Phase-type

Parameters	${\displaystyle S,m\times m}$ subgeneratormatrix ${\displaystyle \mathit{\alpha }}$,probabilityrow vector
Support	${\displaystyle x\in [0;\mathrm{\infty })}$
PDF	${\displaystyle \mathit{\alpha }{e}^{xS}{\mathit{S}}^0}$ See article for details
CDF	${\displaystyle 1-\mathit{\alpha }{e}^{xS}\mathbf{1}}$
Mean	${\displaystyle -\mathit{\alpha }{S}^{-1}\mathbf{1}}$
Median	no simple closed form
Mode	no simple closed form
Variance	${\displaystyle 2\mathit{\alpha }{S}^{-2}\mathbf{1}-(\mathit{\alpha }{S}^{-1}\mathbf{1}{)}^2}$
MGF	${\displaystyle -\mathit{\alpha }(tI+S{)}^{-1}{\mathit{S}}^0+{\alpha }_0}$
CF	${\displaystyle -\mathit{\alpha }(itI+S{)}^{-1}{\mathit{S}}^0+{\alpha }_0}$

Aphase-type distribution is aprobability distribution constructed by a convolution or mixture ofexponential distributions.^[1] It results from a system of one or more inter-relatedPoisson processes occurring insequence, or phases. The sequence in which each of the phases occurs may itself be astochastic process. The distribution can be represented by arandom variable describing the time until absorption of aMarkov process with one absorbing state. Each of thestates of the Markov process represents one of the phases.

It has adiscrete-time equivalent – thediscrete 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.

Consider acontinuous-time Markov process withm + 1 states, wherem ≥ 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 them + 1 phases given by the probability vector (α₀,α) whereα₀ is a scalar andα is a 1 × m vector.

Thecontinuous 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 atransition rate matrix,

whereS is anm × m matrix andS⁰ = –S1. Here1 represents anm × 1 column vector with every element being 1.

The distribution of timeX until the process reaches the absorbing state is said to be phase-type distributed and is denoted PH(α,S).

The distribution function ofX is given by,

${\displaystyle F(x)=1-\mathit{\alpha }\exp (Sx)\mathbf{1},}$

and the density function,

${\displaystyle f(x)=\mathit{\alpha }\exp (Sx){\mathbf{S}}^{\mathbf{0}},}$

for allx > 0, where exp( · ) is thematrix 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

${\displaystyle E[X^n]=(-1{)}^{n}n!\mathit{\alpha }{S}^{-n}\mathbf{1}.}$

TheLaplace transform of the phase type distribution is given by

${\displaystyle M(s)={\alpha }_0+\mathit{\alpha }(sI-S{)}^{-1}{\mathbf{S}}^{\mathbf{0}},}$

whereI is the identity matrix.

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

• Degenerate distribution, point mass at zero or theempty 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.

In all the following examples it is assumed that there is no probability mass at zero, that is α₀ = 0.

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 andα = 1.

The mixture of exponential orhyperexponential distribution with λ₁,λ₂,...,λ_n>0 can be represented as a phase type distribution with

${\displaystyle \mathit{\alpha }=({\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4,...,{\alpha }_n)}$

with${\displaystyle \sum _{i=1}^n{\alpha }_i=1}$ and

This mixture of densities of exponential distributed random variables can be characterized through

${\displaystyle f(x)=\sum _{i=1}^n{\alpha }_i{\lambda }_i{e}^{-{\lambda }_{i}x}=\sum _{i=1}^n{\alpha }_i{f}_{X_i}(x),}$

or its cumulative distribution function

${\displaystyle F(x)=1-\sum _{i=1}^n{\alpha }_i{e}^{-{\lambda }_{i}x}=\sum _{i=1}^n{\alpha }_i{F}_{X_i}(x).}$

with${\displaystyle X_i\sim Exp({\lambda }_i)}$

The Erlang distribution has two parameters, the shape an integerk > 0 and the rate λ > 0. This is sometimes denotedE(k,λ). The Erlang distribution can be written in the form of a phase-type distribution by makingS ak×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),

${\displaystyle \mathit{\alpha }=(1,0,0,0,0),}$

and

For a given number of phases, the Erlang distribution is the phase type distribution with smallest coefficient of variation.^[2]

Thehypoexponential distribution is a generalisation of the Erlang distribution by having different rates for each transition (the non-homogeneous case).

The mixture of two Erlang distributions with parameterE(3,β₁),E(3,β₂) and (α₁,α₂) (such that α₁ + α₂ = 1 and for eachi, α_i ≥ 0) can be represented as a phase type distribution with

${\displaystyle \mathit{\alpha }=({\alpha }_1,0,0,{\alpha }_2,0,0),}$

and

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

and

${\displaystyle \mathit{\alpha }=(1,0,\dots ,0),}$

where 0 <p₁,...,p_k-1 ≤ 1. In the case where allp_i = 1 we have the Erlang distribution. The Coxian distribution is extremely important as any acyclic phase-type distribution has an equivalent Coxian representation.

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

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

BuTools includes methods for generating samples from phase-type distributed random variables.^[3]

Any distribution can be arbitrarily well approximated by a phase type distribution.^[4][5] 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^[2]).

• BuTools aMATLAB andMathematica script for fitting phase-type distributions to 3 specified moments
• momentmatching aMATLAB script to fit a minimal phase-type distribution to 3 specified moments^[6]
• KPC-toolbox a library ofMATLAB scripts to fit empirical datasets to Markovian arrival processes and phase-type distributions.^[7]

Methods to fit a phase type distribution to data can be classified as maximum likelihood methods or moment matching methods.^[8] Fitting a phase type distribution toheavy-tailed distributions has been shown to be practical in some situations.^[9]

• PhFit a C script for fitting discrete and continuous phase type distributions to data^[10]
• EMpht is a C script for fitting phase-type distributions to data or parametric distributions using anexpectation–maximization algorithm.^[11]
• 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.^[12]
• jPhase is a Java library which can also compute metrics for queues using the fitted phase type distribution^[13]

• Discrete phase-type distribution
• Continuous-time Markov process
• Exponential distribution
• Hyper-exponential distribution
• Queueing theory

• M. F. Neuts (1975), Probability distributions of phase type, In Liber Amicorum Prof. Emeritus H. Florin, Pages 173-206, University of Louvain.
• M. F. Neuts.Matrix-Geometric Solutions in Stochastic Models: an Algorithmic 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: Stochastic 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.

• Continuous distributions
• Types of probability distributions