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Scaling Limits for Cumulative Input Processes

Laboratory of Actuarial Mathematics, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark
School of Operations Research and Information Engineering, Cornell University, Ithaca, New York 14853
mikosch{at}math.ku.dk
gennady{at}orie.cornell.edu

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 ((T_n, Z_n))_nZ, where T_n 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 Z_n is the amount of work brought to the system at time T_n. 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 T_n–T_n–1, clustering behavior due to the arrival of an impulse generating a flow of activities, but also dependence between the arrival process (T_n) and the marks (Z_n). 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 T_n, or the number of activities starting at T_n. 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.

Key Words: telecommunications; cumulative input process; marked-point process; stable Lévy motion; fractional Brownian motion; scaling limits; long-range dependence
History: Received: April 11, 2006; revision received: September 6, 2006;