- Phase-type distribution
Probability distribution

name =Phase-type

type =density

pdf_

cdf_

parameters =$S,;\; m\; imes\; m$ subgeneratormatrix

$\backslash boldsymbol\{alpha\}$,probability row vector

support =$x\; in\; [0;\; infty)!$

pdf =$\backslash boldsymbol\{alpha\}e^\{xS\}\backslash boldsymbol\{S\}^\{0\}$

See article for details

cdf =$1-\backslash boldsymbol\{alpha\}e^\{xS\}\backslash boldsymbol\{1\}$

mean =$-1\backslash boldsymbol\{alpha\}\{S\}^\{-1\}mathbf\{1\}$

mode =no simple closed form

variance =$2\backslash boldsymbol\{alpha\}\{S\}^\{-2\}mathbf\{1\}$

median =no simple closed form

skewness =$-6\backslash boldsymbol\{alpha\}\{S\}^\{-3\}mathbf\{1\}/sigma^\{3\}$

kurtosis =$24\backslash boldsymbol\{alpha\}\{S\}^\{-4\}mathbf\{1\}/sigma^\{4\}-3$

entropy =

mgf =$\backslash boldsymbol\{alpha\}(-tI-S)^\{-1\}\backslash boldsymbol\{S\}^\{0\}+alpha\_\{m+1\}$

char =$\backslash boldsymbol\{alpha\}(itI-S)^\{-1\}\backslash boldsymbol\{S\}^\{0\}+alpha\_\{m+1\}$A

**phase-type distribution**is aprobability distribution that results from a system of one or more inter-relatedPoisson process es occurring insequence , or phases. The sequence in which each of the phases occur 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 thestate s 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" ≥ 1. The states 1,...,"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 (**α**,α_{"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,

:$\{Q\}=left\; [egin\{matrix\}\{S\}mathbf\{S\}^0\backslash mathbf\{0\}0end\{matrix\}\; ight]\; ,$

where "S" is an "m"×"m" matrix and

**"S**"^{0}= -S**1**. 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,

:$F(x)=1-\backslash boldsymbol\{alpha\}exp(\{S\}x)mathbf\{1\},$

and the density function,

:$f(x)=\backslash boldsymbol\{alpha\}exp(\{S\}x)mathbf\{S^\{0,$

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. The moments of the distribution function are given by:$E\; [X^\{n\}]\; =(-1)^\{n\}n!\backslash boldsymbol\{alpha\}\{S\}^\{-n\}mathbf\{1\}.$

**pecial 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 α

_{"m"+1}= 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 parameter (α_{1},α_{2},α_{3},α_{4},α_{5}) (such that $sum\; alpha\_i\; =1$ and α_{"i"}> 0 for all "i") and (λ_{1},λ_{2},λ_{3},λ_{4},λ_{5}) can be represented as a phase type distribution with:$\backslash boldsymbol\{alpha\}=(alpha\_1,alpha\_2,alpha\_3,alpha\_4,alpha\_5),$

and

:$\{S\}=left\; [egin\{matrix\}-lambda\_10000\backslash 0-lambda\_2000\backslash 00-lambda\_300\backslash 000-lambda\_40\backslash 0000-lambda\_5\backslash end\{matrix\}\; ight]\; .$

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 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),

:$\backslash boldsymbol\{alpha\}=(1,0,0,0,0),$and:$\{S\}=left\; [egin\{matrix\}-lambdalambda000\backslash 0-lambdalambda00\backslash 00-lambdalambda0\backslash 000-lambdalambda\backslash 0000-lambda\backslash end\{matrix\}\; ight]\; .$

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,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:$\backslash boldsymbol\{alpha\}=(alpha\_1,0,0,alpha\_2,0,0),$

and

:$\{S\}=left\; [egin\{matrix\}-eta\_1eta\_10000\backslash \; 0-eta\_1eta\_1000\backslash 00-eta\_1000\backslash \; 000-eta\_2eta\_20\backslash 0000-eta\_2eta\_2\backslash 00000-eta\_2\backslash end\{matrix\}\; ight]\; .$

**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,:$S=left\; [egin\{matrix\}-lambda\_\{1\}p\_\{1\}lambda\_\{1\}0dots00\backslash \; 0-lambda\_\{2\}p\_\{2\}lambda\_\{2\}ddots00\backslash \; vdotsddotsddotsddotsddotsvdots\backslash \; 00ddots-lambda\_\{k-2\}p\_\{k-2\}lambda\_\{k-2\}0\backslash \; 00dots0-lambda\_\{k-1\}p\_\{k-1\}lambda\_\{k-1\}\backslash \; 00dots00-lambda\_\{k\}end\{matrix\}\; ight]$

and

:$\backslash boldsymbol\{alpha\}=(1,0,dots,0),$

where 0 < "p"

_{1},...,"p"_{"k"-1}≤ 1, in the case where all "p"_{"i"}= 1 we have the 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.**ee also***

Discrete phase-type distribution

*Continuous-time Markov process

*Exponential distribution

*Hyper-exponential distribution

*Queueing model

*Queuing theory **References*** M. F. Neuts. Matrix-Geometric Solutions in Stochastic Models: an Algorthmic 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: Stocahstic 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.

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