Phasetype
Parameters

S,\; m\times m subgenerator matrix
\boldsymbol{\alpha}, probability row vector

Support

x \in [0; \infty)\!

PDF

\boldsymbol{\alpha}e^{xS}\boldsymbol{S}^{0}
See article for details

CDF

1\boldsymbol{\alpha}e^{xS}\boldsymbol{1}

Mean

\boldsymbol{\alpha}{S}^{1}\mathbf{1}

Median

no simple closed form

Mode

no simple closed form

Variance

2\boldsymbol{\alpha}{S}^{2}\mathbf{1}(\boldsymbol{\alpha}{S}^{1}\mathbf{1})^{2}

MGF

\boldsymbol{\alpha}(tI+S)^{1}\boldsymbol{S}^{0}+\alpha_{0}

CF

\boldsymbol{\alpha}(itI+S)^{1}\boldsymbol{S}^{0}+\alpha_{0}

A phasetype distribution is a probability distribution constructed by a convolution or mixture of exponential distributions.^{[1]} It results from a system of one or more interrelated 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 phasetype distribution.
The set of phasetype distributions is dense in the field of all positivevalued distributions, that is, it can be used to approximate any positivevalued distribution.
Contents

Definition 1

Characterization 2

Special cases 3

Examples 4

Exponential distribution 4.1

Hyperexponential or mixture of exponential distribution 4.2

Erlang distribution 4.3

Mixture of Erlang distribution 4.4

Coxian distribution 4.5

Generating samples from phasetype distributed random variables 5

Approximating other distributions 6

Fitting a phase type distribution to data 7

See also 8

References 9
Definition
Consider a continuoustime 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 phasetype 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 vector with every element being 1.
Characterization
The distribution of time X until the process reaches the absorbing state is said to be phasetype 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}.
Special cases
The following probability distributions are all considered special cases of a continuous phasetype distribution:

Degenerate distribution, point mass at zero or the empty phasetype 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 nonidentical 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 nonidentical or a mixture of identical and nonidentical phases, generalises the Erlang.
As the phasetype distribution is dense in the field of all positivevalued distributions, we can represent any positive valued distribution. However, the phasetype is a lighttailed or platikurtic distribution. So the representation of heavytailed 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 nontrivial example of a phasetype distribution is the exponential distribution of parameter λ. The parameter of the phasetype 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_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 phasetype distribution by making S a k×k matrix with diagonal elements λ and superdiagonal 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.^{[2]}
The hypoexponential distribution is a generalisation of the Erlang distribution by having different rates for each transition (the nonhomogeneous case).
Mixture of Erlang distribution
The mixture of two Erlang distribution 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 hypoexponential distribution. Instead of only being able to enter the absorbing state from state k it can be reached from any phase. The phasetype 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_{k2}&p_{k2}\lambda_{k2}&0\\ 0&0&\dots&0&\lambda_{k1}&p_{k1}\lambda_{k1}\\ 0&0&\dots&0&0&\lambda_{k} \end{matrix}\right]
and

\boldsymbol{\alpha}=(1,0,\dots,0),
where 0 < p_{1},...,p_{k1} ≤ 1. In the case where all p_{i} = 1 we have the hypoexponential distribution. The Coxian distribution is extremely important as any acyclic phasetype distribution has an equivalent Coxian representation.
The generalised Coxian distribution relaxes the condition that requires starting in the first phase.
Generating samples from phasetype distributed random variables
BuTools includes methods for generating samples from phasetype distributed random variables.^{[3]}
Approximating other distributions
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 a MATLAB and Mathematica script for fitting phasetype distributions to 3 specified moments

momentmatching a MATLAB script to fit a minimal phasetype distribution to 3 specified moments^{[6]}
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.^{[7]} Fitting a phase type distribution to heavytailed distributions has been shown to be practical in some situations.^{[8]}

PhFit a C script for fitting discrete and continuous phase type distributions to data^{[9]}

EMpht is a C script for fitting phasetype distributions to data or parametric distributions using an expectation–maximization algorithm.^{[10]}

HyperStar was developed around the core idea of making phasetype fitting simple and userfriendly, in order to advance the use of phasetype distributions in a wide range of areas. It provides a graphical user interface and yields good fitting results with only little user interaction.^{[11]}

jPhase is a Java library which can also compute metrics for queues using the fitted phase type distribution^{[12]}
See also
References

^ HarcholBalter, M. (2012). "RealWorld Workloads: High Variability and Heavy Tails". Performance Modeling and Design of Computer Systems. p. 347.

^ ^{a} ^{b}

^ Horváth, G. B.; Reinecke, P.; Telek, M. S.; Wolter, K. (2012). "Efficient Generation of PHDistributed Random Variates". Analytical and Stochastic Modeling Techniques and Applications. Lecture Notes in Computer Science 7314. p. 271.

^ Bolch, Gunter; Greiner, Stefan; de Meer, Hermann; Trivedi, Kishor S. (1998). "SteadyState Solutions of Markov Chains". Queueing Networks and Markov Chains. pp. 103–151.

^

^ Osogami, T.; HarcholBalter, M. (2006). "Closed form solutions for mapping general distributions to quasiminimal PH distributions". Performance Evaluation 63 (6): 524.

^ Lang, Andreas; Arthur, Jeffrey L. (1996). "Parameter approximation for PhaseType distributions". In Chakravarthy, S.; Alfa, Attahiru S. Matrix Analytic methods in Stochastic Models. CRC Press.

^ Ramaswami, V.; Poole, D.; Ahn, S.; Byers, S.; Kaplan, A. (2005). "Ensuring Access to Emergency Services in the Presence of Long Internet DialUp Calls". Interfaces 35 (5): 411.

^ Horváth, András S.; Telek, Miklós S. (2002). "PhFit: A General PhaseType Fitting Tool". Computer Performance Evaluation: Modelling Techniques and Tools. Lecture Notes in Computer Science 2324. p. 82.

^ Asmussen, Søren; Nerman, Olle; Olsson, Marita (1996). "Fitting PhaseType Distributions via the EM Algorithm". Scandinavian Journal of Statistics 23 (4): 419–441.

^ Reinecke, P.; Krauß, T.; Wolter, K. (2012). "Clusterbased fitting of phasetype distributions to empirical data". Computers & Mathematics with Applications 64 (12): 3840.

^ Pérez, J. F.; Riaño, G. N. (2006). "jPhase: an objectoriented tool for modeling phasetype distributions". Proceeding from the 2006 workshop on Tools for solving structured Markov chains (SMCtools '06) (PDF).

M. F. Neuts. MatrixGeometric 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 phasetype distributions. Communications in Statistics: Stochastic Models, 6(1), 157.

C. A. O'Cinneide (1999). Phasetype distribution: open problems and a few properties, Communication in Statistic: Stochastic Models, 15(4), 731757.














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