# Phase-type distribution

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.

## Definition

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 (α$_0$,α) where α$_0$ 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,
    ${Q}=\left[{\begin{matrix}0&\mathbf {0} \\\mathbf {S} ^{0}&{S}\\\end{matrix}}\right],$
where S is an m × m matrix and S$^0$ = –S1. Here 1 represents an m × 1 column 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 PH(α,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( · ) is the matrix exponential. It is usually assumed the probability of process starting in the absorbing state is zero (i.e. α$_0$= 0). The moments of the distribution function are given by
    $E[X^{n}]=(-1)^{n}n!{\boldsymbol {\alpha }}{S}^{-n}\mathbf {1} .$
The Laplace transform of the phase type distribution is given by
    $M(s)=\alpha _{0}+{\boldsymbol {\alpha }}(sI-S)^{-1}\mathbf {S^{0}} ,$
where I is the identity matrix.

## 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.
* Hyperexponential 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 platykurtic 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 α$_0$ = 0.

### Exponential distribution

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.

### Hyperexponential or mixture of exponential distribution

The mixture of exponential or hyperexponential distribution with λ$_1$,λ$_2$,...,λ$_n$>0 can be represented as a phase type distribution with
    ${\boldsymbol {\alpha }}=(\alpha _{1},\alpha _{2},\alpha _{3},\alpha _{4},...,\alpha _{n})$
with $\sum _{i=1}^{n}\alpha _{i}=1$ and
    ${S}=\left[{\begin{matrix}-\lambda _{1}&0&0&0&0\\0&-\lambda _{2}&0&0&0\\0&0&-\lambda _{3}&0&0\\0&0&0&-\lambda _{4}&0\\0&0&0&0&-\lambda _{5}\\\end{matrix}}\right].$
This mixture of densities of exponential distributed random variables can be characterized through
    $f(x)=\sum _{i=1}^{n}\alpha _{i}\lambda _{i}e^{-\lambda _{i}x}=\sum _{i=1}^{n}\alpha _{i}f_{X_{i}}(x),$
or its cumulative distribution function
    $F(x)=1-\sum _{i=1}^{n}\alpha _{i}e^{-\lambda _{i}x}=\sum _{i=1}^{n}\alpha _{i}F_{X_{i}}(x).$
with $X_{i}\sim Exp(\lambda _{i})$

### Erlang distribution

The Erlang distribution has two parameters, the shape an integer k > 0 and the rate λ > 0. This is sometimes denoted E(k,λ). The Erlang distribution can be written in the form of a phase-type distribution by making S a k×k matrix with diagonal elements -λ and super-diagonal elements λ, with the probability of starting in state 1 equal to 1. For example, E(5,λ),
    ${\boldsymbol {\alpha }}=(1,0,0,0,0),$
and
    ${S}=\left[{\begin{matrix}-\lambda &\lambda &0&0&0\\0&-\lambda &\lambda &0&0\\0&0&-\lambda &\lambda &0\\0&0&0&-\lambda &\lambda \\0&0&0&0&-\lambda \\\end{matrix}}\right].$
For a given number of phases, the Erlang distribution is the phase type distribution with smallest coefficient of variation.
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 distributions with parameter E(3,β$_1$), E(3,β$_2$) and (α$_1$,α$_2$) (such that α$_1$ + α$_2$ = 1 and for each i, α$_i$ ≥ 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 Erlang distribution. 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$_1$,...,p$_{k-1}$ ≤ 1. In the case where all p$_i$ = 1 we have the Erlang distribution. The Coxian distribution is extremely 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.

## Properties
### Minima of Independent PH Random Variables

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

## Generating samples from phase-type distributed random variables

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

## Approximating other distributions

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).
* BuTools a MATLAB and Mathematica script for fitting phase-type distributions to 3 specified moments
* momentmatching a MATLAB script to fit a minimal phase-type distribution to 3 specified moments
* KPC-toolbox a library of MATLAB scripts to fit empirical datasets to Markovian arrival processes and phase-type distributions.

## Fitting a phase type distribution to data

Methods to fit a phase type distribution to data can be classified as maximum likelihood methods or moment matching methods. Fitting a phase type distribution to heavy-tailed distributions has been shown to be practical in some situations.
* PhFit a C script for fitting discrete and continuous phase type distributions to data
* EMpht is a C script for fitting phase-type distributions to data or parametric distributions using an expectation–maximization algorithm.
* HyperStar was developed around the core idea of making phase-type fitting simple and user-friendly, in order to advance the use of phase-type distributions in a wide range of areas. It provides a graphical user interface and yields good fitting results with only little user interaction.
* jPhase is a Java library which can also compute metrics for queues using the fitted phase type distribution