Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-10T19:16:59.119Z Has data issue: false hasContentIssue false
  • English
  • Français

Event coupling and performance sensitivity analysis of generalized semi-Markov processes

Part of: Stochastic processes Special processes

Published online by Cambridge University Press:  01 July 2016

Xi-Ren Cao
Xi-Ren Cao*
Affiliation:
The Hong Kong University of Science and Technology
*
* Postal address: Department of Electrical and Electronic Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong.
Get access Rights & Permissions [Opens in a new window]

Abstract

We study a fundamental feature of the generalized semi-Markov processes (GSMPs), called event coupling. The event coupling reflects the logical behavior of a GSMP that specifies which events can be affected by any given event. Based on the event-coupling property, GSMPs can be classified into three classes: the strongly coupled, the hierarchically coupled, and the decomposable GSMPs. The event-coupling property on a sample path of a GSMP can be represented by the event-coupling trees. With the event-coupling tree, we can quantify the effect of a single perturbation on a performance measure by using realization factors. A set of equations that specifies the realization factors is derived. We show that the sensitivity of steady-state performance with respect to a parameter of an event lifetime distribution can be obtained by a simple formula based on realization factors and that the sample-path performance sensitivity converges to the sensitivity of the steady-state performance with probability one as the length of the sample path goes to infinity. This generalizes the existing results of perturbation analysis of queueing networks to GSMPs.

Keywords

PERTURBATION ANALYSIS

MSC classification

Primary: 60G20: Generalized stochastic processes
Secondary: 60G17: Sample path properties 60K25: Queueing theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 27 , Issue 3 , September 1995 , pp. 741 - 769
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

Research supported in part by the grant HKUST 599/94E.

References

Billingsley, P. (1979) Probability and Measure. Wiley, New York.Google Scholar
Borovkov, A. A. (1976) Stochastic Processes in Queueing Theory. Springer-Verlag, New York.Google Scholar
Borovkov, A. A. (1978) Ergodicity and stability theorems for a class of stochastic equations and their application. Theory Prob. Appl. 23, 227–247.Google Scholar
Cao, X. R. (1985) Convergence of parameter sensitivity estimates in a stochastic experiment. IEEE Trans. Autom. Control 30, 845–853.Google Scholar
Cao, X. R. (1988) A sample performance function of Jackson queueing networks. Operat. Res. 36, 128–136.Google Scholar
Cao, X. R. (1989) The static property of a perturbed multiclass closed queueing network and decomposition. IEEE Trans. Autom. Control 34, 246–249.Google Scholar
Cao, X. R. (1990) Realization factors and sensitivity analysis of queueing networks with state dependent service rates. Adv. Appl. Prob. 22, 178–210.Google Scholar
Cao, X. R. (1992) A new method of performance sensitivity analysis for non-Markovian queueing networks. Queueing systems: Theory and Applications 10, 313–350.CrossRefGoogle Scholar
ÇInlar, E. (1975) Introduction to Stochastic Processes. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
Coyle, A. J. and Taylor, P. G. (1992) Bounds on the sensitivity of generalised semi-Markov processes with a single generally distributed lifetime. Math. Operat. Res. 17, 132–147.Google Scholar
Glasserman, P. (1990) The limiting values of derivative estimates based on perturbation analysis. Commun. Statist. Stoch. Models 6, 229–257.Google Scholar
Glasserman, P. (1991) Gradient Estimation Via Perturbation Analysis. Kluwer, Dordrecht.Google Scholar
Glynn, P. W. (1989) A GSMP formalism for discrete event systems. IEEE Proc. 77, 14–23.Google Scholar
Haas, P. J. and Shedler, G. S. (1987) Regenerative generalized semi-Markov processes. Stoch. Models 3, 409–438.CrossRefGoogle Scholar
Ho, Y. C. and Cao, X. R. (1991) Perturbation Analysis of Discrete Event Dynamic Systems. Kluwer, Dordrecht.CrossRefGoogle Scholar
Schassberger, R. (1976) On the equilibrium distribution of a class of finite-state generalized semi-Markov processes. Math. Operat. Res. 1, 395–406.Google Scholar
Schassberger, R. Insensitivity of steady-state distributions of generalized semi-Markov processes. Ann. Prob. 5, 87–99.Google Scholar
Sigman, K. (1989) Notes on the stability of closed queueing networks. J. Appl. Prob. 26, 678–681.CrossRefGoogle Scholar
Suri, R. (1987) Infinitesimal perturbation analysis for general discrete event systems. J. ACM 34, 686–717.Google Scholar
Whitt, W. (1980) Continuity of generalized semi-Markov processes. Math. Operat. Res. 5, 494–501.Google Scholar