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Stationary Point Processes

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

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.

Keywords

Point Process Arrival Process Departure Process Poisson Point Process Marked Point Process 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

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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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