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Analysis of a large number of Markov chains competing for transitions

Emmanuelle Anceaume 1, 2 François Castella 3, 4 Bruno Sericola 5
1 CIDER
IRISA-D1 - SYSTÈMES LARGE ÉCHELLE
2 CIDRE - Confidentialité, Intégrité, Disponibilité et Répartition
CentraleSupélec, Inria Rennes – Bretagne Atlantique , IRISA-D1 - SYSTÈMES LARGE ÉCHELLE
3 IPSO - Invariant Preserving SOlvers
Inria Rennes – Bretagne Atlantique , IRMAR - Institut de Recherche Mathématique de Rennes
5 DIONYSOS - Dependability Interoperability and perfOrmance aNalYsiS Of networkS
IRISA-D2 - RÉSEAUX, TÉLÉCOMMUNICATION ET SERVICES, Inria Rennes – Bretagne Atlantique
Abstract : We consider the behavior of a stochastic system composed of several identically distributed, but non independent, discrete-time absorbing Markov chains competing at each instant for a transition. The competition consists in determining at each instant, using a given probability distribution, the only Markov chain allowed to make a transition. We analyze the first time at which one of the Markov chains reaches its absorbing state. When the number of Markov chains goes to infinity, we analyze the asymptotic behavior of the system for an arbitrary probability mass function governing the competition. We give conditions for the existence of the asymptotic distribution and we show how these results apply to cluster-based distributed systems when the competition between the Markov chains is handled by using a geometric distribution.
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Submitted on : Sunday, September 30, 2012 - 9:36:22 PM
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Emmanuelle Anceaume, François Castella, Bruno Sericola. Analysis of a large number of Markov chains competing for transitions. International Journal of Systems Science, Taylor & Francis, 2014, 45 (3), pp.232--240. ⟨10.1080/00207721.2012.704090⟩. ⟨hal-00736916⟩

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