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The Maclaurin series for performance functions of Markov chains

Part of: Stochastic processes Inference from stochastic processes Markov processes

Published online by Cambridge University Press:  01 July 2016

Xi-Ren Cao
Xi-Ren Cao*
Affiliation:
Hong Kong University of Science and Technology
*
∗ Postal address: The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong. Email address: eecao@usthk.ust.hk
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Abstract

We derive formulas for the first- and higher-order derivatives of the steady state performance measures for changes in transition matrices of irreducible and aperiodic Markov chains. Using these formulas, we obtain a Maclaurin series for the performance measures of such Markov chains. The convergence range of the Maclaurin series can be determined. We show that the derivatives and the coefficients of the Maclaurin series can be easily estimated by analysing a single sample path of the Markov chain. Algorithms for estimating these quantities are provided. Markov chains consisting of transient states and multiple chains are also studied. The results can be easily extended to Markov processes. The derivation of the results is closely related to some fundamental concepts, such as group inverse, potentials, and realization factors in perturbation analysis. Simulation results are provided to illustrate the accuracy of the single sample path based estimation. Possible applications to engineering problems are discussed.

Keywords

Perturbation analysisfundamental matrixsample path estimation

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60G17: Sample path properties 60J45: Probabilistic potential theory 62M05: Markov processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 30 , Issue 3 , September 1998 , pp. 676 - 692
Copyright
Copyright © Applied Probability Trust 1998 

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Footnotes

Supported in part by Hong Kong UPGC under grant HKUST 690/95E.

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