The Scaling Phenomena

Scaling behaviour is observed in many elds and its ubiquity has been amply witnessed in telecommunications traac data. Whereas the notion of scaling can be attached to diierent features such as self-similarity, long-range dependence or (multi-)fractality, it leads in all cases to challenging problems in terms of analysis, estimation and synthesis. These issues are addressed here by using the naturally suited multiresolution approach, based on the wavelet transform, which allows the elaboration of powerful and versatile tools within a general and uniied framework. The proposed methods are discussed in detail with respect to their rationale, statistical performance, computational eeciency, robustness, relation with other existing approaches, and use in practice. Theoretical claims are supported by both numerical simulations and selected examples of traac data. The presence of scaling behaviour in telecommunications traac is striking not only in its ubiquity, appearing in almost every kind of packet data, but also in the wide range of scales over which the scaling holds (see e.g. 32], 14], 54]). It is rare indeed that a physical phenomenon obeys a consistent law over so many orders of magnitude. This may well extend further, as increases in network bandwidth over time progressively`reveal' higher scales. While the presence of scaling is now well established, its impact on teletraf-c issues and network performance is still the subject of some confusion and uncertainty. Why is scaling in traac important for networking? It is clear, as far as modelling of the traac itself is concerned, that a feature as prominent as scaling should be built into models at a fundamental level, if these are to be both accurate and parsimonious. Scaling, therefore, has immediate implications for the choice of classes of traac models, and consequently on the choice, and subsequent estimation, of model parameters. Such estimation is required for initial model veriication, for tting purposes, as well as for traac monitoring. Traac modelling however, does not occur in isolation but in the context of performance issues. Depending on the performance metric of interest, and the model of the network element in question, the impact and therefore the relevance of scaling behaviour will vary. As a simple example, it is known that in certain innnite buuer uid queues fed by long range dependent (LRD) On/OO sources, that the stationary queueing distribution has innnite mean, a radically non-classical result. Such innnite moments disappear however if the buuer is nite, intuitively because a nite …