Stationary Point Processes

Abstract

A queueing system can be seen as an operator on arrival processes. If the sequence of the arrival times of customers is (t _n ) and (S _n ) is the sequence of their respective sojourn times in the queue (the nth customer arrives at time t _n and leaves at t _n + S _n ). The queue transforms a point process {t _n } (the arrival process) in another point process {t _n + S _n } (the departure process). In this setting, it is quite natural to investigate the properties of point processes that are preserved by such a transformation. In fact, very few properties remain unchanged. Most of the independence properties are lost for the departure process (the examples of the M/M/1 queue or some product form networks seen in Chapter 4 are remarkable exceptions to this general rule). For example, if the arrival process is a renewal process, the departure process is not, in general, a renewal process.