# Phase-type distribution

Parameters Probability density function Cumulative distribution function $S,\;m\times m$ subgenerator matrix${\boldsymbol {\alpha }}$ , probability row vector $x\in [0;\infty )\!$ ${\boldsymbol {\alpha }}e^{xS}{\boldsymbol {S}}^{0}$ See article for details $1-{\boldsymbol {\alpha }}e^{xS}{\boldsymbol {1}}$ $-1{\boldsymbol {\alpha }}{S}^{-1}\mathbf {1}$ no simple closed form no simple closed form $2{\boldsymbol {\alpha }}{S}^{-2}\mathbf {1}$ $-6{\boldsymbol {\alpha }}{S}^{-3}\mathbf {1} /\sigma ^{3}$ $24{\boldsymbol {\alpha }}{S}^{-4}\mathbf {1} /\sigma ^{4}$ ${\boldsymbol {\alpha }}(-tI-S)^{-1}{\boldsymbol {S}}^{0}+\alpha _{m+1}$ ${\boldsymbol {\alpha }}(itI-S)^{-1}{\boldsymbol {S}}^{0}+\alpha _{m+1}$ A phase-type distribution is a probability distribution that 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 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 phase-type distribution is dense in the field of all positive-valued distributions, that is, it can be used to approximate any positive valued distribution.

## Definition

There exists a continuous-time Markov process with $m+1$ states, where $m\geq 1$ . The states $1,\dots ,m$ are transient states and state $m+1$ is an absorbing state. The process has an initial probability of starting in any of the $m+1$ phases given by the probability vector $({\boldsymbol {\alpha }},\alpha _{m+1})$ .

The continuous phase-type distibution is the distribution of time from the processes starting until absorption in the absorbing state.

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

${Q}=\left[{\begin{matrix}{S}&\mathbf {S} ^{0}\\\mathbf {0} &0\end{matrix}}\right],$ where ${S}$ is a $m\times m$ matrix and $\mathbf {S} ^{0}=-{S}\mathbf {1}$ . Here $\mathbf {1}$ represents an $m\times 1$ vector with every element being 1.

## Characterization

The distribution of time $X$ until the process reaches the absorbing state is said to be phase-type distributed and is denoted $\operatorname {PH} ({\boldsymbol {\alpha }},{S})$ .

The distribution function of $X$ is given by,

$F(x)=1-{\boldsymbol {\alpha }}\exp({S}x)\mathbf {1} ,$ and the density function,

$f(x)={\boldsymbol {\alpha }}\exp({S}x)\mathbf {S^{0}} ,$ for all $x>0$ , where $\exp(\cdot )$ is the matrix exponential. It is usually assumed the probability of process starting in the absorbing state is zero. The moments of the distribution function are given by,

$E[X^{n}]=(-1)^{n}n!{\boldsymbol {\alpha }}{S}^{-n}\mathbf {1} .$ ## Special cases

The following probability distributions are all considered special cases of a continuous phase-type distribution:

• Degenerate distribution, point mass at zero or the empty phase-type distribution - 0 phases.
• Exponential distribution - 1 phase.
• Erlang distribution - 2 or more identical phases in sequence.
• Deterministic distribution (or constant) - The limiting case of an Erlang distribution, as the number of phases become infinite, while the time in each state becomes zero.
• Coxian distribution - 2 or more (not necessarily identical) phases in sequence, with a probability of transitioning to the terminating/absorbing state after each phase.
• Hyper-exponential distribution (also called a mixture of exponential) - 2 or more non-identical phases, that each have a probability of occurring in a mutually exclusive, or parallel, manner. (Note: The exponential distribution is the degenerate situation when all the parallel phases are identical.)
• Hypoexponential distribution - 2 or more phases in sequence, can be non-identical or a mixture of identical and non-identical phases, generalises the Erlang.

As the phase-type distribution is dense in the field of all positive-valued distributions, we can represent any positive valued distribution. However, the phase-type is a light-tailed or platikurtic distribution. So the representation of heavy-tailed or leptokurtic distribution by phase type is an approximation, even if the precision of the approximation can be as good as we want.

### Examples

In all the following examples it is assumed that there is no probability mass at zero, that is $\alpha _{m+1}=0$ .

#### Exponential distribution

The simplest non-trivial example of a phase-type distribution is the exponential distribution of parameter $\lambda$ . The parameter of the phase-type distribution are : ${\boldsymbol {S}}=-\lambda$ and ${\boldsymbol {\alpha }}=1$ #### Hyper-exponential or mixture of exponential distribution

The mixture of exponential or hyper-exponential distribution with parameter $(\alpha _{1},\alpha _{2},\alpha _{3},\alpha _{4},\alpha _{5})$ (such that $\sum \alpha _{i}=1$ and $,\alpha _{i}>0\forall i$ ) and $(\lambda _{1},\lambda _{2},\lambda _{3},\lambda _{4},\lambda _{5})$ can be represented as a phase type distribution with

${\boldsymbol {\alpha }}=(\alpha _{1},\alpha _{2},\alpha _{3},\alpha _{4},\alpha _{5}),$ and

${S}=\left[{\begin{matrix}-\lambda _{1}&0&0&0&0\\0&-\lambda _{2}&0&0&0\\0&0&-\lambda _{3}&0&0\\0&0&0&-\lambda _{4}&0\\0&0&0&0&-\lambda _{5}\\\end{matrix}}\right].$ The mixture of exponential can be characterized through its density

$f(x)=\sum _{i=1}^{5}\alpha _{i}\lambda _{i}e^{-\lambda _{i}x}$ or its distribution function

$F(x)=1-\sum _{i=1}^{5}\alpha _{i}e^{-\lambda _{i}x}.$ This can be generalized to a mixture of $n$ exponential distributions.

#### Erlang distribution

The Erlang distribution has two parameters, the shape an integer $k>0$ and the rate $\lambda >0$ . This is sometimes denoted $E(k,\lambda )$ . The Erlang distribution can be written in the form of a phase-type distribution by making ${S}$ a $k\times k$ matrix with diagonal elements $-\lambda$ and super-diagonal elements $\lambda$ , with the probability of starting in state 1 equal to 1. For example $E(5,\lambda )$ ,

${\boldsymbol {\alpha }}=(1,0,0,0,0),$ and

${S}=\left[{\begin{matrix}-\lambda &\lambda &0&0&0\\0&-\lambda &\lambda &0&0\\0&0&-\lambda &\lambda &0\\0&0&0&-\lambda &\lambda \\0&0&0&0&-\lambda \\\end{matrix}}\right].$ The hypoexponential distribution is a generalisation of the Erlang distribution by having different rates for each transition (the non-homogeneous case).

#### Mixture of Erlang distribution

The mixture of two Erlang distribution with parameter $E(3,\beta _{1})$ , $E(3,\beta _{2})$ and $(\alpha _{1},\alpha _{2})$ (such that $\alpha _{1}+\alpha _{2}=1$ and $\forall i,\alpha _{i}\geq 0$ ) can be represented as a phase type distribution with

${\boldsymbol {\alpha }}=(\alpha _{1},0,0,\alpha _{2},0,0),$ and

${S}=\left[{\begin{matrix}-\beta _{1}&\beta _{1}&0&0&0&0\\0&-\beta _{1}&\beta _{1}&0&0&0\\0&0&-\beta _{1}&0&0&0\\0&0&0&-\beta _{2}&\beta _{2}&0\\0&0&0&0&-\beta _{2}&\beta _{2}\\0&0&0&0&0&-\beta _{2}\\\end{matrix}}\right].$ #### Coxian distribution

The Coxian distribution is a generalisation of the hypoexponential. Instead of only being able to enter the absorbing state from state k it can be reached from any phase. The phase-type representation is given by,

$S=\left[{\begin{matrix}-\lambda _{1}&p_{1}\lambda _{1}&0&\dots &0&0\\0&-\lambda _{2}&p_{2}\lambda _{2}&\ddots &0&0\\\vdots &\ddots &\ddots &\ddots &\ddots &\vdots \\0&0&\ddots &-\lambda _{k-2}&p_{k-2}\lambda _{k-2}&0\\0&0&\dots &0&-\lambda _{k-1}&p_{k-1}\lambda _{k-1}\\0&0&\dots &0&0&-\lambda _{k}\end{matrix}}\right]$ and

${\boldsymbol {\alpha }}=(1,0,\dots ,0),$ where $0 , in the case where all $p_{i}=1$ we have the hypoexponential distribution. The Coxian distribution is extremly important as any acyclic phase-type distribution has an equivalent Coxian representation.

The generalised Coxian distribution relaxes the condition that requires starting in the first phase. 