Phase-type distribution

Phase-type

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 phase-type distribution is a probability distribution that 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.

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 λ₁,λ₂,...,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 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

${S}=\left[\begin{matrix}-\lambda&\lambda&0&0&0\\0&-\lambda&\lambda&0&0\\0&0&-\lambda&\lambda&0\\0&0&0&-\lambda&\lambda\\0&0&0&0&-\lambda\\\end{matrix}\right].$

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 (α₁,α₂) (such that α₁ + α₂ = 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. 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,...,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.

Approximation

Theoretically, any distribution can be arbitrarily well approximated by a phase type distribution. In practice, however, approximations can be poor. Approximating a deterministic distribution of time 1 with 10 phases, each of average length 0.1 will have variance 0.1.

See also

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

References

• 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: 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.