Markov renewal process

A semi - Markov process (SMP ), also known as a Markov renewal process, is a generalization of a Markov process. In contrast to a Markov process, the changes of state take place at equal time intervals, in this case the residence time is given in a state of another stochastic process.

Definition

In the theory of stochastic processes, a semi- Markov process is given by a pair of processes. is a Markov chain with state space and transition matrix (so-called controlled chain). is a process for, and dependent only on. The distribution function is given by this.

The semi - Markov process is the one process whose state is determined at the time accordingly. The residence time of up is then given by.

Properties

Since the properties of both depends on the current state and next state of the Markov property is generally not fulfilled. However, the process is a Markov process. This also explains the name semi - Markov process.

Applications

Systems, for example, in queuing theory have properties that can not always be mapped using simple Markov processes. One example is called the autocorrelation. To achieve this, semi- Markov process for modeling the arrival rates are often used.