There has been recently a big increase in the number of studies related to the statistical analysis and mathematical modeling of traffic measurements from modern communication networks. While many of the measured traffic traces differ from each other in their high-frequency behavior, they typically exhibit a number of statistical characteristics that tend to be insensitive with respect to the constant changes that real-life data networks experience over time. Such robust characteristics are sometimes referred to as "traffic invariants", and include such phenomena as long-range dependence and/or heavy tails. Long-range dependence occurs when the covariances decay slowly to zero slowly as the lag increases. Heavy tails refer to the power decrease of the marginal distributions. In the networking context, heavy tails abound and have been observed at practically all layers in the networking hierarchy. This research focuses on these traffic invariants: how to detect and measure them, how to explain their presence in realistic networking situations, and also how to identify other potential invariants, especially within the apparent chaotic structure in the high-frequency domain.

One of the open problems in understanding the dynamic nature of network traffic is when the underlying traffic (e.g., packet rate process), in addition to exhibiting strong temporal dependencies, is itself non-Gaussian and heavy-tailed. This research aims to develop physical models that can explain the joint presence of long-range dependence and heavy tails at the microscopic level. These models should be useful in practice, give rise to efficient traffic generation methods that result in synthetic traces with realistic features, and provide novel insights into the wide area of network performance analysis.

This project involves the collaborative effort of the P.I. (Murad S. Taqqu) at Boston University and the Co-P.I. (Walter Willinger) at ATT Research.

The focus of this research is on a special class of stochastic processes with the following three properties: stationary increments, self-similar and stable non-Gaussian probability laws (stable sssi processes, in short). Unlike the Gaussian case, there are infinitely many different stable sssi processes. This overwhelming variety may be regarded as a fundamental problem. One now has to understand how these processes are different or what it is that they have in common. However, non-Gaussianity also brings to the picture new tools that were unavailable in the Gaussian case. It has been known for quite some time now that non-Gaussian stable processes having some invariance property, like self-similarity or stationarity of the increments, can be associated with nonsingular flows. It is then based on some properties of these flows that one can describe the structure of the corresponding stable processes. The focus will be at first on an important subclass of stable sssi processes called self-similar mixed moving averages. The connection of self-similar mixed moving averages to nonsingular flows allows one to decompose them into separate, independent processes and then explore each part in the decomposition separately.

The purpose of this research is to better understand a class of random processes that have characteristics that one encounters in many areas of applications. Examples of such processes include the limit of the so-called renewal reward processes applied in telecommunications and ``random wavelet expansions'' introduced in probabilistic modeling of images. These processes are fractal-like. They display scale invariance and tend to take often extreme values that deviate greatly from the mean. Their mathematical structure is complex. The goal of this research is to develop tools that can be used to analyze that structure.