Fractional Brownian Motions and Levy Motions as Limits of Stochastic Traffic Models

摘要

This article offers an overview of some approximating schemes and their performance analysis for communication networks driven by traffic processes possessing long-range dependence. The schemes lead to the so-called fractional Brownian motion models or Levy motion models. Under certain conditions, the load of a network with bursty inputs can be mathematically well approximated by a reflected multi-dimensional Levy motion with certain drift. Although fractional Brownian motion models are mathematically fundamental in describing long-range dependence and self-similarity, it is argued that a particular type of Levy models is a good candidate for explaining phenomena observed in networking practice. Unlike fractional Brownian motions, Levy motions are semimartingales, and hence stochastic calculus plays an important role in investigating the stationary regime and quality of service of the network. We point out that there is a systematic way for writing evolution equations and equations for the stationary regime. En passant, we deal with questions of existence and uniqueness of a stationary regime and the question of existence and construction of a suitable topology that makes the multi-dimensional reflection mapping (which is responsible for the precise characterization of the limiting evolution equations) continuous, thereby allowing us to pass the limit inside the reflection operator.