Discrete phase-type distribution

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 . The distribution can be represented by a random variable describing the time until absorption of an absorbing 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.

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 ${\displaystyle m}$ transient states is

where ${\displaystyle {T}}$ is a ${\displaystyle m\times m}$ matrix and ${\displaystyle \mathbf {T} ^{0}+{T}\mathbf {1} =\mathbf {1} }$. The transition matrix is characterized entirely by its upper-left block ${\displaystyle {T}}$.

...
Wikipedia