Scaling Limits for Cumulative Input Processes

Abstract

We study different scaling behavior of very general telecommunications cumulative input processes. The activities of a telecommunication system are described by a marked-point process ((Tn, Zn))n∈Z, where Tn is the arrival time of a packet brought to the system or the starting time of the activity of an individual source, and the mark Zn is the amount of work brought to the system at time Tn. This model includes the popular ON/OFF process and the infinite-source Poisson model. In addition to the latter models, one can flexibly model dependence of the interarrival times Tn−Tn−1, clustering behavior due to the arrival of an impulse generating a flow of activities, but also dependence between the arrival process (Tn) and the marks (Zn). Similarly to the ON/OFF and infinite-source Poisson model, we can derive a multitude of scaling limits for the input process of one source or for the superposition of an increasing number of such sources. The memory in the input process depends on a variety of factors, such as the tails of the interarrival times or the tails of the distribution of activities initiated at an arrival Tn, or the number of activities starting at Tn. It turns out that, as in standard results on the scaling behavior of cumulative input processes in telecommunications, fractional Brownian motion or infinite-variance Lévy stable motion can occur in the scaling limit. However, the fractional Brownian motion is a much more robust limit than the stable motion, and many other limits may occur as well.

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