Linear Positive Systems and Phase-type Representations

In this paper we try to put a bridge between the theory of phase-type distributions and the general theory of positive linear systems. The phase-type distributions correspond to the random hitting time of an absorbing Markov chain. The representation problem which consists of finding a Markov chain associated with some phase-type distribution is a positive realization problem. It turns out that this realization problem may be seen as concerning a large class of positive problems. Therefore a lot of results obtained in the field of phase-type distributions may be extended to a large class of positive systems.

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