, probability row vector
See article for details
|median:||no simple closed form|
|mode:||no simple closed form|
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.
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.
Consider a continuous-time Markov process with m+1 states, where m ≥ 1, such that the states 1,...,m are transient states and state m+1 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,αm+1).
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,
where S is an m×m matrix and S0 = -S1. Here 1 represents an m×1 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(α,S).
The distribution function of X is given by,
and the density function,
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
The following probability distributions are all considered special cases of a continuous phase-type distribution:
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.
In all the following examples it is assumed that there is no probability mass at zero, that is αm+1 = 0.
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.
The mixture of exponential or hyper-exponential distribution with parameter (α1,α2,α3,α4,α5) (such that and αi > 0 for all i) and (λ1,λ2,λ3,λ4,λ5) can be represented as a phase type distribution with
The mixture of exponential can be characterized through its density
or its cummulative distribution function
This can be generalized to a mixture of n exponential distributions.
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),
The hypoexponential distribution is a generalisation of the Erlang distribution by having different rates for each transition (the non-homogeneous case).
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, αi ≥ 0) can be represented as a phase type distribution with
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,
where 0 < p1,...,pk-1 ≤ 1. In the case where all pi = 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.