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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
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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