On the canonical representation of phase type distributions

Abstract

The characterization and the canonical representation of order-n phase type distributions (PH(n)) is an open research problem.This problem is solved for n = 2, since the equivalence of the acyclic and the general PH distributions has been proven for a long time. However, no canonical representations have been introduced for the general PH distribution class so far for n > 2. In this paper we summarize the related results for n = 3. Starting from these results we provide a canonical representation of the PH(3) class (that is a minimal representation, too) and present a symbolical transformation procedure to obtain the canonical representation based on any (not only Markovian) vectormatrix representation of the distribution. We show that -using the same approachno symbolical results can be derived for the order-4 PH distributions, thus probably the PH(3) class is the highest order PH class for which a symbolical canonical transformation exists.Using the transformation method to canonical form for PH(3) we numerically evaluate the moment bounds of the PH(3) distribution set, compare it to the order-3 acyclic PH distribution (APH(3)) class, and present other possible applications of the canonical form.Email address: {ghorvath,telek}@hit.bme.hu (Gábor Horváth and Miklós Telek).

Moment Matching-Based Distribution Fitting with Generalized Hyper-Erlang Distributions

“…In [14] a PH of size 2 is used to match 3 moments, while [10] is able to match 5 moments with 3 states (it is known that the canonical form of 2-state PH distributions have 3, the one of 3-state PH distributions have 5 parameters). In [6] a hyperexponential distribution of size N is obtained based on 2N − 1 moments.…”

A heuristic procedure for compact Markov representation of PH distributions

“…The easiest solution for these technical issues is the use of canonical representation [6,9] -a conveniently defined, minimal, unique Markov chain representation. Unfortunately, canonical representations are available only for order 2 and order 3 general PH distributions [9] and some special subclasses of higher order PH distributions, e.g. acyclic PH distributions [6].…”

Moments Characterization of Order 3 Matrix Exponential Distributions

“…Unfortunately there are no tools to investigate the relation of the ME(n) and the PH(n) classes for n > 2. However, recent results on ME(3) [5] and PH(3) [6] distributions make it possible to investigate the relation of the ME(3) and the PH(3) classes.…”

“…Related PH(3) results: Another important preliminary work is [6] which provides a canonical representation of PH(3) distributions. More precisely, [6] presents an algorithm that transforms any order 3 matrix exponential function to PH(3) canonical form if it is possible.…”

“…More precisely, [6] presents an algorithm that transforms any order 3 matrix exponential function to PH(3) canonical form if it is possible. In this paper, this algorithm is used to characterize the borders of the PH(3) class.…”