Small and large scale behavior of moments of Poisson cluster processes

Abstract

Poisson cluster processes are special point processes that find use in modeling Internet traffic, neural spike trains, computer failure times and other real-life phenomena. The focus of this work is on the various moments and cumulants of Poisson cluster processes, and specifically on their behavior at small and large scales. Under suitable assumptions motivated by the multiscale behavior of Internet traffic, it is shown that all these various quantities satisfy scale free (scaling) relations at both small and large scales. Only some of these relations turn out to carry information about salient model parameters of interest, and consequently can be used in the inference of the scaling behavior of Poisson cluster processes. At large scales, the derived results complement those available in the literature on the distributional convergence of normalized Poisson cluster processes, and also bring forward a more practical interpretation of the so-called slow and fast growth regimes. Finally, the results are applied to a real data trace from Internet traffic.

Likelihood-based inference for modelling packet transit from thinned flow summaries

“…However, we have shown that, even on real data, closed-form approximations can be utilised without severe consequences. Optimising a likelihood is slower than methods that rely on simple arithmetic computation (such as the moments-based estimators of Hohn et al (2003) and Antunes et al (2017)). Computation also increases with higher degrees of packet thinning (i.e.…”

“…Renewal processes form a natural context for internet traffic and have previously been used extensively in traffic analysis (e.g. Pipiras, 2011, 2016;Antunes et al, 2017;González-Arévalo and Roy, 2010;Hohn et al, 2003;Hohn and Veitch, 2006;Williamson, 2001). However, Williamson (2001) argues that simple renewal processes are limited in their ability to jointly model packet-level burstiness and the interaction between flows.…”

“…González-Arévalo and Roy (2010) provide an algorithm to generate this process. These models were utilised by Antunes et al (2017), and so we adopt them here for our simulated analyses. Characterising flow-level traffic as Poisson processes admits many attractive features, including independence of flows, memorylessness for successive flows, and uniformity of flow arrivals.…”

“…While computing convolutions of random variables can be expensive, the Gamma distribution is closed under convolutions. The parameters used to generate our simulated datasets are adopted from the empirical analyses of Antunes et al (2017). We restrict our attention to assessing packet level characteristics in order to avoid large computational overheads.…”

“…The Zeta distribution is often used to model the flow sizes since network traffic is typically dominated by many small mouse flows and few large elephant flows (Hohn et al, 2003;Hohn and Veitch, 2006;Antunes et al, 2017). The empirical analysis of Antunes et al (2017) yields a flow size shape parameter of k = 1.02. However, it utilises a distributional approximation by defining the true mean flow size to be k(k−1) −1 .…”