Multiplicative Cascades

Abstract

In this chapter, we investigate, the mathematical techniques that we will be using for the analysis and modeling of broadband network traffic. We describe the multiplicative cascade model which is a paradigm shift in the modeling of random processes from traditional linear additive models. The analysis of the binomial cascade process is carried out and the expression for the multifractal spectrum is derived. We also examine the interpretation of the multifractal spectrum curve in an intuitive manner involving fractal dimensions, and then using large deviation principles. The relationships of the multifractal spectrum with Renyi’s entropy and generalized dimensions are also brought out. Multiplicative cascades were first proposed by Kolmogorov in the modeling of turbulence [101]. Turbulence flows are characterized by a scaling relation which controls the transfer of energy from one scale to another in a cascade structure [102], [103],[104], [105],[106],[107].