Encyclopedia of Operations Research and Management Science

2013 Edition
| Editors: Saul I. Gass, Michael C. Fu

Phase-type Probability Distributions

  • Marcel F. Neuts
Reference work entry
DOI: https://doi.org/10.1007/978-1-4419-1153-7_755

The probability distributions of phase-type, or PH-distributions, form a useful general class for the representation of nonnegative random variables. A comprehensive discussion of their basic properties is given in Neuts (1981). There are parallel definitions and properties of discrete and continuous PH-distributions, but the discussion here emphasizes the continuous case.

The simplest example is the Erlang random variable, which can be expressed as the sum of independent exponentially distributed random variables. As a result, one can construct a realization of an Erlang random variable by going through a series of phases, one for each exponential random variable; hence, the Erlang distribution is a phase-type distribution. Generalizing this phase-type idea governs the movement through the phases by a Markov chain that permits movement back and forth between the interior phases, with the final stage being an absorbing barrier.

More specifically, a probability distribution F(⋅) on [0,...
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  1. Asmussen, S. (1992). Phase-type representations in random walk and queueing problems. Annals of Probability, 20, 772–789.CrossRefGoogle Scholar
  2. Asmussen, S., Haggström, O., & Nerman, O. (1992). EMPHT — A program for fitting phase-type distributions (Studies in Statistical Quality Control and Reliability, Mathematical Statistics). Sweden: Chalmers University and University of Göteborg.Google Scholar
  3. Johnson, M. A. (1993a). Selecting parameters of phase distributions: Combining nonlinear programming, heuristics, and Erlang distributions. ORSA Journal on Computing, 5, 69–83.CrossRefGoogle Scholar
  4. Johnson, M. A. (1993b). An empirical study of queueing approximations based on phase-type distributions. Stochastic Models, 9, 531–561.CrossRefGoogle Scholar
  5. Neuts, M. F. (1981). Matrix-geometric solutions in stochastic models: An algorithmic approach. Baltimore: The Johns Hopkins University Press (Reprinted by Dover Publications, 1994).Google Scholar
  6. O'Cinneide, C. A. (1990). Characterization of phase-type distributions. Stochastic Models, 6, 1–57.CrossRefGoogle Scholar
  7. Pagano, M. E., & Neuts, M. F. (1981). Generating random variates from a distribution of phase type. In T. I. Oren, C. M. Delfosse, & C. M. Shub (Eds.), 1981 Winter simulation conference proceedings (pp. 381–387). New Jersey: Institute of Electrical and Electronics Engineers.Google Scholar
  8. Schmickler, L. (1992). MEDA: Mixed Erlang distributions as phase-type representations of empirical distribution functions. Stochastic Models, 8, 131–156.CrossRefGoogle Scholar

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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Systems & Industrial EngineeringThe University of ArizonaTucsonUSA