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]. The simplest characterization is through the interarrival times Xi, since they are IID. Each arrival epoch Sn is simply the sum X1 + X2 ++ Xn of n IID rv s. The characterization of greatest interest in this Chapter is the Renewal counting process, {N(t); t > 0}. Recall from ( ) and ( ) that the arrival epochs and the counting rv s are related in each of the following equivalent ways.)

158 CHAPTER 4. RENEWAL PROCESSES In most situations, we use the words arrivals and renewals interchangably, but for this type of example, the word arrival is used for the counting process {N(t); t > 0} and the word renewal is used for {Nr(t); t > 0}.The reason for being interested in {Nr(t); t > 0} is that it allows us to analyze very complicated queues such as this in two stages.

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  Processes, Stochastic, Stochastic processes

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