Markov renewal process
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In probability and statistics a Markov renewal process is a random process 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).
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Definition
Consider a state space Consider a set of random variables , where are the jump times and are the associated states in the Markov chain (see Figure). Let the inter-arrival time, . Then the sequence is called a Markov renewal process if
Relation to other stochastic processes
- If we define a new stochastic process for , then the process is called a semi-Markov process. Note the main difference between an MRP and a semi-Markov process is that the former is defined as a two-tuple of states and times, whereas the latter is the actual random process that evolves over time and any realisation of the process has a defined state for any given time. The entire process is not Markovian, i.e., memoryless, as happens in a CTMC. Instead the process is Markovian only at the specified jump instants. This is the rationale behind the name, Semi-Markov.[1][2][3] (See also: hidden semi-Markov model.)
- A semi-Markov process (defined in the above bullet point) where all the holding times are exponentially distributed is called a continuous time Markov chain/process (CTMC). 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 CTMC.
- The sequence in the MRP is a discrete-time Markov chain. In other words, if the time variables are ignored in the MRP equation, we end up with a DTMC.
- If the sequence of s are independent and identically distributed, and if their distribution does not depend on the state , then the process is a renewal process. So, if the states are ignored and we have a chain of iid times, then we have a renewal process.
See also
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