Limits of on/off hierarchical product models for data transmission

Abstract

A hierarchical product model seeks to model network traffic as a product of independent on/off processes. Previous studies have assumed a Markovian structure for component processes amounting to assuming that exponential distributions govern on and off periods, but this is not in good agreement with traffic measurements. However, if the number of factor processes grows and input rates are stabilized by allowing the on period distribution to change suitably, a limiting on/off process can be obtained which has exponentially distributed on periods and whose off periods are equal in distribution to the busy period of an $M/G/\infty$ queue. We give a fairly complete study of the possible limits of the product process as the number of factors grows and offer various characterizations of the approximating processes. We also study the dependence structure of the approximations.