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Asymptotic rate of a Markov process

Published online by Cambridge University Press:  01 July 2016

B. D. Craven
B. D. Craven*
Affiliation:
University of Melbourne
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Abstract

For a Markov process in discrete time, both (a) the time-dependent behaviour of the process at time t, and (b) the behaviour of moments (if finite) of the process, whether or not stationary, as a function of time-lag t, are determined by solutions to a single equation, the key equation. The state space of the process is required only to be both a measure space, and a locally convex topological vector space, which may have infinite dimension. The asymptotic behaviour of both (a) and (b), for large times t, is shown to be (nearly) the same, being proportional to tξe−βt, where the exponent β is calculable from an eigenvalue of a linear operator, describing the Markov process. For a large class of Markov processes, the spectrum of this operator is shown to consist of discrete eigenvalues only. Queueing applications are discussed.

Keywords

ASYMPTOTIC RATEKEY EQUATIONDISCRETE TIME MARKOV PROCESSAUTOCORRELATION
Type
Research Article
Information
Advances in Applied Probability , Volume 6 , Issue 4 , December 1974 , pp. 732 - 746
Copyright
Copyright © Applied Probability Trust 1974 

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References

[1] Cohen, J. W. (1967) On two integral equations of queueing theory. J. Appl. Prob. 4, 343–355.Google Scholar
[2] Craven, B. D. (1963) Asymptotic transient behaviour of the bulk service queue. J. Austral. Math. Soc. 3, 503–512.CrossRefGoogle Scholar
[3] Craven, B. D. (1965) Serial dependence of a Markov process. J. Austral. Math. Soc. 5, 299–314.CrossRefGoogle Scholar
[4] Craven, B. D. (1969) Asymptotic correlation in a queue. J. Appl. Prob. 6, 573–585.CrossRefGoogle Scholar
[5] Copson, E. T. (1935) Theory of Functions of a Complex Variable. Oxford University Press.Google Scholar
[6] Kendall, D. G. (1959) Geometric ergodicity and the theory of queues. Mathematical Methods in the Social Sciences. (Eds. Arrow, K. J., Karlin, S., and Suppes, P.) Stanford University Press, California. 176–195.Google Scholar
[7] Krasnoselskii, M. A., Zabreiko, P. P., Pustylnik, E. I. and Sobolevski, P. E. (1966) Integral Operators in Spaces of Summable Functions. (In Russian) Nauka, Moscow.Google Scholar
[8] Robertson, A. P. and Robertson, W. (1964) Topological Vector Spaces. Cambridge University Press.Google Scholar
[9] Vere-Jones, D. (1962) Geometric ergodicity in denumerable Markov chains. Quart. J. Math. Oxford 2nd. Ser. 13, 7–28.CrossRefGoogle Scholar
[10] Vere-Jones, D. (1963) On the spectra of some linear operators associated with queueing systems. Z. Wahrscheinlichkeitsth. 2, 12–21.Google Scholar