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Book Reviews: A Summary of Multifractal Traffic Modeling

Nelson Liu, Institute for Systems Research

One of the most important discoveries about Internet traffic is the scaling phenomenon. Different from the traditional traffic pattern, primarily the Poisson model, this type of traffic shows a slowly decaying dependence structure and can be approximately characterized by a self-similar process. Multiscale behavior is a more complex version of self-similarity, indicating that the scaling property of the traffic process isn't uniform across multiple time scales and orders of statistics. It's often modeled with the multifractal process. Since the mid-'80s, researchers have conducted numerous studies on self-similar and multiscale traffic and have made tremendous progress. Scaling phenomena affect network performance significantly and have important implications for many aspects of network design and engineering, such as traffic engineering, network planning and management, and quality-of-service control. Multifractal Based Network Traffic Modeling provides an accessible but somewhat narrow summary of the research on self-similar and multiscale traffic.

Murali Krishna P, Vikram M. Gadre, and Uday B. Desai give a concise account of the background and basic mathematical tools for scaling traffic modeling. This includes a brief historical review of teletraffic modeling followed by definitions of several types of random processes, such as counting, self-similar, and heavy-tailed processes. Notably, the authors describe in detail the fractional Brownian motion as a representative self-similar process, including its correlation analysis and wavelet-based analysis. They present analysis techniques for self-similar processes, such as the R/S method, logscale variance plot, and Hurst parameter estimation. They emphasize wavelet-based analysis, summarizing the moment analysis of wavelet coefficients and the Abry-Veitch estimator for the Hurst parameter derived from it. This section's strengths include a subsection titled "The Need for Multifractal Processes" and an explanation of the difference between self-similar and multifractal processes following the multifractal formulism. The authors provide tutorials about self-similar processes and multifractality based on recently published literature.

They go on to review some well-known self-similar and multiscale traffic models, such as the on-off model, the multifractal-wavelet model, the multiplicative cascade, and self-similar models for MPEG video traffic. They also extend the Poisson model by adding a parameter to characterize the correlation between adjacent values explicitly. However, the authors seem to have ignored the significant advancements that have been made in formal queuing analysis or the physical interpretation of multifractal traffic. Without this information, important questions such as why scaling behaviors matter and how to control them are largely left unanswered.

The rest of the book—more than half of it—discusses the authors' proposed VVGM (Variable Varianse Gaussian Multiplicative) model. In chapters 5-8, they describe this model and use it to analyze traffic aggregation, video traffic, queuing performance, and burstiness control. As the description shows, the VVGM method is essentially derived from Anja Feldman, Anna Gilbert, and Walter Willinger's multifractal-cascade model (reference 39 in the book). It would be interesting to see a comparison between the two models.


Nevertheless, beginners might find useful information in Multifractal Based Network Traffic Modeling because the authors apply traffic models to real data. They present most of their results by comparing metrics for real traffic traces and the traces their model generated. They provide some interesting perspectives. For example, they define an entropy metric and relate it with the ergodic property of aggregate traffic. Also, they use the effective-bandwidth theory to analyze their model's queuing performance, which partially remedies the lack of information on queuing analysis in the first part of the book. The book would also be useful for graduate students preparing for research in telecommunications and network areas and for network engineers who are interested in knowing some basic concepts and techniques involved in recent traffic modeling.

Related Links

  • "An Empirical Study of the Multiscale Predictability of Network Traffic",
  • "A Multiplicative Multifractal Model for TCP Traffic",
  • "Wavelet Multifractal Modeling for Network Traffic and Queueing Analysis" ,

Cite this article: Nelson Liu, "A Summary of Multifractal Traffic Modeling," review of Multifractal Based Network Traffic Modeling by Murali Krishna P, Vikram M. Gadre, and Uday B. Desai, IEEE Distributed Systems Online, vol. 7, no. 3, 2006, art. no. 0603-03008.

About the Authors

Nelson Liu is a research assistant professor at the Institute for Systems Research. Contact him at
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