A phase-type distribution is a probability distribution constructed by a convolution or mixture of exponential distributions. It 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 occurs 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 = [ 0 0 S 0 S ] , {\displaystyle {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 column 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 − α exp ( S x ) 1 , {\displaystyle F(x)=1-{\boldsymbol {\alpha }}\exp({S}x)\mathbf {1} ,}
and the density function,
f ( x ) = α exp ( S x ) S 0 , {\displaystyle 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 ! α S − n 1 . {\displaystyle E[X^{n}]=(-1)^{n}n!{\boldsymbol {\alpha }}{S}^{-n}\mathbf {1} .}
The Laplace transform of the phase type distribution is given by
M ( s ) = α 0 + α ( s I − S ) − 1 S 0 , {\displaystyle M(s)=\alpha _{0}+{\boldsymbol {\alpha }}(sI-S)^{-1}\mathbf {S^{0}} ,}
where I is the identity matrix.
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. Hyperexponential 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 platykurtic 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.
Hyperexponential or mixture of exponential distribution The mixture of exponential or hyperexponential distribution with λ1,λ2,...,λn>0 can be represented as a phase type distribution with
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