Scaling, Fractals and Wavelets

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same logic is also valid for various methods of multiplexing relevant to circuit switching A second question lies in the possibility that multiplicative, rather than additive, mechanisms exist in the network, potentially allowing the realization of one mathematical path (following cascades) for the generation of multifractal properties It was suggested that the hierarchy of protocols can ful ll this role [FEL 98] by recursive subdivision of source data However, the true cause (or causes) of multifractality observed remains to be determined If, in low load, multiplexing, switching and demultiplexing operations are well understood in terms of linear operations, at high load non-linearities, mainly due to buffers, electronic queues, are inevitable From strong smoothing we expect elimination of scale invariance over a certain scale range, however at a large scale the in uence of heavy tails, a property of great robustness, will persist However, the non-linear mechanisms potentially involved are richer than a simple truncation of what is otherwise a simple linear superposition If a control mechanism regulates a ow resulting from a given source, such as for example in TCP connections, there is a coupling between the source and the network, a feedback, which modi es the transmission depending on the state of the network, controlled by example by the level of loss detected Thus, network queues generate an indirect coupling between different sources, producing a highly non-linear dynamics capable of very signi cantly modifying the nature of traf c This effect is stronger as the proportion of sources thus regulated is large Such dynamics, and its potential capacity to generate scale invariance such as self-similarity and multifractality, has begun to generate considerable excitement in the networking research community Finally, it is interesting to note a return to dynamic system approaches, which were considered by Erramilli and Singh early in the history of fractal traf c [ERR 90, ERR 95] but which did not evolve thereafter 124 New models, new behaviors 1241 Character of a model By a good model we understand, rst of all, that the statistics of the data are well captured by the random structure of the model It is imperative to insist on the principle of parsimony, in other words, that only the minimal number of parameters necessary to cover the essential degrees of freedom be used An excess of parameters is the sign of a model which is over- tted to a speci c dataset, which does not therefore capture any generality, or hold structural validity In this case, the majority of the parameters lack physical signi cance, and as a result their estimation is likely to be dif cult and arbitrary Finally, to measure the degree of adequacy of a model, good metrics should be chosen In the context of telecommunications, these will not only refer to the statistics of a ow, but to the system as a whole Thus, the model

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Scale Invariance in Computer Network Traf c

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should be judged by its capacity to predict some measure of quality of service, of which there are a number to chose from Among the metrics which are precisely de ned, and yet reasonably close to the concerns of users, we count loss rate and average packet delay, whereas an example which is more focused on engineering questions of network dimensioning is the distribution function of queue contents, which is the marginal of the waiting process However, we also work under the constraint of considering problems for which we can hope to nd solutions Often, we impose simple idealizations, for example queues with in nite waiting rooms We then commonly use the fact that the probability Q(x), that the level of an in nite queue exceeds x, bounds from above the probability of a loss in a corresponding system where the queue is of nite size x From the rst studies on the impact of fractal traf c we have seen that the behavior of systems can deviate notably from traditional intuition In 1993, Veitch [VEI 93] emphasized this fact by presenting a simple system in which a fractal renewal process, with an average incoming rate of zero, could produce a dynamic non-trivial queue In this section, we consider three model classes representing the state of the art and corresponding performance studies, essentially the form of Q(x) for large x Each class considered itself exhibits untraditional behavior, though of very different types Each of the models allows an interpretation in terms of a linear superposition of on/off sources, though they were not necessarily proposed in that light, and in each case other motivations are possible 1242 The fractional Brownian motion family Often, instead of studying traf c via its rate X(k), we turn to the series k Y (k) = i=1 X(i), measuring the mass of data accumulated over the interval (0, k] Passing over to continuous time, if X(t) is stationary, Y (t) has stationary increments, that is the distributions of the increments {Y (t + ) Y (t), t } do not depend on t We can decompose this process as Y (t) = Y t + Y W (t) (126)

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where W (t) has zero-mean If the rate process exhibits long-range dependence, the natural choice to model W (t) is the fractional Brownian motion (FBM) BH (t), t , 0 < H < 1 This canonical process is the unique self-similar Gaussian process with stationary increments Thus, it has a perfect scale invariance simultaneously in all its statistics across all scales, for example its variance obeys Var[BH (t)] = |t|2H If we differentiate fractional Brownian motion with H < 1 and = 1, we obtain fractional Gaussian noise with = 2(1 H), which has long memory if H > 1 2 In 1994, Norros [NOR 94] examined a system called fractional Brownian storage consisting of an in nite reservoir with a constant drainage rate of C, fed by Y (t) This type of system is known as a uid queue, as the data ows into the reservoir which

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