Mathematical Preliminaries

Abstract

In this chapter, we present basic background material regarding stochastic processes for the modeling of broadband traffic. We start with the basic theory on random processes and introduce counting processes that are traditionally used as models for traffic processes. A detailed coverage of the matter discussed in this chapter is available in [44],[45],[46]. We then introduce self similar models that were proposed as an alternative for the counting processes and also introduce the reader to some of the terms associated with them. In particular we present concepts of self similarity, long range dependence and heavy tailedness in stochastic processes. These processes can be broadly termed mono fractal processes. The criteria for such a classification of these processes are mentioned and the relationships between these phenomena are looked into. As examples, we also bring out certain stochastic processes that fall into each category and state their properties. It has been observed that measured network traffic exhibits scale invariance or statistical self similarity [11],[23],[47]. This calls for the application of self similar stochastic processes for the modeling and study of network traffic processes. The initial models proposed for broadband network traffic process modeling were based on the variations of the above mentioned classes of stochastic processes. We also mention the inherent limitations in using these monofractal processes for modeling traffic related phenomena thus making strong, the case for applying multifractals for the same.