Phase-type distribution

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: 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.
 * 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.

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.