On approximate renewal models for the superposition of renewal processes

It is well known that the superposition of a finite number of renewal processes is not renewal anymore. A renewal approximation can be obtained by simply ignoring the interarrival dependencies and using the interarrival distribution. We show that this simple approximation is also rate-optimal, i.e., it defines a rate process that minimizes the mean-squared rate error functional over the set of all renewal processes. We also show that the optimal approximation is closely related to the rate of a new process, called the recurrence process, which is constructed by sampling the recurrence times from the original process. Applications to traffic analysis are discussed.