Phase-Type Distributions: Open Problems and a Few Properties

A phase-type distribution is the distribution of a killing time in a finite-state Markov chain. This paper describes some conjectures concerning these distributions. Some partial proofs and other evidence are offered in support of the conjectures. Some of the conjectures are structural in nature, and concern the geometry of the set of phase-type distributions, while others are quantitative, and concern the properties of their densities. 1 Introduction A phase-type distribution is simply the distribution of a killing time in a finite-state Markov chain, and, as such, is a fundamental object in probability. It is surprising how little is known about them beyond elementary closure properties [5, 34, 39] and the simple characterization theorem given as Theorem 0 below [43]. Consider a phase-type distribution associated with an n-state continuous time Markov chain. It is fairly easy to see that its density has properties along the following lines (to be made precise later): (a) it cannot..