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Discrete Stochastic Processes, Chapter 4: Renewal Processes

Chapter 4 Renewal Processes Introduction Recall that a Renewal process is an arrival process in which the interarrival intervals are positive,1 independent and identically distributed (IID) random variables (rv s). Renewal Processes (since they are arrival Processes ) can be specified in three standard ways, first, by the joint distributions of the arrival epochs S1, S2, .. , second, by the joint distributions of the interarrival times X1, X2, .. , and third, by the joint distributions of the counting rv s, N(t) for t > 0. Recall that N(t) represents the number of arrivals to the system in the interval (0, t].)

Example 4.1.1 (Visits to a given state for a Markov chain). Suppose a recurrent finite-state Markov chain with transition matrix [P] starts in state i at time 0. Then on the first return to state i, say at time n, the Markov chain, from time n on, is a probabilistic replica of the chain starting at time 0. That is, the state at time 1 is j ...

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