Use of α-stable self-similar stochastic processes for modeling traffic in broadband networks

Fast calculation of α-stable density functions based on off-line precomputations. Application to ML parameter estimation

“…An α-stable CDF can be computed as the cumulative sum of the corresponding PDF. This approach has been used by researchers in several fields where α-stable PDFs are needed (see [11], for instance). Mittnik et al [18] propose a detailed algorithm to compute α-stable densities using this approach which is O (n log 2 n), with n a sufficiently large number of points to be calculated.…”

“…Now, let the three vertices of the found containing triangle be located at (α 1 , β 1 ), (α 2 , β 2 ) and (α 3 , β 3 ). In each of these vertices, an α-stable density has been previously approximated and tabulated with a tolerance of Q x , as described in (11) where w i , i ∈ {1, 2, 3}, are the respective weights of each neighboring precalculated density.…”

Non-asymptotic Delay Bounds for Networks with Heavy-Tailed Traffic

“…The latter two groups include certain self-similar processes, in particular, those governed by fractional Brownian motion [23], but do not extend to heavy-tailed distributions. There are also efforts for extending the network calculus to heavy-tailed distributions [12], [13], [16], [17], which are discussed in more detail in the next section.…”

On convergence of aggregated traffic to totally positively skewed -stable distribution

“…denote the aggregated network traffic, defined as number of packets arriving per unit time [1]. The characteristic function of S(t) can be calculated [5]:…”

“…Considering that the impulsive nature is determined by the low frequency components (|v| ≪ 1) of the characteristic function, we consider the expansion of the above formula (8) around v 0, as in [4,5], and then we obtain…”