Poisson type models and descriptive statistics of computer network information flows

Many contemporary publications on network traffic gravitate to ideas of self-similarity and long-range dependence. The corresponding elegant and parsimonious mathematical techniques proved to be efficient for the description of a wide class of aggregated processes. Sharing the enthusiasm about the above ideas the authors also believe that whenever it is possible any problem must be considered at the most basic level in an attempt to understand the driving forces of the processes under analysis. Consequently the authors try to show that some behavioral patterns of descriptive statistics which are typical for long-memory processes (a particular case of long-range dependence) can also be explained in the framework of the traditional Poisson process paradigm. Applying the concepts of inhomogeneity, compoundness and double stochasticity they propose a simple and intuitively transparent approach of explaining the expected shape of the observed histograms of counts and the expected behavior of the sample covariance functions. Matching the images of these two descriptive statistics allows them to infer the presence of trends or double stochasticity in analyzed time series. They considered only statistics which are based on counts. A similar approach may be applied to waiting or inter-arrival time sequences and will be discussed in other publications. They hope that combining the reported results with the statistical methods based on aggregated models may lead to computationally affordable on-line techniques of compact and visualized data analysis of network flows.