Is Network Traffic Appriximated by Stable Lévy Motion or Fractional Brownian Motion?

Abstract

Cumulative broadband network traffic is often thought to be well modeled by fractional Brownian motion (FBM). However, some traffic measurements do not show an agreement with the Gaussian marginal distribution assumption. We show that if connection rates are modest relative to heavy tailed connection length distribution tails, then stable Lévy motion is a sensible approximation to cumulative traffic over a time period. If connection rates are large relative to heavy tailed connection length distribution tails, then FBM is the appropriate approximation. The results are framed as limit theorems for a sequence of cumulative input processes whose connection rates are varying in such a way as to remove or induce long range dependence.

Scaling Limits for Cumulative Input Processes

“…Such limits quite clearly depend on the relative speed at which the number of input processes n and the timescale T grow. For two special cases of (3), the ON/OFF model and the infinite-source Poisson model (see §3), it has been established in Mikosch et al [28] that if the number of input processes grows relatively slowly, under a proper normalization the sequence D n T converges in distribution (in the M 1 topology) to a stable Lévy motion. If, on the other hand, the number of input processes grows relatively fast, a properly normalized process D n T converges in distribution (in the J 1 topology) to a fractional Brownian motion.…”

“…This is perhaps the most popular model for teletraffic. Consider a single ON/OFF source such as a workstation as described in Heath et al [15], Leland et al [24], Taqqu et al [37], Pipiras and Taqqu [31], Levy and Taqqu [25], Mikosch et al [28], Stegeman [35], Gaigalas and Kaj [12], and Pipiras et al [32]. During an ON period, the source generates traffic at a constant rate one, for example, one byte per time unit.…”

“…In particular, limit theory for the scaled and centered cumulative input process has been employed to prove some fundamental results about teletraffic; see Leland et al [24], Taqqu et al [37], Mikosch et al [28], and Gaigalas and Kaj [12]. One of the aims of this line of research was to show that the LRD of the individual sources (which is due to the Pareto-like ON/OFF times) can be inherited by the limit process of the cumulative input.…”

“…However, depending on the number of superimposed ON/OFF processes, one can get quite different limit processes. Mikosch et al [28] show that if the number of ON/OFF sources increases "fast" with time, one gets fractional Brownian motion with Hurst coefficient H ∈ 0 5 1 , i.e., a Gaussian process with LRD in its increments. If this number grows "slowly" one gets an -stable Lévy motion as weak limit of the scaled cumulative input process.…”

Prediction in a Poisson cluster model

“…More recently, shot noise processes have been used for modeling large computer networks such as the Internet; see for example Konstantopoulos and Lin [12], Kurtz [13] for some early work. In the context of the workload of large computer networks, shot noise processes arise as aggregated versions of the ON/OFF or infinite source Poisson models, also known as M/G/∞ model; see for example Levy and Taqqu [16], Pipiras and Taqqu [24], Mikosch et al [22] and for further extensions Faÿ et al [2], Mikosch and Samorodnitsky [23]. Other applications include finance (Samorodnitsky [25], Klüppelberg and Kühn [8]) and physics (Giraitis et al [3]).…”