Towards Describing Multi-fractality of Traffic Using Local Hurst Function

Abstract

Long-range dependence and self-similarity are two basic properties of network traffic time series. Fractional Brownian motion (fBm) and its increment process fractional Gaussian noise (fGn) are commonly used to model network traffic with the Hurst index H that determines both the regularity of the sample paths and the long memory property of traffic. However, it appears too restrictive for traffic modeling since it can only model sample paths with the same smoothness for all time parameterized by a constant H. A natural extension of fBm is multifractional Brownian motion (mBm), which is indexed by a time-dependent Hurst index H(t). The main objective of this paper is to model multi-fractality of traffic using H(t), i.e., mBm, on a point-by-point basis instead of an interval-by-interval basis as traditionally done in computer networks. The numerical results for H(t) of real traffic, which are demonstrated in this paper, show that H(t) of traffic is time-dependent, which not only provide an alternative evidence of the multifractal phenomena of traffic but also reveal an challenging issue in traffic modeling: multi-fractality modeling of traffic.