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Transient phenomena for Markov chains and applications

Part of:Markov processes

Published online by Cambridge University Press:  01 July 2016

A. A. Borovkov,
G. Fayolle and
D. A. Korshunov
A. A. Borovkov*
Affiliation:
Institute of Mathematics, Novossibirsk
G. Fayolle*
Affiliation:
INRIA
D. A. Korshunov*
Affiliation:
Institute of Mathematics, Novossibirsk
*
∗ Postal address: Academy of Sciences, Institute of Mathematics, Novossibirsk, 90, 630090 USSR.
∗∗ Postal address: INRIA, Domaine de Voluceau-Rocquencourt, B.P 105, 78153 Le Chesnay, France.
∗ Postal address: Academy of Sciences, Institute of Mathematics, Novossibirsk, 90, 630090 USSR.
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Abstract

We consider a family of irreducible, ergodic and aperiodic Markov chains X(ε) = {X(ε) n, n ≧0} depending on a parameter ε > 0, so that the local drifts have a critical behaviour (in terms of Pakes' lemma). The purpose is to analyse the steady-state distributions of these chains (in the sense of weak convergence), when ε↓ 0. Under assumptions involving at most the existence of moments of order 2 + γ for the jumps, we show that, whenever X (0) is not ergodic, it is possible to characterize accurately these limit distributions. Connections with the gamma and uniform distributions are revealed. An application to the well-known ALOHA network is given.

Keywords

ERGODICITYRANDOM ACCESSSTABILITYWEAK CONVERGENCE

MSC classification

Primary:60J27: Continuous-time Markov processes on discrete state spaces

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 24 , Issue 2 , June 1992, pp. 322 - 342
Copyright
Copyright © Applied Probability Trust 1992 

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Footnotes

All correspondence should be addressed to this author.

References

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