Scaling, Fractals and Wavelets in Visual Studio .NET

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Scaling, Fractals and Wavelets
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thus widespread, although it suffers from a subtle, but signi cant, problem of slow convergence which is under-appreciated [ROU 99] A variance which is in nite may appear unrealistic, and be seen as a weak device for generating long-range dependence, unrelated to empirical observations How can we claim to observe an in nite variance when in practice one can only measure and handle nite quantities The answer, like elsewhere in science, lies in the fact that a model does not claim absolute truth, but elegant utility If when measuring the distribution of values of a quantity, we observe that they follow relation (124) across a wide range of t, up to and including the largest available, an in nite variance model becomes entirely relevant as an idealization It was in this spirit that Cunha, Bestavros and Crovella declared in 1995 [CUN 95] that they observed heavy tails, in nite variances, in many characteristics of web documents, particularly their sizes They observed this same property in the sizes of UNIX les, thus revealing an orebody rich in power laws, capable of contributing to the existence of on/off type sources with in nite variance In [WIL 95], Willinger, Taqqu and Sherman went further in their analysis of local Ethernet traf c and also external Ethernet traf c, which consists of traf c offered to (and received from) the Internet Not only did they observe evidence of on/off behavior in individual ows, de ned as traf c owing between unique emission and reception address pairs, but in most cases the estimated values of were indeed well within the in nite variance range 1233 Chemistry In general, the addition of independent processes induces the addition of their temporal structures More precisely, if the independent processes Xi (t) have Xi ( ) as their covariance functions, then the covariance function of the process X = Xi is just X ( ) = Xi ( ) It follows that the presence of long-range dependence in at least one of the Xi induces a superposition, with an exponent equal to the minimum of those of the long-range dependent components This is reminiscent of rules governing the fractal dimension of a union of fractal sets [FAL 90] If, for example, the Xi (t) are independent identically distributed (iid) copies of a LRD process of parameters (c , ), the parameters of the superposition are simply (c m, ) The persistence of long-range dependence applies in particular to a nite superposition of N identically distributed and independent on/off sources As for in nite superpositions, models as interesting as they are signi cant emerge depending on the precise way in which the normalization is carried out Let us initially examine a normalization relating to the instanteous rate h (during the on states), leaving the structure of individual sources untouched (notably and ) By increasing N , we can show [KUR 96, TAQ 97] that there is convergence both in the distributional and weak senses to a Gaussian process It is not surprising that, to obtain this result, it is necessary to impose a normalization proportional to N after having rst subtracted the mean This limiting process has long memory and, by aggregating we
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Scale Invariance in Computer Network Traf c
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can obtain fractional Gaussian noise in a second limit operation, this time operating on time This bond between on/off sources and the canonical scale invariant process will be explored further in the next section Here we emphasize that the order of the limit operations, rst in rate and then over time, is of central importance If we try reversing these we obtain a completely different result, the L vy ight, a stable process with stationary and independent increments which does not, for the moment, correspond to a model applicable to traf c The interpretation of fractional Gaussian noise to be a combination of on/off sources reveals it to be an example of a process where the source of scale invariance lies in the linear primitive components themselves, to which a linear superposition does not add anything essential On the other hand, there is another normalization which provides an example of where scale invariance is emergent In this case, we leave the peak rate h constant, and lengthen the silent periods with N so as to maintain the total arrival rate constant More precisely, we set = /N , with xed , and obtain in the limit N = h / and an arrival process of bursts that obeys a Poisson process of parameter In this limit, each source contributes only a single active period because silences before and after extend to in nity The sole burst that remains to every source can be interpreted as the transfer of a single le at constant rate The number of simultaneously active sources is described by a Poisson law of parameter / For nite N , as well as in nite limit N , the rate process has long memory In contrast, for the limit process, long memory is no longer ascribable to individual sources but to heavy tailed distribution of the size of the transfers which remain individually constant, without any scale invariance of their own An aggregate of such sources can be regarded as a random and independent model of le transfers across a network The question of non-linear mixtures is, not unexpectedly, much more complicated, little studied, and beyond the scope of this chapter In its broad sense, it implies that sources can in uence each other, in other words that there exists feedback between the superposition and its components We will brie y return to this later 1234 Mechanisms A coherent way of explaining scale invariance has already emerged: long-range dependence is generated by the heavy tailed property, is preserved by superposition, and is well idealized by a canonical Gaussian process Are there network mechanisms capable of carrying out these linear mathematical operations The answer is yes From the discrete nature of packets, ows can inter-penetrate in switches andmultiplexers (traf c concentrators), thereby effectively adding, in an approximate sense, their instantaneous rates Moreover, since the packets remain identi able in the mixture, this quasi-additivity also acts in the inverse direction, in the demultiplexers where ows leave a large link to move away from the core of network, or in switches where ows are extracted from one superposition to be integrated into others The
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