# 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 $m\geq1$. 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 $0<p_{1},\dots,p_{k-1}\leq 1$, 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.

## References

• M. F. Neuts. Matrix-Geometric Solutions in Stochastic Models: an Algorthmic Approach, Chapter 2: Probability Distributions of Phase Type; Dover Publications Inc., 1981.
• G. Latouche, V. Ramaswami. Introduction to Matrix Analytic Methods in Stochastic Modelling, 1st edition. Chapter 2: PH Distributions; ASA SIAM, 1999.
• C. A. O'Cinneide (1990). Characterization of phase-type distributions. Communications in Statistics: Stocahstic Models, 6(1), 1-57.
• C. A. O'Cinneide (1999). Phase-type distribution: open problems and a few properties, Communication in Statistic: Stochastic Models, 15(4), 731-757.