Representation of a Stochastic Traffic Bound

Abstract

This paper presents a theoretical representation of a stochastic traffic bound (σ',ρ') that consists of two items, the burstiness bound σ' and the bound of long-term average rate ρ'. The novelty of the suggested representation is that the burstiness bound and the bound of long-term average rate are separately connected to the fractal dimension D that is the measure of the local self-similarity together with the small-scale factor r and the Hurst parameter H that is the measure of the long-range dependence (LRD) together with the large-scale factor a of traffic. More precisely, we obtain σ' = r <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2D-5</sup> σ and ρ' = a <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-H</sup> ρ, where σ is the conventional bound of burstiness and ρ the conventional bound of long-term average rate, respectively. Thus, the present bound (σ',ρ') takes the conventional bound, say (σ,ρ), as a special case when r = 1 and a = 1. Hence, the proposed representation provides us with a flexible way to tighten a traffic bound. Since we study the stochastically bounded modeling of traffic by taking into account the parameters in stochastic modeling, namely, D, H, r, and a, as well as the parameters in the deterministic modeling of traffic, i.e., σ and ρ, a new outlook regarding the stochastically bounded modeling of traffic is revealed. In addition, we open a problem to estimate r and a with respect to the possible applications of the proposed bound to the practice.