On the canonical representation of phase type distributions

Gábor Horváth, Miklós Telek

Research output: Contribution to journalArticle

18 Citations (Scopus)


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) vector-matrix representation of the distribution. We show that-using the same approach-no 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.

Original languageEnglish
Pages (from-to)396-409
Number of pages14
JournalPerformance Evaluation
Issue number8
Publication statusPublished - Aug 1 2009



  • Canonical representation
  • Moment bounds
  • Phase-type distribution

ASJC Scopus subject areas

  • Software
  • Modelling and Simulation
  • Hardware and Architecture
  • Computer Networks and Communications

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