On the Canonical Representation of Order 3 Discrete Phase Type Distributions

In spite of the fact that discrete phase type (DPH) distributions are used almost as often as continuous phase type (CPH) distributions canonical representation is not available for general (cyclic) order 3 DPH distributions yet. In this paper we investigate the canonical representation of DPH distributions of order 3. During the course of this investigation we find that the problem of canonical representation of order 3 DPH distributions is far more complex than the one of order 3 CPH distribution. As a result we needed to distinguish 8 different subclasses of order 3 DPH distributions, while it was enough to distinguish 3 subclasses of order 3 CPH distributions for their canonical representation. Additionally, we were not able to prove all subclasses of DPH distributions with the relatively simple methodology which was sufficient for the canonical representation of order 3 DPH distributions.