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parsimonious manner for both the global or large-time scale as well as local or smalltime scale characteristics observed in measured WAN traf c. While the global scaling behavior is already part of Kurtz's original model (via the relationship between heavy-tailed sizes or durations of the individual sessions and the asymptotic self-similarity of the aggregate packet stream) and is captured by the Hurst parameter H, the original model does not incorporate local scaling behavior. However, we have seen earlier that by choosing an appropriate generator for the generic underlying conservative binomial cascade for the within-connection traf c rate process, we are able to obtain the same overall multifractal scaling as captured by the multifractal spectrum associated with the generic cascade model for the individual TCP connections. The practical relevance for such a structural workload model is that it allows for a more complete description of network traf c than exists to date in cases where higher-order stastistics or multiplicative aspects of the traf c play an important role but cannot be adequately accounted for by traditional, strictly second-order descriptions of network traf c. By aiming for a complete description of traf c, a more comprehensive analysis of network performance-related problem becomes feasible and desirable. In the past, thorough analytical studies of which aspects of network traf c are imortant for which aspects of network performance have often been prevented due to a lack of models that provide provably complete descriptions of the traf c processes under study. This situation can lead to misconceptions and misunderstandings of the relevance of certain aspects of traf c for certain aspects of performance (e.g., see Grossglauser and Bolot [15], Heyman and Lakshman [16], and Ryu and Elwalid [32].) In a rst attempt to allow for a more complete description of network traf c, Riedi at al. [31] (see also Ribeiro et al. [26]) emphasize performance aspects of description traf c models with additive and multiplicative structures. Working in the wavelet domain, they discuss [31] a multiplicative model based on binomial cascades, which exhibits the multifractal properties observed in measured network traf c at small scales and, in addition, matches the self-similar behavior of traf c over large time scales. Their model becomes approximately additive at large scales, as the variance of the cascade generator decreases with increasing scale, explaining why a purely multiplicative model can be consistent with an additive property in the limit of large scales. Riedi et al. [31] also provide initial evidence that models allowing for a more complete description of network traf c, in particular its multifractal behavior, typically outperform additive Gaussian models in the context of speci c performance problems [26]. 20.5 CONCLUSION
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One of the implications of the discovery of self-similar or multifractal scaling behavior in measured network traf c has been the realization that network traf c modeling and performance analysis can and should no longer be viewed as exercises in data tting and queueing theory or simulations. Instead, relevant traf c modeling
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NETWORK TRAFFIC DYNAMICS
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has become a natural by-product of a renewed effort that aims at gaining a physical (i.e., network-related) understanding of the empirically observed scaling phenomena. Moreover, the novel insights gained from such a physical-based understanding of actual network traf c dynamics often allows for a qualitative assessment of their potential impact on network performance, when more quantitative methods appear to be mathematically intractable or are not yet available. While traditional performance modeling has mainly lived in the con nes of mathematically tractable queueing models, the observed scaling properties of measured network traf c and the constantly changing nature of today's networks strongly suggest a shift away from focusing exclusively on quantitative methods for assessing the wide range of network performance-related problems toward achieving instead a more qualitative understanding of the implications of the dominant features of measured network traf c on relevant networking issues. While supporting such a qualitative knowledge where possible through a quantitative analysis is clearly desirable, we believe that the development of an ubiquitous, stable, robust, and high-performance networking infrastructure of the future will depend crucially on a qualitative rather than quantitative understanding of networks and network traf c dynamics. Finally, in terms of practical relevance, we also argue that by incorporating via multifractals local scaling characteristics of the traf c into a workload model, it may become in fact feasible to adequately describe traf c in a closed system (like the Internet) with an open model. The vast majority of currently used models for network traf c completely ignore the fact that the dynamic nature of packet traf c over a given link is the result of a combination of source=user behavior and highly nonlinear interactions between the individual users and the network. The search for a physical explanation of the observed multifractal nature of measured traf c at the packet level is intimately related to trying to sort out these complicated interactions and to abstract them to a level that is intuitively appealing, conforms to networking reality, and captures and explains in a mathematically rigorous manner empirically observed phenomena. Clearly, a prerequisite for succeeding in this endeavor is a close collaboration with networking experts who are familiar with the details of the various protocols and control mechanisms that operate at the different layers within the hierarchical structure of modern-day data networks and who are aware of the problems that are associated with the highly dynamic, constantly changing, and extremely heterogeneous nature of today's communication networks.
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