Aggregation of network traffic and anisotropic scaling of random fields

We discuss joint spatial-temporal scaling limits of sums A _λ,γ (indexed by (x, y) ∈ R ²) of large number O(λ ^γ ) of independent copies of integrated input process X = {X(t),t ∈ R} at time scale λ, for any given γ > 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 _λ,γ tend to an α-stable Lévy sheet (1 < α < 2) if γ < γ ₀, and to a fractional Brownian sheet if γ > γ ₀, for some γ ₀ > 0. We also prove an ‘intermediate’ limit for γ = γ ₀. 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. Rootzén and A. Stegeman [Ann. Appl. Probab. 12 (2002), no. 1, 23–68] and other papers to more general and new input processes.