Abstract-We consider a family of stochastic processes built from infinite sums of independent positive random functions on . Each of these functions increases linearly between two consecutive negative jumps, with the jump points following a Poisson point process on . The motivation for studying these processes stems from the fact that they constitute simplified models for TCP traffic. Such processes bear some analogy with Lévy processes, but are more complex since their increments are neither stationary nor independent. In the work of Barral and Lévy Véhel, the Hausdorff multifractal spectrum of these processes was computed. We are interested here in their Large Deviation and Legendre multifractal spectra. These "statistical" spectra are seen to give, in this case, a richer information than the "geometrical" Hausdorff spectrum. In addition, our results provide a firm theoretical basis for the empirical discovery of the multifractal nature of TCP traffic. Index Terms-Hölder regularity, internet traffic control protocol, Large Deviation multifractal spectrum, Legendre multifractal spectrum, multifractal processes. I. BACKGROUND AND MOTIVATIONS E STUDY in this paper a family of stochastic processes built from infinite sums of independent positive random functions on . Each of these functions increases linearly between two consecutive negative jumps, with the jump points following a Poisson point process on . The interest of this class of processes is twofold. The first is theoretical: They provide examples of additive processes with nonstationary and correlated increments that have a rich multifractal behavior. More precisely, it was shown in [3] that their Hausdorff multifractal spectrum is nontrivial and is similar to the one of Lévy processes. We compute here their Large Deviation and Legendre multifractal spectra, and we show that they give an even more precise information than the Hausdorff spectrum. The second interest stems from applications: The motivation for studying the processes considered here is that they constitute simplified but realistic models for Traffic Control Protocol (TCP) traffic on the Internet. Empirical studies, beginning with [19] and [26], have shown that traffic on the Internet generated by the TCP is, under wide conditions, multifractal. This property has important consequences in practice. For instance, one may show that the queuing behavior of a multifractal traffic is significantly worse that the one of a nonfractal traffic (see [6] for details). It is therefore desirable to understand which features of TCP are responsible for multifractality, and maybe reduce their negative impact on, e.g., the queuing behavior. "Explaining" the multifractality of traffic traces from basic features of the Internet is a difficult task. Models investigated so far have been based on the paradigm of multiplicative cascades [6] , [20] . Indeed, with few exceptions (notably [14] , [16], and [17]), multifractal analysis has mainly been applied to multiplicative processes. An obvious reason is that a multiplicative structure often leads naturally to multifractal properties [22] , [23] . However, there exists a number of real-world processes for which there is convincing experimental evidence of multifractality, but which do not display an associated multiplicative structure. Among these, a major example is Internet traffic: Multiplicative models for TCP are not really convincing because there is no physical evidence that genuine traffic actually behaves as a cascading or multiplicative process. As a matter of fact, TCP traffic is rather an additive process, where the contributions of individual sources of traffic are merged in a controlled way. The analysis developed below shows that merely adding sources managed by TCP does lead to a multifractal behavior. This result provides a theoretical confirmation to the empirical finding that TCP traffic is multifractal. Furthermore, it sheds light on the possible causes of this multifractality. Indeed, it indicates that it may be explained from the very nature of the protocol, with no need to invoke a hypothetical multiplicative structure: It appears that multifractality in TCP arises from the interplay between the additive increase multiplicative decrease (AIMD) mechanism and the random nonsynchronism of the sources. In addition, comparing the multifractal spectrum estimated numerically from TCP logs to our theoretical findings should allow to describe the fine structure of the hierarchy of sources. 1 Indeed, our computations permit to trace back, in a quantitative way, the main multifractal features of traces to specific mechanisms of TCP. The exact details of TCP seem too intricate to allow for a tractable mathematical analysis. We consider a simplified model that captures the main ingredients of the congestion avoidance 1 We note in passing that the spectra computed in this paper are the ones that are estimated from numerical data, in contrast to the Hausdorff spectrum obtained in [3] , which is purely theoretical. 1063-6692 © 2012 IEEE
Lévy-Véhel and Rams REF showed that adding sources managed by TCP can lead to multifractal behavior.
Large Deviation Multifractal Analysis of a Class of Additive Processes With Correlated Nonstationary Increments
IEEE/ACM Transactions on Networking
IEEE/ACM Transactions on Networking
Mathematics Computer Science