AGGREGATION OF NETWORK TRAFFIC AND ANISOTROPIC SCALING OF RANDOM FIELDS

We discuss joint spatial-temporal scaling limits of sums A(?,?) (indexed by (x, y)? R-+(2)) of large number O(?(?)) of independent copies of integrated input process X = {X (t), t ? R} at time scale ?, for any given-y > 0. We consider two classes of inputs X: (I) Poisson shot-noise with (random) pulse process, and (II) regenerative process with random pulse process and regeneration times following a heavy-tailed stationary renewal process. The above classes include several queueing and network traffic models for which joint spatial-temporal limits were previously discussed in the literature. In both cases (I) and (II) we find simple conditions on the input process in order that the normalized random fields A?,? tend to an a-stable Le ' vy sheet (1 < a < 2) if-y <-y0, and to a fractional Brownian sheet if-y >-y0, for some-y0 > 0. We also prove an 'intermediate' limit for-y =-y0. Our results extend the previous works of R. Gaigalas and I. Kaj [Bernoulli 9 (2003), no. 4, 671-703] and T. Mikosch, S. Resnick, H. Rootze ' n and A. Stegeman [Ann. Appl. Probab. 12 (2002), no. 1, 23-68] and other papers to more general and new input processes.