Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-22T04:05:10.689Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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 

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

∗∗

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

References

[1] Brockwell, P. J. (1977) Stationary distributions for dams with additive input and content-dependent release rate. Adv. Appl. Prob. 9, 645–663.Google Scholar
[2] Cheong, C. K. and Heathcote, C. R. (1965) On the rate of convergence of waiting times. J. Austral. Math. Soc. 5, 365–373.Google Scholar
[3] Çinlar, E. and Pinsky, M. (1971) A stochastic integral in storage theory. Z. Wahrscheinlichkeitsth. 17, 227–240.Google Scholar
[4] Çinlar, E. and Pinsky, M. (1972) On dams with additive inputs and general release rule. J. Appl. Prob. 9, 422–429.Google Scholar
[5] Daley, D. J. (1968) Stochastically monotone Markov chains. Z. Wahrscheinlichkeitsth. 10, 305–317.Google Scholar
[6] Feller, W. (1971) An Introduction to Probability Theory and Its Applications , Vol. II, 2nd edn. Wiley, New York.Google Scholar
[7] Harrison, J. M. and Resnick, S. I. (1976) The stationary distribution and first exit probabilities of a storage process with general release rule. Math. Operat. Res. 1, 347–358.Google Scholar
[8] Laslett, G. M., Pollard, D. B. and Tweedie, R. L. (1978) Techniques for establishing ergodic and recurrence properties of continuous valued Markov chains. Nav. Res. Logist. Quart. 25, 455–472.CrossRefGoogle Scholar
[9] Miller, H. D. (1965) Geometric ergodicity in a class of denumerable Markov chains. Z. Wahrscheinlichkeitsth. 4, 354–373.Google Scholar
[10] Neuts, M. and Teugels, J. (1969) Exponential ergodicity of the M/G/1 queue. SIAM J. Appl. Math. 17, 921–929.Google Scholar
[11] Nummelin, E. (1976) A splitting technique for f-recurrent Markov chains. Report-HTTK-MAT A 80, Helsinki University of Technology.Google Scholar
[12] Nummelin, E. and Tweedie, R. L. (1978) Geometric ergodicity and R -positivity for general Markov chains. Ann. Prob. 6, 404–420.CrossRefGoogle Scholar
[13] O'brien, G. L. (1975) The comparison method for stochastic processes. Ann. Prob. 3, 80–88.Google Scholar
[14] Orey, S. (1971) Lecture Notes on Limit Theorems for Markov Chain Transition Probabilities. Van Nostrand Reinhold, London.Google Scholar
[15] Rogozin, B. A. (1969) Distribution of the maximum of a process with independent increments. Siberian Math. J. 10, 989–1010.Google Scholar
[16] Schäl, M. (1971) The analysis of queues with state-dependent parameters by Markov renewal processes. Adv. Appl. Prob. 3, 155–175.Google Scholar
[17] Shtatland, E. S. (1966) The asymptotic behaviour of the distribution of a maximum for a class of processes with independent increments. Soviet Math. 7, 1297–1299.Google Scholar
[18] Tuominen, P. and Tweedie, R. L. (1979) Exponential decay and ergodicity of general Markov processes and their discrete skeletons. Adv. Appl. Prob. 11,.Google Scholar
[19] Tweedie, R. L. (1976) Criteria for classifying general Markov chains. Adv. Appl. Prob. 8, 737–771.Google Scholar
[20] Widder, D. V. (1946) The Laplace Transform. Princeton University Press, Princeton, N.J. Google Scholar