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A phase type distribution is a probability distribution constructed by a convolution or mixture of exponential distributions. It results from a system of one or more inter related Poisson processes occurring in sequence, or phases. The sequence in which each of the phases occurs may itself be a stochastic process. The distribution can be represented by a random variable describing the time until absorption of a Markov process with one absorbing state. Each of the states of the Markov process represents one of the phases. It has a discrete time equivalent the discrete phase type distribution. The set of phase type distributions is dense in the field of all positive valued distributions, that is, it can be used to approximate any positive valued distribution.

In all the following examples it is assumed that there is no probability mass at zero, that is α0 = 0.

The simplest non trivial example of a phase type distribution is the exponential distribution of parameter λ. The parameter of the phase type distribution are : S = λ and α = 1.

Similarly to the exponential distribution, the class of PH distributions is closed under minima of independent random variables. A description of this is here.

BuTools includes methods for generating samples from phase type distributed random variables.

Any distribution can be arbitrarily well approximated by a phase type distribution. In practice, however, approximations can be poor when the size of the approximating process is fixed. Approximating a deterministic distribution of time 1 with 10 phases, each of average length 0.1 will have variance 0.1 (because the Erlang distribution has smallest variance
KPC toolbox a library of MATLAB scripts to fit empirical datasets to Markovian arrival processes and phase type distributions.