Aggregation of network traffic and anisotropic scaling of random fields

Abstract

We discuss joint spatial-temporal scaling limits of sums A λ , γ A_{\lambda ,\gamma } (indexed by ( x , y ) ∈ R + 2 (x,y) \in \mathbb {R}^2_+ ) of large number O ( λ γ ) O(\lambda ^{\gamma }) of independent copies of integrated input process X = { X ( t ) , t ∈ R } X = \{X(t), t \in \mathbb {R}\} at time scale λ \lambda , for any given γ > 0 \gamma >0 . We consider two classes of inputs X 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 A_{\lambda ,\gamma } tend to an α \alpha -stable Lévy sheet ( 1 > α > 2 ) (1> \alpha >2) if γ > γ 0 \gamma > \gamma _0 , and to a fractional Brownian sheet if γ > γ 0 \gamma > \gamma _0 , for some γ 0 > 0 \gamma _0>0 . We also prove an ‘intermediate’ limit for γ = γ 0 \gamma = \gamma _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.