Aggregation of network traffic and anisotropic scaling of random fields

Abstract

We discuss joint spatial-temporal scaling limits of sums $A_{λ,γ}$ (indexed by $(x,y) \in \mathbb{R}^2_+$) of large number $O(λ^γ)$ of independent copies of integrated input process $X = \{X(t), t \in \mathbb{R}\}$ 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 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 previous works Mikosch et al. (2002), Gaigalas, Kaj (2003) and other papers to more general and new input processes.