Triangular order of triangular phase-type distributions ∗

The order of a phase-type distribution is the least number of states needed to represent it. This quantity is not well understood. In this paper we introduce a simpler quantity in the context of triangular (acyclic) phase-type distributions, called triangular order.This is the least number of states needed for a triangular representation. A fairly complete theory of triangular order is developed. It is hoped that this simple theory will suggest natural approaches to the study of order in the general setting. The concept of span-minimality is introduced here, and plays a role in the argument. As an aside, we prove that the set of phase-type distributions of a given order is closed under weak convergence