# Phase-type distribution

Probability density function | |

Cumulative distribution function | |

Parameters | subgenerator matrix , probability row vector |
---|---|

Support | |

Probability density function (pdf) | See article for details |

Cumulative distribution function (cdf) | |

Mean | |

Median | no simple closed form |

Mode | no simple closed form |

Variance | |

Skewness | |

Excess kurtosis | |

Entropy | |

Moment-generating function (mgf) | |

Characteristic function |

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 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.

## Contents

## Definition

There exists a continuous-time Markov process with states, where . The states are transient states and state is an absorbing state. The process has an initial probability of starting in any of the phases given by the probability vector .

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,

where is a matrix and . Here represents an vector with every element being 1.

## Characterization

The distribution of time until the process reaches the absorbing state is said to be phase-type distributed and is denoted .

The distribution function of is given by,

and the density function,

for all , where 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,

## 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 .

#### Exponential distribution

The simplest non-trivial example of a phase-type distribution is the exponential distribution of parameter . The parameter of the phase-type distribution are : and

#### Hyper-exponential or mixture of exponential distribution

The mixture of exponential or hyper-exponential distribution with parameter (such that and ) and can be represented as a phase type distribution with

and

The mixture of exponential can be characterized through its density

or its distribution function

This can be generalized to a mixture of exponential distributions.

#### Erlang distribution

The Erlang distribution has two parameters, the shape an integer and the rate . This is sometimes denoted . The Erlang distribution can be written in the form of a phase-type distribution by making a matrix with diagonal elements and super-diagonal elements , with the probability of starting in state 1 equal to 1. For example ,

and

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 , and (such that and ) can be represented as a phase type distribution with

and

#### 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,

and

where , in the case where all we have the hypoexponential distribution. The Coxian distribution is extremly 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.

## See 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.