Semi-Markov process

In probability and statistics a Markov renewal process is a that generalizes the notion of Markov jump processes. Other random processes like Markov chain, Poisson process, and renewal process can be derived as a special case of an MRP (Markov renewal process).

Consider a state space ${\displaystyle \mathrm {S} .}$ Consider a set of random variables ${\displaystyle (X_{n},T_{n})}$, where ${\displaystyle T_{n}}$ are the jump times and ${\displaystyle X_{n}}$ are the associated states in the Markov chain (see Figure). Let the inter-arrival time, ${\displaystyle \tau _{n}=T_{n}-T_{n-1}}$. Then the sequence ${\displaystyle (X_{n},T_{n})}$ is called a Markov renewal process if

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