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The probabilistic significance of the rate matrix in matrix-geometric invariant vectors

Published online by Cambridge University Press:  14 July 2016

Marcel F. Neuts
Marcel F. Neuts*
Affiliation:
University of Delaware
*
∗Postal address: Department of Mathematical Sciences, 501 Kirkbride Office Building, University of Delaware, Newark, DE 19711, U.S.A.
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Abstract

Many queueing models have embedded Markov chains, whose invariant probability vector is of a (modified) matrix-geometric form. The rate matrix R is shown to have the following probabilistic interpretation:

The element Rvj is the expected number of visits to the state (i + 1, j), before the first return to the set i = {(i, 1), …,(i, m)}, in a chain starting in the state (i, v).

Keywords

MARKOV CHAINSTABOO PROBABILITIESMATRIX-GEOMETRIC SOLUTIONQUEUEING THEORYCOMPUTATIONAL PROBABILITY
Type
Short Communications
Information
Journal of Applied Probability , Volume 17 , Issue 1 , March 1980 , pp. 291 - 296
Copyright
Copyright © Applied Probability Trust 1980 

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Footnotes

This research was sponsored by the Air Force Office of Scientific Research Air Force Systems Command USAF, under Grant No. AFOSR–77–3236.

References

[1] Chung, K. L. (1967) Markov Chains with Stationary Transition Probabilities, 2nd edn. Springer-Verlag, New York.Google Scholar
[2] Evans, R. V. (1967) Geometric distribution in some two-dimensional queueing systems. Operat. Res. 15, 830–846.Google Scholar
[3] Neuts, M. F. (1978) Markov chains with applications in queueing theory, which have a matrix-geometric invariant vector. Adv. Appl. Prob. 10, 185–212.CrossRefGoogle Scholar
[4] Winsten, C. B. (1959) Geometric distributions in the theory of queues. J. R. Statist. Soc. 21, 1–35.Google Scholar