Aggregation of network traffic and anisotropic scaling of random fields

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