Phase-type distribution

Phase-type

Parameters	${\displaystyle S,\;m\times m}$ subgenerator matrix ${\displaystyle {\boldsymbol {\alpha }}}$, probability row vector
Support	${\displaystyle x\in [0;\infty )\!}$
PDF	${\displaystyle {\boldsymbol {\alpha }}e^{xS}{\boldsymbol {S}}^{0}}$ See article for details
CDF	${\displaystyle 1-{\boldsymbol {\alpha }}e^{xS}{\boldsymbol {1}}}$
Mean	${\displaystyle -{\boldsymbol {\alpha }}{S}^{-1}\mathbf {1} }$
Median	no simple closed form
Mode	no simple closed form
Variance	${\displaystyle 2{\boldsymbol {\alpha }}{S}^{-2}\mathbf {1} -({\boldsymbol {\alpha }}{S}^{-1}\mathbf {1} )^{2}}$
MGF	${\displaystyle -{\boldsymbol {\alpha }}(tI+S)^{-1}{\boldsymbol {S}}^{0}+\alpha _{0}}$
CF	${\displaystyle -{\boldsymbol {\alpha }}(itI+S)^{-1}{\boldsymbol {S}}^{0}+\alpha _{0}}$

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 occur may itself be a . 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.

Consider a continuous-time Markov process with m + 1 states, where m ≥ 1, such that the states 1,...,m are transient states and state 0 is an absorbing state. Further, let the process have an initial probability of starting in any of the m + 1 phases given by the probability vector (α₀,α) where α₀ is a scalar and α is a 1 × m vector.

The continuous phase-type distribution is the distribution of time from the above process's starting until absorption in the absorbing state.

This process can be written in the form of a transition rate matrix,

where S is an m × m matrix and S⁰ = –S1. Here 1 represents an m × 1 vector with every element being 1.

The distribution of time X until the process reaches the absorbing state is said to be phase-type distributed and is denoted PH(α,S).

The distribution function of X is given by,

and the density function,

for all x > 0, where exp( · ) is the matrix exponential. It is usually assumed the probability of process starting in the absorbing state is zero (i.e. α₀= 0). The moments of the distribution function are given by

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