Gaussian Processes in Communication Networks

Telecommunication systems have drastically changed in the past 20 years mainly because they allow to transfer data, video, voice in the same support. The traffic in telecommunications networks does not behave as the classical models of telephony based on Poisson processes with exponential input. Instead, self-similarity and long-range dependence of the Internet traffic were revealed. Self-similarity means invariance in distribution under suitable scaling of time and space, long-range dependence means slow decay of autocorrelation function.

Gaussian models became popular to describe long-range dependence when this property of the network traffic was revealed. The best understood Gaussian process is the fractional Brownian motion (fBm) is now very popular in Internet traffic modeling. This is justified by the theoretical result that the sum of a large number of on-off inputs, with either on-times or off-times having a heavy-tailed distribution with infinite variance, converges to fBm, after rescaling time appropriately.

In the talk, we present main notions and basic features of Gaussian self-similar and long-range dependent processes arising in the modeling of modern broadband communication networks. We concentrate on the mathematical properties of the models. However, Gaussian approximation cannot be relied on a priory (there are many situations in which Gaussianity does not apply). That is why Gaussian approximation in the context of communication networking should be quantitatively justified. By this reason we also give a survey for some methods to detect whether the one-dimensional marginal distribution of traffic increments satisfies normal approximation. These methods are based on statistical tests which are also presented.