Aggregation of network traffic and anisotropic scaling of random fields

Abstract: 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 normalized random fields $A_{\lambda,\gamma}$ tend to an $\alpha$-stable L\'evy 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 previous works Mikosch et al. (2002), Gaigalas, Kaj (2003) and other papers to more general and new input processes.