Discrete phase-type distribution
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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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[edit] Definition
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],](../../../../math/8/4/2/84273ffaabd08179fe75700e073bebb4.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.
[edit] Characterization
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 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},](../../../../math/0/b/2/0b26e749bc7046b5f7a576545e782650.png)
where I is the appropriate dimension identity matrix.
[edit] Special cases
Just as the continuous time distribution is a generalisation of exponential, the discrete time is a genearlisation of the geometric distribution, for example:
- Degenerate distribution, point mass at zero or the empty phase-type distribution - 0 phases.
- 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.
- Negative binomial distribution - 2 or more identical phases in sequence.
[edit] See also
- Phase-type distribution
- Geometric distribution
- Markov chains
- Negative binomial distribution
- Queueing model
- Queuing theory
[edit] 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.
