Markov renewal processes are a class of random processes in probability and statistics that generalize the class of Markov jump processes. Other classes of random processes, such as Markov chains and Poisson processes, can be derived as special cases among the class of Markov renewal processes, while Markov renewal processes are special cases among the more general class of renewal processes.

Definition

to be considered a Markov renewal process the following condition should hold:

Relation to other stochastic processes

1. is called a semi-Markov process as it happens in a continuous-time Markov chain. The process is Markovian only at the specified jump instants, justifying the name semi-Markov. (See also: hidden semi-Markov model.)

2. A semi-Markov process (defined in the above bullet point) in which all the holding times are exponentially distributed is called a continuous-time Markov chain. In other words, if the inter-arrival times are exponentially distributed and if the waiting time in a state and the next state reached are independent, we have a continuous-time Markov chain.

3. in the Markov renewal process is a discrete-time Markov chain. In other words, if the time variables are ignored in the Markov renewal process equation, we end up with a discrete-time Markov chain.

4. , then the process is a renewal. So, if the states are ignored and we have a chain of iid times, then we have a renewal process.

See also

• Markov process

• Renewal theory

• Variable-order Markov model

• Hidden semi-Markov model

References