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Some theorems on the transient covariance of Markov chains

Published online by Cambridge University Press:  14 July 2016

John F. Reynolds
John F. Reynolds*
Affiliation:
University College, Cardiff
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Abstract

Several authors have considered the covariance structure of continuous parameter Markov chains. Most of this work has dealt with particular process ses, notably Morse (1955) who analysed the simple M/M/1 queue and Bene-(1961) who considered a telephone trunking model. Furthermore, the results obtained apply only when the process has attained its limiting (stationary) distribution. A recent paper by Reynolds (1968) gave some general results for finite chains, still assuming stationarity. This note generalises the results obtained therein, and considers the covariance structure during the transient period prior to attaining the stationary distribution where this exists. In the case where no such distribution exists, the results are valid throughout the whole lifetime of the process.

Keywords

FINITE MARKOV CHAINSCOVARIANCE FUNCTIONS
Type
Short Communications
Information
Journal of Applied Probability , Volume 9 , Issue 1 , March 1972 , pp. 214 - 218
Copyright
Copyright © Applied Probability Trust 1972 

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References

Beneš, V. E. (1961) The covariance function of a simple trunk group, with applications to traffic measurement. Bell Syst. Tech. J. 40, 117–148.Google Scholar
Cox, D. R. and Miller, H. D. (1965) The Theory of Stochastic Processes. Methuen. London.Google Scholar
Morse, P. M. (1955) Stochastic properties of waiting lines, Operat. Res. 3, 255–261.Google Scholar
Reynolds, J. F. (1968) On the autocorrelation and spectral functions of queues. J. Appl. Prob. 5, 467–475.Google Scholar