Discrete phase-type distribution: Wikis

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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.

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

${P}=\left[\begin{matrix}{T}&\mathbf{T}^0\\\mathbf{0}&1\end{matrix}\right],$

where T is a $m\times m$ matrix and $\mathbf{T}^0+{T}\mathbf{1}=\mathbf{1}$. 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.

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 $\mathrm{PH}_{d}(\boldsymbol{\tau},{T})$ or $\mathrm{DPH}(\boldsymbol{\tau},{T})$.

Its cumulative distribution function is

$F(k)=1-\boldsymbol{\tau}{T}^{k}\mathbf{1},$

for k = 0,1,2,..., and its density function is

$f(k)=\boldsymbol{\tau}{T}^{k-1}\mathbf{T^{0}},$

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,

$E[K(K-1)...(K-n+1)]=n!\boldsymbol{\tau}(I-{T})^{-n}{T}^{n-1}\mathbf{1},$

where I is the appropriate dimension identity matrix.

Special cases

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:

References

• M. F. Neuts. Matrix-Geometric Solutions in Stochastic Models: an Algorithmic 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.