Aggregation of network traffic and anisotropic scaling of random fields

TL;DR

This paper investigates the joint spatial-temporal scaling limits of aggregated network traffic modeled by different stochastic processes, revealing conditions under which the limits are stable Lévy or fractional Brownian sheets.

Contribution

It extends previous results by establishing new conditions for the convergence of aggregated traffic to stable or Gaussian limits for more general input processes.

Findings

Normalized sums tend to an $ ext{α}$-stable Lévy sheet for certain scaling.

Normalized sums tend to a fractional Brownian sheet for other scalings.

Identifies an intermediate limit case at a critical scaling threshold.

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.