The discrete phase-type distribution is a probability distribution that results from a system of one or more inter-related geometric distributions 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 represents one of the phases.
It has continuous time equivalent in the phase-type distribution.
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A terminating Markov chain is a Markov chain where all states are transient, except one which is absorbing. Reordering the states, the transition probability matrix of a terminating Markov chain with m transient states is
![{P}=\left[\begin{matrix}{T}&\mathbf{T}^0\\\mathbf{0}&1\end{matrix}\right],](http://images-mediawiki-sites.thefullwiki.org/07/2/9/4/3141552851801931.png)
where T is a 
matrix and 
.
The transition matrix is characterized entirely by its upper-left
block T.
Definition. A distribution on {0,1,2,...} is a discrete phase-type distribution if it is the distribution of the first passage time to the absorbing state of a terminating Markov chain with finitely many states.
Fix a terminating Markov chain. Denote T the upper-left block of its
transition matrix and τ the initial
distribution. The distribution of the first time to the absorbing
state is denoted 
or 
.
Its cumulative distribution function is
for k = 0,1,2,..., and its density function is
for k = 1,2,.... It is assumed the probability of process starting in the absorbing state is zero. The factorial moments of the distribution function are given by,
![E[K(K-1)...(K-n+1)]=n!\boldsymbol{\tau}(I-{T})^{-n}{T}^{n-1}\mathbf{1},](http://images-mediawiki-sites.thefullwiki.org/09/2/8/1/57523333537782350.png)
where I is the appropriate dimension identity matrix.
Just as the continuous time distribution is a generalisation of the exponential distribution, the discrete time distribution is a generalisation of the geometric distribution, for example:
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