Revisiting an old friend: On the observability of the relation between Long Range Dependence and Heavy Tail

Taqqu's Theorem \cite{leland94,willinger97} plays a fundamental role in Internet traffic modeling, for two reasons: First, its theoretical formulation matches closely and in a meaningful manner some of the key network mechanisms controlling traffic characteristics; Second, it offers a plausible explanation for the origin of the long range dependence property in relation with the heavy tail nature of the traffic components. Numerous attempts have since been made to observe its predictions empirically, either from real Internet traffic data or from numerical simulations based on popular traffic models, yet rarely has this resulted in satisfactory quantitative agreements. This raised in the literature a number of comments and questions, ranging from the adequacy of the theorem to real world data to the relevance of the statistical tools involved in practical analyses. The present contribution aims at studying under which conditions this fundamental theorem can be actually seen at work on real or simulated data. To do so, numerical simulations based on standard traffic models are analyzed in a wavelet framework. The key time scales involved are derived, enabling a discussion of the origin and nature of the difficulties encountered in attempts to empirically observe Taqqu's Theorem.