Semi-Markov process
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In the mathematical theory of stochastic processes, a semi-Markov process Z, also known as a Markov renewal process, is associated with and can be constructed from a pair of processes W = (X,Y), where X is a Markov chain with state space S and transition probability matrix P, whereas Y is a process for which Y(n) depends only on r = X(n − 1) and s = X(n), and whose distribution function is Frs.
The semi-Markov process Z is then the process that chooses its sites (on S) according to X(n), and that chooses the transition time from X(n − 1) to X(n) according to Y(n).
Since the properties of Y (such as mean transition time) may depend on which site X chooses next, the processes Z are in general not a Markov process. Yet, the associated process W(n) = (X(n),Y(n)) is a Markov process. Hence the name semi-Markov.
