Proof of a fundamental result in self-similar traffic modeling

We state and prove the following key mathematical result in self-similar traffic modeling: the superposition of many ON/OFF sources (also known as packet trains) with strictly alternating ON- and OFF-periods and whose ON-periods or OFF-periods exhibit the Noah Effect (i.e., have high variability or infinite variance) can produce aggregate network traffic that exhibits the Joseph Effect (i.e., is self-similar or long-range dependent). There is, moreover, a simple relation between the parameters describing the intensities of the Noah Effect (high variability) and the Joseph Effect (self-similarity). This provides a simple physical explanation for the presence of self-similar traffic patterns in modern high-speed network traffic that is consistent with traffic measurements at the source level. We illustrate how this mathematical result can be combined with modern high-performance computing capabilities to yield a simple and efficient linear-time algorithm for generating self-similar traffic traces.We also show how to obtain in the limit a Lévy stable motion, that is, a process with stationary and independent increments but with infinite variance marginals. While we have presently no empirical evidence that such a limit is consistent with measured network traffic, the result might prove relevant for some future networking scenarios