Murad S. Taqqu

Stable non-Gaussian random processes : stochastic models with infinite variance

Stable random variables on the real line Multivariate stable distributions Stable stochastic integrals Dependence structures of multivariate stable distributions Non-linear regression Complex stable stochastic integrals and harmonizable processes Self-similar processes Chentsov random fields Introduction to sample path properties Boundedness, continuity and oscillations Measurability, integrability and absolute continuity Boundedness and continuity via metric entropy Integral representation Historical notes and extensions.

Self-similarity through high-variability: statistical analysis of Ethernet LAN traffic at the source level

A number of empirical studies of traffic measurements from a variety of working packet networks have demonstrated that actual network traffic is self-similar or long-range dependent in nature-in sharp contrast to commonly made traffic modeling assumptions. We provide a plausible physical explanation for the occurrence of self-similarity in local-area network (LAN) traffic. Our explanation is based on convergence results for processes that exhibit high variability and is supported by detailed statistical analyzes of real-time traffic measurements from Ethernet LANs at the level of individual sources. This paper is an extended version of Willinger et al. (1995). We develop here the mathematical results concerning the superposition of strictly alternating ON/OFF sources. Our key mathematical result states that the superposition of many ON/OFF sources (also known as packet-trains) with strictly alternating ON- and OFF-periods and whose ON-periods or OFF-periods exhibit the Noah effect produces aggregate network traffic that exhibits the Joseph effect. There is, moreover, a simple relation between the parameters describing the intensities of the Noah effect (high variability) and the Joseph effect (self-similarity). An extensive statistical analysis of high time-resolution Ethernet LAN traffic traces confirms that the data at the level of individual sources or source-destination pairs are consistent with the Noah effect. We also discuss implications of this simple physical explanation for the presence of self-similar traffic patterns in modern high-speed network traffic.

Long-range dependence in variable-bit-rate video traffic

We analyze 20 large sets of actual variable-bit-rate (VBR) video data, generated by a variety of different codecs and representing a wide range of different scenes. Performing extensive statistical and graphical tests, our main conclusion is that long-range dependence is an inherent feature of VBR video traffic, i.e., a feature that is independent of scene (e.g., video phone, video conference, motion picture video) and codec. In particular, we show that the long-range dependence property allows us to clearly distinguish between our measured data and traffic generated by VBR source models currently used in the literature. These findings give rise to novel and challenging problems in traffic engineering for high-speed networks and open up new areas of research in queueing and performance analysis involving long-range dependent traffic models. A small number of analytic queueing results already exist, and we discuss their implications for network design and network control strategies in the presence of long-range dependent traffic. >

ESTIMATORS FOR LONG-RANGE DEPENDENCE: AN EMPIRICAL STUDY

Various methods for estimating the self-similarity parameter and/or the intensity of long-range dependence in a time series are available. Some are more reliable than others. To discover the ones t...

Proof of a fundamental result in self-similar traffic modeling

We state and prove the following key mathematical result in self-similar traffic modeling: the superposition of many ON/OFF sources (also known as packet trains) with strictly alternating ON- and OFF-periods and whose ON-periods or OFF-periods exhibit the Noah Effect (i.e., have high variability or infinite variance) can produce aggregate network traffic that exhibits the Joseph Effect (i.e., is self-similar or long-range dependent). There is, moreover, a simple relation between the parameters describing the intensities of the Noah Effect (high variability) and the Joseph Effect (self-similarity). This provides a simple physical explanation for the presence of self-similar traffic patterns in modern high-speed network traffic that is consistent with traffic measurements at the source level. We illustrate how this mathematical result can be combined with modern high-performance computing capabilities to yield a simple and efficient linear-time algorithm for generating self-similar traffic traces.We also show how to obtain in the limit a Lévy stable motion, that is, a process with stationary and independent increments but with infinite variance marginals. While we have presently no empirical evidence that such a limit is consistent with measured network traffic, the result might prove relevant for some future networking scenarios.

Self-similarity through high-variability: statistical analysis of ethernet LAN traffic at the source level

A number of recent empirical studies of traffic measurements from a variety of working packet networks have convincingly demonstrated that actual network traffic is self-similar or long-range dependent in nature (i.e., bursty over a wide range of time scales) - in sharp contrast to commonly made traffic modeling assumptions. In this paper, we provide a plausible physical explanation for the occurrence of self-similarity in high-speed network traffic. Our explanation is based on convergence results for processes that exhibit high variability (i.e., infinite variance) and is supported by detailed statistical analyses of real-time traffic measurements from Ethernet LANs at the level of individual sources.Our key mathematical result states that the superposition of many ON/OFF sources (also known as packet trains) whose ON-periods and OFF-periods exhibit the Noah Effect (i.e., have high variability or infinite variance) produces aggregate network traffic that features the Joseph Effect (i.e., is self-similar or long-range dependent). There is, moreover, a simple relation between the parameters describing the intensities of the Noah Effect (high variability) and the Joseph Effect (self-similarity). An extensive statistical analysis of two sets of high time-resolution traffic measurements from two Ethernet LANs (involving a few hundred active source-destination pairs) confirms that the data at the level of individual sources or source-destination pairs are consistent with the Noah Effect. We also discuss implications of this simple physical explanation for the presence of self-similar traffic patterns in modern high-speed network traffic for (i) parsimonious traffic modeling (ii) efficient synthetic generation of realistic traffic patterns, and (iii) relevant network performance and protocol analysis.

Convergence of integrated processes of arbitrary Hermite rank

SummaryLet {X(s), −∞<s<∞} be a normalized stationary Gaussian process with a long-range correlation. The weak limit in C[0,1] of the integrated process

Stock Market Prices and Long-Range Dependence

Z_x \left( t \right) = \frac{1}{{d\left( x \right)}}\mathop \smallint \limits_0^{xt} G\left( {X\left( s \right)} \right)ds,{\text{ }}x \to \infty