions. In this paper we present a method for finding tight bounds on the sensitivity of performance measures for the class of GSMPs with a single generally distributed lifetime. Using this method we can find upper and lower bounds for the value of a function of the stationary distribution as the distribution of the general lifetime ranges over a set of distributions with fixed mean. The method is applied to find bounds on the average queue length of the Engset queue and the time congestion in the GI/M/n/n queueing system.

Keywords

SENSITIVITY BOUNDSPERFORMANCE MEASURESQUEUEING SYSTEMS

MSC classification

Primary: 60K20: Applications of Markov renewal processes (reliability, queueing networks, etc.)
Secondary: 90B22: Queues and service
Type
Research Papers
Information
Journal of Applied Probability , Volume 32 , Issue 1 , March 1995 , pp. 63 - 73
Copyright
Copyright © Applied Probability Trust 1995 

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.)

Footnotes

This work was supported by Australian Research Council Grant A69132151.

References

[1] Burman, D. (1981) Insensitivity in queueing systems. Adv. Appl. Prob. 13, 846–859.Google Scholar
[2] Coyle, A. J. and Taylor, P. G. (1992) Bounds on the sensitivity bounds of generalised semi-Markov processes with a single generally distributed lifetime. Math. Operat. Res. 17, 132–147.Google Scholar
[3] Eckberg, A. E. (1977) Sharp bounds on Laplace-Stieltjes transforms, with applications to various queueing problems. Math. Operat. Res. 2, 135–142.Google Scholar
[4] Henderson, W. (1983) Insensitivity and reversed Markov processes. Adv. Appl. Prob. 15, 752–768.Google Scholar
[5] Holtzman, J. M. (1973) The accuracy of the equivalent random method with renewal input. Bell. Syst. Tech. J. 52, 1673–1679.CrossRefGoogle Scholar
[6] Klincewicz, J. G. and Whitt, W. (1984) On approximations for queues, II: Shape constraints. AT&T Bell Lab. Tech. J. 63, 139–161.Google Scholar
[7] 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. Th., Stat. Dec. Fns. and Rand. Proc. 335–343.Google Scholar
[8] Krein, M. G. and Nudel'Man, A. A. (1977) The Markov Moment Problem and Extremal Problems. Translations of Mathematical Monographs 50, American Mathematical Society, Providence, RI.Google Scholar
[9] Schassberger, R. (1977) Insensitivity of steady-state distributions of generalized semi-Markov processes. Part I. Ann. Prob. 5, 87–89.Google Scholar
[10] Schassberger, R. (1978a) Insensitivity of steady-state distributions of generalized semi-Markov processes. Part II. Ann. Prob. 6, 85–93.Google Scholar
[11] Schassberger, R. (1978b) Insensitivity of steady-state distributions of generalized semi-Markov processes with speeds. Adv. Appl. Prob. 10, 836–851.Google Scholar
[12] Taylor, P. G. (1989) Insensitivity of processes with zero speeds. Adv. Appl. Prob. 21, 612–628.CrossRefGoogle Scholar
[13] Whitt, W. (1984) On approximations for queues, I: extremal distributions. AT&T Bell Lab. Tech. J. 63, 115–138.Google Scholar
[14] Whittle, P. (1985) Partial balance and insensitivity. J. Appl. Prob. 22, 168–176.CrossRefGoogle Scholar
[15] Whittle, P. (1986) Partial balance, insensitivity and weak coupling. J. Appl. Prob. 18, 706–723.Google Scholar