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A phase-type distribution is a probability distribution that results from a system 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 Chain with one absorbing state. Each of the states of the Markov Chain represent one of the phases.

The phase-type distribution is said to be dense in the field of all positive valued distributions, that is it can be used approximate any positive valued distribution.

Contents 1 Continuous phase-type distribution 1.1 Definition 1.2 Special cases 1.3 Examples 1.3.1 Erlang distribution 2 Discrete phase-type distribution 2.1 Definition 2.2 Special cases 3 See also 4 References

Assume there exists a continuous-time Markov process with m + 1 states, where , 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 .

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.

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

The distribution function of X is given by,

and the density function,

for all x > 0. If it is assumed the probability of process starting in the absorbing state is zero, the moments of the distribution function are given by,

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

• 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 - 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.)

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 a 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),

and

The discrete phase-type distribution is defined similar to the continuous time.

Assume there exists a discrete-time Markov chain with m + 1 states, where , 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 .

This process can be written in the form of a stochastic matrix,

where is a matrix and .

The distribution of the number of steps K until the process reaches the absorbing state is said to be discretely phase-type distributed and is denoted PH.

The distribution function of K is given by,

for k = 0,1,2,..., and the density function,

for k = 1,2,.... If it is assumed the probability of process starting in the absorbing state is zero, the factorial moments of the distribution function are given by,

where I is the appropriate dimension identity matrix.

Just as the continuous time distribution is a generalisation of exponential, the discrete time is a genearlisation of the geometric distribution, for example:

• Geometric distribution - 1 phase.
• Hypergeometric distribution - 2 or more non-identical phases, that each have a probability of occurring in a mutually exclusive, or parallel, manner.

• Continuous-time Markov process
• Erlang distribution
• Exponential distribution
• Markov chains
• Queuing theory

• Sheldon M Ross. Introduction to Probability Models, 8th edition. Chapter 6: Continuous Time Markov Chains, Academic Press; December, 2002
• G. Latouche, V. Ramaswami. Introduction to Matrix Analytic Methods in Stochastic Modelling, 1st edition. Chapter 2: PH Distributions; ASA SIAM, 1999.

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