Stationary Point Processes

  • Philippe Robert
Part of the Applications of Mathematics book series (SMAP, volume 52)


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.


Point Process Arrival Process Departure Process Poisson Point Process Marked Point Process 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Philippe Robert
    • 1
  1. 1.Domaine de VoluceauINRIALe ChesnayFrance

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