Superposition of Diffusions and its Multifractal Limit Process, A working model for high-speed network traffic data

Computer network traffic has recently been the subject of various types of statistical studies including fractal analysis, and in particular, measuring and modeling Long-Range Dependence (LRD), investigating self-similarity, and showing multifractal properties. The common agreement among several empirical findings about the general properties of traces is summarized in as follows.

(1) Many signals show significant LRD, but behavior inconsistent with strict self-similarity.

(2) For many signals, the scaling behavior of moments as the signal is aggregated is a nontrivial function of the moment order.

3) Many signals have increments that are inherently positive, skewed and hence non-Gaussian; (4) There are some additional properties motivated by our experimental study of ATM traces, providing strong evidence of the presence of Gamma distribution and real-valued bispectrum. Therefore there are two additional requirements.

(4) The marginal distribution of signals of ATM traces is close to Gamma distribution.

(5) Signals of ATM traces have a real-valued bispectrum.

Having these properties in mind we have studied a certain nonlinear diffusion process, superposition of such processes with random coefficients, the limit of the centralized integral processes of the superposition processes and its increment process. The main objective is to find a multifractal model which has an analytically and statistically tractable higher order cumulant structure. We have applied our model to real data. The time series of ATM traces measured in SUNET fits our model very well. The feasibility of carrying out parameter estimation utilizing the dilative stability is also discussed to some extent.