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 to the constant changes that real-life data networks experience over time. Such robust characteristics are sometimes called "traffic invariants", and include such phenomena as long-range dependence and/or heavy tails. Long-range dependence occurs when the correlations of the traffic processes at hand decay to zero slowly as the lag increases (i.e., power-law decay) and causes traffic to exhibit pronounced larger-than average ``bursts'' and lower-than-average ``lulls''. Heavy tails refer to the power-law decay of the underlying probability distributions and capture the notions of extreme variability and ``intermittancy''. In the networking context, long-range dependence and 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, what is their effect on queuing/loss performance, 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 exhibits heavy tails. This research aims to develop physical models that can explain the joint presence of long-range dependence and heavy tails at the macroscopic 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. Another problem area concerns the nature of network traffic at the microscopic level, where we will build upon the recent discovery of the presence of multiplicative mechanisms that cause network traffic over fine time scales to exhibit multifractal scaling properties. This research also focuses on the Web and plans to track how self-similarity, heavy-tails, and multifractals fare in a constantly changing Internet. How does the ever increasing variability in access speeds (e.g., traditional modems on one hand, cable modems, 100 Mbps Ethernet on the other) and the more and more heterogeneous nature of access technologies (e.g., phone modems, cable modems, ADSL) will affect the nature of currently considered workload models and how they will impact the scaling (i.e., self-similarity) properties of aggregate packet traffic. This project involves the collaborative effort of the P.I. (Murad S. Taqqu) at Boston University and the Co-P.I. (Walter Willinger) at AT&T-Labs Research. For more details please refer to the web site AREF="http://math.bu.edu/people/murad"> http://math.bu.edu/people/murad.
