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A positive recurrence criterion associated with multidimensional queueing processes

  • Zvi Rosberg (a1)

Abstract

A criterion is given for positive recurrence of a multidimensional, aperiodic, irreducible Markov chain with a denumerable state space. This criterion extends to the multidimensional case Foster's one-dimensional criterion. The multidimensional criterion consists of several conditions, one for each coordinate of the process. The usefulness of this criterion is shown through a queueing network example.

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Corresponding author

Postal address: Department of Business Administration, University of Illinois at Urbana-Champaign, 350 Commerce Building (West), Urbana, IL 61801, U.S.A.

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This research was done while the author was visiting at C.O.R.E., Université de Louvain.

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References

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Foster, F.G. (1953) On the stochastic matrices associated with certain queueing processes. Ann. Math. Statist. 24, 355360.
Kingman, J. F. C. (1961) The ergodic behaviour of random walks. Biometrika 48, 391396.
Kushner, H. (1971) Introduction to Stochastic Control. Holt, Rinehart and Winston, New York.
Lippman, S. A. (1975) Applying a new device in the optimization of exponential queueing systems. Operat. Res. 23, 687710.
Marlin, P. G. (1973) On the ergodic theory of Markov chains. Operat. Res. 21, 617622.
Pakes, A. G. (1969) Some conditions of ergodicity and recurrence of Markov chains. Operat. Res. 17, 10581061.
Rosberg, Z. (1978) Service Policies in a Queueing Network with Exponential Service Stations (in Hebrew). Ph.D. Thesis, The Hebrew University of Jerusalem.

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A positive recurrence criterion associated with multidimensional queueing processes

  • Zvi Rosberg (a1)

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