Encyclopedia of Operations Research and Management Science

2001 Edition
| Editors: Saul I. Gass, Carl M. Harris

Phase-type probability distributions

Reference work entry
DOI: https://doi.org/10.1007/1-4020-0611-X_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. This discussion emphasizes the continuous case.

A probability distribution F(⋅) on [0, ∞)is ofphasetype if it can arise as the absorption time distribution of an (m + 1)-state Markov chain with m transient states 1,..., m and an absorbing state 0. The generator Qv of such a Markov chain is written as
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References

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    Asmussen, S. (1992). “Phase-type representations in random walk and queueing problems,” Annals Probability, 20, 772–789.Google Scholar
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    Asmussen, S., Haggström, O., and Nerman, O. (1992). “EMPHT — A program for fitting phase-type distributions,” in Studies in Statistical Quality Control and Reliability, Mathematical Statistics, Chalmers University and University of Göteborg, Sweden.Google Scholar
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    O'Cinneide, C.A. (1990). “Characterization of phase-type distributions,” Stochastic Models, 6, 1–57.Google Scholar
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    Johnson, M.A. (1993). “Selecting parameters of phase distributions: Combining nonlinear programming, heuristics, and Erlang distributions,” ORSA Jl. Computing, 5, 69–83.Google Scholar
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    Johnson, M.A. (1993). “An empirical study of queueing approximations based on phase-type distributions,” Stochastic Models, 9, 531–561.Google Scholar
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    Neuts, M.F. (1981). Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. The Johns Hopkins University Press, Baltimore. Reprinted by Dover Publications, 1994.Google Scholar
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    Pagano, M.E. and Neuts, M.F. (1981). “Generating Random Variates from a Distribution of Phase Type,” 1981 Winter Simulation Conference Proceedings, T.I. Oren, C.M. Delfosse, C.M. Shub, eds., 381–387. Google Scholar
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    Schmickler, L. (1992). “MEDA: Mixed Erlang distributions as phase-type representations of empirical distribution functions,” Stochastic Models, 8, 131–156.Google Scholar

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  1. 1.The University of ArizonaTucsonUSA