Multi-fractional generalized Cauchy process and its application to teletraffic

Abstract

The contributions given in this paper are in two aspects. The first is to introduce a novel random function, which we call the multi-fractional generalized Cauchy (mGC) process. The second is to dissertate its application to network traffic for studying the multi-fractal behavior of traffic on a point-by-point basis. The introduced mGC process is with the time varying fractal dimension D(t) and the time varying Hurst parameter H(t). The representations of the autocorrelation function (ACF) and the power spectrum density (PSD) of the mGC process are proposed. Besides, the asymptotic expressions of the ACF and PSD of the mGC process are presented. The computation formula of D(t) is given. The mGC model may be a new tool to describe the multi-fractal behavior of traffic. Precisely, it may be used to reveal the local irregularity or local self-similarity (LSS), which is a small-time scale behavior of traffic, and global long-term persistence or long-range dependence (LRD), which is a large-time scale behavior of traffic, on a point-by-point basis. The cast study with real traffic traces exhibits that the variance of D(t) is much greater than that of H(t). Thus, the present mGC model may provide a novel way to explain the fact that traffic has highly local irregularity while its LRD is robust.