Phase-type distributions and representations: Some results and open problems for system theory

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Abstract
In this paper we consider phase-type distributions. These distributions correspond to the random hitting time of an absorbing Markov chain. They are used for modelling various random times, in particular, those which appear in manufacturing systems as processing times, times to failure, repair times, etc. The Markovian nature of these distributions allows the use of very efficient matrix based computer methods for performance evaluation. In this paper we give a system theory oriented introduction to phase-type distributions. We concentrate mainly on the representation problem which consists of finding a Markov chain associated with some phase-type distribution. This is a realization problem in the sense of system theory with a lot of links with the classical linear system theory but also with a number of constraints which make the problem harder but more interesting. Indeed this problem has strong connections with the positive realization problem in control theory. The paper recalls known results, gives some new results, and points out the main remaining problems.

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