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Phase-Type (PH) Distributions

  1. Rafael Pérez-Ocón

Published Online: 15 FEB 2011

DOI: 10.1002/9780470400531.eorms0659

Wiley Encyclopedia of Operations Research and Management Science

Wiley Encyclopedia of Operations Research and Management Science

How to Cite

Pérez-Ocón, R. 2011. Phase-Type (PH) Distributions. Wiley Encyclopedia of Operations Research and Management Science. .

Author Information

  1. University of Granada, Department of Statistics and Operational Research, Granada, Spain

Publication History

  1. Published Online: 15 FEB 2011


The phase-type (PH) distributions were introduced by Neuts (1981). They are playing an important role in modeling random phenomena due to their interesting properties, and it has a widespread use in applied probability. A continuous PH distribution is the distribution of the absorption time in a Markov chain in continuous time with a finite-state space and one absorbent state. The cumulative distribution function (CDF) has exponential form, and it is deduced from the underlying Markov process. The class of PH distributions has important closure properties which make it very versatile in applications; it is closed for finite convolution, finite mixture, formation of coherent systems, and order statistics. On the other hand, it has interesting properties of approximation. This class is dense, in the sense of distributions, in the class of distribution functions defined on the positive real half line, so it is possible to approach general distributions by elements of this class. The use of PH distributions in stochastic modeling leads to results that can be presented in algorithmic form and the results are susceptible to computational implementation. The PH distributions together with the Markovian arrival processes are the basic elements in what is known as matrix-analytic methods.


  • phase-type distribution;
  • Markov process;
  • matrix-exponential distribution;
  • PH-renewal process;
  • shock model