Semi-Markov process

Semi-Markov process

A continuous-time stochastic process is called a semi-Markov process or 'Markov renewal process' if the embedded jump chain (the discrete process registering what values the process takes) is a Markov chain, and where the holding times (time between jumps) are random variables with any distribution, whose distribution function may depend on the two states between which the move is made. A semi-Markov process where all the holding times are exponentially distributed is called a continuous time Markov chain/process (CTMC).

A semi-Markov 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[clarification needed].

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.

See also

References and Further Reading

  • Ross, S.M. (1995) Stochastic Processes. Wiley. ISBN 978-0471120629
  • Barbu, V.S, Limnios, N. (2008) Semi-Markov Chains and Hidden Semi-Markov Models toward Applications: Their Use in Reliability and DNA Analysis. ISBN 978-0387731711

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