Self‐Similar Network Traffic: An Overview

Abstract

SELF-SIMILAR NETWORK TRAFFIC: AN OVERVIEWto the shape of the whole, i.e., it is self-similar. Of course, this is not too surprising since the constructive process-by its recursive action-endows the limiting object with the scale-invariance property.The 1-dimensional Cantor set, e.g., as obtained by projecting the 2-D Cantor set onto the line, can be given an interpretation as a traffic series Xt 2 f 0; 1g-call it "Cantor traffic"-where Xt = 1 means that there is a packet transmission at time t. This is depicted in Figure 1. 1.2 (left). If the constructive process is terminated at iteration n 0, then the contiguous line segments of length 1=3 n may be interpreted as on-periods or packet trains of duration 1=3 n , and the segments between successive on-periods as off-periods or absence of traffic activity. Nonuniform traffic intensities may be imparted by generalizing the constructive framework via the use of probability measures. For example, for the 1-dimensional Cantor set, instead of letting the left and right components after scaling have identical "mass," they may be assigned different mass, subject to the constraint that the total mass be preserved at each stage of the iterative construction. This modification corresponds to defining a probability measure on the Borel subsets of 0; 1 and distributing the measure at each iteration nonuniformly left and right. Note that the classical Cantor set construction-viewed as a map-is not measure-preserving. Figure 1shows such a construction with weights L = 2 =3, R = 1 =3 for the left and right components, respectively. The probability measure is represented by "height"; we observe that scale-invariance is exactly preserved. In general, the traffic patterns producible with fixed weights L , R are limited, but one can extend the framework by allowing possibly different weights associated with every edge in the weighted viii SELF-SIMILAR NETWORK TRAFFIC: AN OVERVIEW processes which are nondecreasing processes whose differences-also called increment process-constitute the original process. For example, for the on/off Cantor traffic construction (cf. Figure 1.1.2 (left)), let us assign the interpretation that time is discrete such that at step n 0, it ranges over the values t = 0; 1=3 n ; 2=3 n ; : : : ; 3 n , 1=3 n ; 1. Thus we can equivalently index the discrete time steps by i = 0; 1; 2; : : : ; 3 n . With a slight abuse of notation, let us redefine X as Xi = 1 if, and only if, in the original process Xi=3 n = 1 and Xi=3 n , " = 1 for all 0 " 1=3 n . That is, for i values for which an on-period in the original process Xt begins at t = i=3 n , Xi is defined to be zero. Thus, in the case of n = 2 , we have

Workload Modeling for Computer Systems Performance Evaluation

“…To read more: Papers with good background material on heavy tails include the survey by Newman [513], the classic regarding network traffic by Paxson and Floyd [539], and the overview by Park and Willinger [537]. The volume edited by Adler et al [16] is devoted to dealing with heavy tails, but despite its name (A Practical Guide to Heavy Tails) it is still more on the mathematical than the practical side.…”

“…A good place to start is the book edited by Park and Willinger [536], in particular the introductory chapter by the editors that reviews the field and the underlying mathematics [537]. Surveys include Cappé et al [106], Abry et al [3], and Samorodnitsky [588].…”

“…In general we may have one without the other, and in particular, one can construct self-similar processes that are not long-range dependent. But in the practical situations of interest these two phenomena turn out to coincide, and the self-similarity is typically a result of long-range dependence, and not a result of heavy-tailed distributions [537].…”

“…For example, such evaluations lead to larger and more realistic estimates of required buffer space and other parameters [214]. They also cast doubt on the feasibility of various schemes for providing guaranteed levels of service, because the high variability of the load prevents the system from being able to estimate how much extra capacity is available [537].…”

“…But in the practical situations of interest they turn out to coincide, and the self-similarity is typically a result of long-range dependence and not of heavy-tailed distributions [537]. This starts with Hurst's own work, which demonstrated that self-similarity is observed even if the individual samples are normally distributed [353].…”