Equilibrium distributions of discrete phase type

Abstract

It is well known that the family of discrete phase-type distributions is closed under convolutions, mixtures, and the unary “geometric mixture” operation. In this article, we show that the equilibrium distributions of this family are again of phase type. A simple result is obtained for the limiting equilibrium distributions; that is, the equilibrium distributions of any discrete phase-type distribution with an irreducible representation always converge to a geometric distribution. Numerical examples are provided in three different cases. At the end of the article, a more general type of equilibrium distribution is considered to which a discount factor is introduced.