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Operator-geometric stationary distributions for markov chains, with application to queueing models

Published online by Cambridge University Press:  01 July 2016

R. L. Tweedie
R. L. Tweedie*
Affiliation:
CSIRO Division of Mathematics and Statistics
*
∗Present address: SIROMATH Pty Ltd, 1 York St., Sydney, NSW 2000, Australia.
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Abstract

This paper considers a class of Markov chains on a bivariate state space , whose transition probabilities have a particular ‘block-partitioned' structure. Examples of such chains include those studied by Neuts [8] who took E to be finite; they also include chains studied in queueing theory, such as (Nn , Sn ) where Nn is the number of customers in a GI/G/1 queue immediately before, and Sn the remaining service time immediately after, the nth arrival.

We show that the stationary distribution Πfor these chains has an ‘operator-geometric' nature, with , where the operator S is the minimal solution of a non-linear operator equation. Necessary and sufficient conditions for Πto exist are also found. In the case of the GI/G/1 queueing chain above these are exactly the usual stability conditions.

G1/G/1 QUEUE; PHASE-TYPE; INVARIANT MEASURE; FOSTER'S CONDITIONS

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 14 , Issue 2 , June 1982 , pp. 368 - 391
Copyright
Copyright © Applied Probability Trust 1982 

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References

[1] Chung, K. L. (1967) Markov Chains with Stationary Transition Probabilities, 2nd edn. Springer-Verlag, New York.Google Scholar
[2] Cogburn, R. (1975) A uniform theory for sums of Markov chain transition probabilities. Ann. Prob. 3, 191–214.Google Scholar
[3] Feller, W. (1971) An Introduction to Probability Theory and its Applications, volume II, 2nd edn. Wiley, New York.Google Scholar
[4] Finch, P. D. (1961) On the busy period in the queueing system GI/G/1. J. Austral. Math. Soc. 2, 217–228.Google Scholar
[5] Neuts, M. F. (1967) Two Markov chains arising from examples of queues with state dependent service times. Sankhya A 29, 255–264.Google Scholar
[6] Neuts, M. F. (1969) The queue with Poisson input and general service times, treated as a branching process. Duke Math. J. 36, 215–231.CrossRefGoogle Scholar
[7] Neuts, M. F. (1975) Probability distributions of phase type. In Liber Amicorum Professor Emeritus H. Florin, Department of Mathematics, University of Louvain, Belgium, 173–206.Google Scholar
[8] Neuts, M. F. (1978) Markov chains with applications in queueing theory, which have a matrix-geometric invariant probability vector. Adv. Appl. Prob. 10, 185–212.Google Scholar
[9] Neuts, M. F. (1980) The probabilistic significance of the rate matrix in matrix-geometric invariant vectors. J. Appl. Prob. 17, 291–296.CrossRefGoogle Scholar
[10] Neuts, M. F. (1981) Matrix-Geometric Solutions in Stochastic Models–An Algorithmic Approach. The Johns Hopkins University Press, Baltimore, MD.Google Scholar
[11] Orey, S. (1975) Lecture Notes on Limit Theorems for Markov Chain Transition Probabilities. Van Nostrand-Reinhold, London.Google Scholar
[12] Revuz, D. (1975) Markov Chains. North-Holland, Amsterdam.Google Scholar
[13] Tuominen, P. and Tweedie, R. L. (1979) The recurrence structure of general Markov processes. Proc. Lond. Math. Soc. (3) 39, 554–576.CrossRefGoogle Scholar
[14] Tweedie, R. L. (1976) Criteria for classifying general Markov chains. Adv. Appl. Prob. 8, 737–771.CrossRefGoogle Scholar
[15] Tweedie, R. L. (1974) R-theory for Markov chains on a general state space I. Ann. Prob. 2, 840–864.Google Scholar