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# Phase-type distribution

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 Title: Phase-type distribution Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### Phase-type distribution

 Parameters 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} -\boldsymbol{\alpha}{S}^{-1}\mathbf{1} no simple closed form no simple closed form 2\boldsymbol{\alpha}{S}^{-2}\mathbf{1}-(\boldsymbol{\alpha}{S}^{-1}\mathbf{1})^{2} -\boldsymbol{\alpha}(tI+S)^{-1}\boldsymbol{S}^{0}+\alpha_{0} -\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 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.

## Contents

• Definition 1
• Characterization 2
• Special cases 3
• Examples 4
• Exponential distribution 4.1
• Hyper-exponential or mixture of exponential distribution 4.2
• Erlang distribution 4.3
• Mixture of Erlang distribution 4.4
• Coxian distribution 4.5
• Generating samples from phase-type distributed random variables 5
• Approximating other distributions 6
• Fitting a phase type distribution to data 7
• See also 8
• References 9

## 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 (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,

{Q}=\left[\begin{matrix}0&\mathbf{0}\\\mathbf{S}^0&{S}\\\end{matrix}\right],

where S is an m × m matrix and S0 = –S1. Here 1 represents an m × 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 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. α0= 0). 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 α0 = 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.

### Hyper-exponential or mixture of exponential distribution

The mixture of exponential or hyper-exponential distribution with λ12,...,λn>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_iF_{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 distribution with parameter E(3,β1), E(3,β2) and (α12) (such that α1 + α2 = 1 and for each i, αi ≥ 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 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 < p1,...,pk-1 ≤ 1. In the case where all pi = 1 we have the hypoexponential 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.

## 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

## 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

## See also

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