Poisson Approximations for Traffic Processes in Telecommunications Networks

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 it can be argued that 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 to reality. 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. Further, they allow one to establish an approximate version of Ben Melamed's loop criterion for identifying which traffic flows in Markovian networks are Poisson.

I shall conclude by sketching more recent work which has some surprising implications, even for the folklore of when the Poisson distribution is a good approximation to the Binomial distribution.

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

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Last modified: 16th July 1995