Abstract
We consider a family of stochastic processes built from infinite sums of independent positive random functions on $R_+$. Each of these functions increases linearly between two consecutive negative jumps, with the jump points following a Poisson point process on $R_+$. The motivation for studying these processes stems from the fact that they constitute simplified models for TCP traffic on the Internet. Such processes bear some analogy with Lévy processes, but they are more complex in the sense that their increments are neither stationary nor independent. Nevertheless, we show that their multifractal behavior is very much the same as that of certain Lévy processes. More precisely, we compute the Hausdorff multifractal spectrum of our processes, and find that it shares the shape of the spectrum of a typical Lévy process. This result yields a theoretical basis to the empirical discovery of the multifractal nature of TCP traffic.
Highlights
Background and MotivationsWe study in this work a family of stochastic processes built from infinite sums of independent positive random functions on R+
The first is theoretical: It will be seen that the infinite sums of independent random positive functions that we study, though they have non-stationary and correlated increments, have connections with Levy processes
Though the strategy developed in [17, 18] to study the multifractal nature of functions with a dense countable set of jump points applies partly here, our more complex setting requires different and/or refined arguments at key points of the study
Summary
We study in this work a family of stochastic processes built from infinite sums of independent positive random functions on R+ Each of these functions increases linearly between two consecutive negative jumps, with the jump points following a Poisson point process on R+. Recent empirical studies, beginning with [23, 29], have shown that traffic on the Internet generated by the Traffic Control Protocol (TCP) is, under wide conditions, multifractal This property has important consequences in practice. The analysis developed below shows that merely adding sources managed by TCP does lead to a multifractal behavior This result provides a theoretical confirmation to the empirical finding that TCP traffic is multifractal. Our computations allow to trace back, in a quantitative way, the main multifractal features of traces to specific mechanisms of TCP This may have practical consequence in traffic engineering
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