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Consider a discrete time absorbing Markov chain with state space Xn, n = 0, 1, 2, ∙ ∙ ∙ , with Xn = 0, 1, 2, ∙ ∙ ∙ , N, where state 0 is an absorbing state. The transition matrix of this chain can be written as

where with for at least one i. Also define t = 1 – T1, where 1 is a column of ones.

There is a discrete random variable Y, which is said to have a PH distribution ( α,T) if one can write

Several well-known discrete distributions can be represented as PH distributions. Examples include the geometric distribution, the negative binomial distribution, to name just a few. In addition, most discrete distributions can be reasonably approximated by discrete PH distribution (see Mészáros et al. 2014 and references therein).

It was shown in Alfa (2004) that any discrete distribution, with finite support, can be represented by PH distribution with elapsed time or remaining time format. For example, consider the interarrival A and let . In the elapsed time format, the matrix T has only its superdiagonal elements that are non-zero and every other element is zero, and the vector α = [1, 0, 0, ∙∙∙ , ], and the matrix T is written as

where

For the remaining time representation, the vector is the vector of the discrete distribution and the matrix T is written as