# 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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## References

1. Asmussen, S. (1992). Phase-type representations in random walk and queueing problems. Annals of Probability, 20, 772–789.
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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
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