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Exponential ergodicity in Markovian queueing and dam models

Published online by Cambridge University Press:  14 July 2016

Pekka Tuominen  and
Richard L. Tweedie
Pekka Tuominen*
Affiliation:
University of Western Australia
Richard L. Tweedie
Affiliation:
University of Western Australia
*
∗Permanent address: Department of Mathematics, University of Helsinki, Hallituskatu 15, 00100 Helsinki 10, Finland.
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Abstract

We investigate conditions under which the transition probabilities of various Markovian storage processes approach a stationary limiting distribution π at an exponential rate. The models considered include the waiting time of the M/G/1 queue, and models for dams with additive input and state-dependent release rule satisfying a ‘negative mean drift' condition. A typical result is that this exponential ergodicity holds provided the input process is ‘exponentially bounded'; for example, in the case of a compound Poisson input, a sufficient condition is an exponential bound on the tail of the input size distribution. The results are proved by comparing the discrete-time skeletons of the Markov process with the behaviour of a random walk, and then showing that the continuous process inherits the exponential ergodicity of any of its skeletons.

Keywords

RATE OF CONVERGENCELIMIT DISTRIBUTIONM/G/1 QUEUEWAITING TIMESADDITIVE INPUTSSTOCHASTIC COMPARISONRANDOM WALK
Type
Research Papers
Information
Journal of Applied Probability , Volume 16 , Issue 4 , December 1979 , pp. 867 - 880
Copyright
Copyright © Applied Probability Trust 1979 

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Footnotes

∗∗

Present address: CSIRO Division of Mathematics and Statistics, P.O.Box 310, South Melbourne, Victoria 3205, Australia.

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