Phase-type distributions in stochastic automata networks

Abstract

International audienceStochastic automata networks (Sans) are high-level formalisms for modeling very large and complex Markov chains in a compact and structured manner. To date, the exponential distribution has been the only distribution used to model the passage of time in the evolution of the different San components. In this paper we show how phase-type distributions may be incorporated into Sans thereby providing the wherewithal by which arbitrary distributions can be used which in turn leads to an improved ability for more accurately modeling numerous real phenomena. The approach we develop is to take a San model containing phase-type distributions and to translate it into another, stochastically equivalent, San model having only exponential distributions. In the San formalism, it is the events that are responsible for firing transitions and our procedure is to associate a stochastic automaton with each event having a phase-type distribution. This automaton models the distribution of time until the event occurs. In this way, the size of the elementary matrices remain small, because the size of the automata are small: the automata are either those of the original San, or are those associated with the phase-type events and are of size k, the number of phases in the representation of the distribution

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“…One reason for large component state spaces is the modeling of nonexponential distributions by phasetype (PH) distributions (Neuts 1981) with a large number of phases. PH distributions are often implicitly used in structured Markovian models as, for example, in Buchholz et al (2000), Buchholz (1992), and Hermanns (2002), but the integration of PH distribution in compositional models is still an open topic (e.g., Sbeity et al 2008). Often the reason for a large number of phases are distributions with a small coefficient of variation or density functions with a large number of local maxima and minima.…”

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“…Markovian queueing networks have played a very significant role in a number of physical systems [1,[12][13][14]18,22,27]. A queueing network is studied under two different situations.…”

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