Poisson Approximations for Traffic Processes

Phil Pollett

Abstract: In modelling teletraffic systems it is frequently assumed that the offered traffic is a Poisson process. This assumption arises largely in order that the models can be analysed simply, but it is certainly appropriate when the numbers of arrivals in a given time interval are independent of past arrivals, and has a Poisson distribution. As an immediate consequence of the Poisson assumption, the arrival rate is deterministic (non-random), conditional on the past. This seemingly unremarkable property provides the key to the modern theory of traffic processes, for it actually characterizes Poisson processes. That deterministic past-conditional arrival rate implies Poisson traffic is a celebrated theorem of Watanabe, and I shall begin with a very elementary proof of this result.

On many occasions the Poisson assumption is, at best, only an approximation. I shall present a simple method which allows one to assess the accuracy of this approximation. The method involves establishing bounds on the degree of deviation from Poisson traffic. These bounds enable one to make precise predictions as to the circumstances in which the approximation is good.

Acknowledgement: This worked was funded by the Australian Research Council.

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Last modified: 6th June 2000