Finding upper-triangular representations for phase-type distributions with 3 distinct real poles

In this paper we seek particular representations for absolutely continuous Phase-type distributions with 3 distinct real poles. First, we define subsets of these Phase-type distributions given the 3 distinct poles. One subset contains distributions that have upper triangular PH representation of order n, but do not have a triangular one of PH order n−1. This is done by using the invariant polytope approach. For any distribution in our subsets we give an invariant polytope containing the corresponding distribution by finding the vertices of the polytope. Second, we propose a method that actually constructs the generator matrix of the required PH representation from the invariant polytope. Consequently, our method constructs an upper triangular PH representation that has minimal order among the upper triangular PH representations given the probability density function of a PH distribution.