An invariant of representations of phase-type distributions and some applications

In this paper we consider phase-type distributions, their Laplace transforms which are rational functions and their representations which are finite-state Markov chains with an absorbing state. We first prove that, in any representation, the minimal number of states which are visited before absorption is equal to the difference of degree between denominator and numerator in the Laplace transform of the distribution. As an application, we prove that when the Laplace transform has a denominator with n real poles and a numerator of degree less than or equal to one the distribution has order n. We show that, in general, this result can be extended neither to the case where the numerator has degree two nor to the case of non-real poles.