Scaling, Fractals and Wavelets

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Logscale Diagram, N=2 [ (j1,j2)= (3,15) Q= 0011384 ] Logscale Diagram, N=2 [ (j1,j2)= (3,15) Q= 0011384 ]

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Figure 122 Wavelet analysis of scale invariance The Logscale diagram, an estimated wavelet spectrum (in log-log coordinates), is shown for the data Left: Ethernet pAug The slope gives an estimate of 1 with good properties, which is reliable from j = 3 We obtain 1 = 059 001 Right: the number of new TCP connections in 10 ms bins on an Internet link We see two scale ranges, from j = 1 to 8 and j = 8 to 19 Daubechies 2 wavelets were used

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right side in Figure 122) For more details on this method, its use and robustness, see [ABR 00, VEI 99] In the left plot of Figure 122, the wavelet method is applied to pAug We see a general alignment across all scales in this time series of length n = 218 which, starting from j = 3, justi es the estimate of the slope (a weighted regression is used which gives more weight to small scales where there is more data, as indicated by the con dence intervals displayed) Thus, the self-similarity visible in Figure 121 can be objectively revealed and qualitatively measured 1223 Beyond the revolution Despite considerable resistance, the concept of the fractality spread quickly In fact, after a short span of time, as frequently as resistance was found, the opposite attitude was also encountered, that is, the idea that we had only to measure the value to capture the essence of fractal traf c In fact, a model based on relation (123), comprising only three parameters, X , X and , cannot claim, except in particular cases, to describe the essence of an object as rich as traf c Even if we are ready to accept a Gaussian hypothesis, which eliminates the need for dealing with statistics of orders higher than two, models that allow more exibility are required For example, it is necessary to think about the constant c of equation (122), which gives the size of the law of which expresses the nature There is no reason, a priori, to assume that its value is equal to that of standard fractional Gaussian noise, where it takes

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Scale Invariance in Computer Network Traf c

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the parametric form c = 1 (1 )(2 ) The same comments are valid for short 2 range correlations, which are very important for sources such as video In order for a traf c which presents long-range dependence but which also notably deviates from fractional Gaussian noise to be well-modeled by the latter, it is necessary to have a high value of m; however, the time scales thereby neglected can have signi cant impacts on performance In 1997, L vy V hel and Riedi [LEV 97] observed that in TCP data, not only can the behavior over small scales be far from that of a fractional Gaussian noise, but we can even nd another form of scale invariance, that of multifractality These observations were con rmed on other TCP data by Feldman et al [FEL 98] In Figure 122, the plot on the right side shows a second order analysis of a time series which corresponds to the number of new TCP connections arriving in bins of width 10 ms In addition to the slope, on the right, corresponding to long-range dependence, we observe a second slope at small scales, a second zone in which an invariance lies However, multifractality means much more than a simple fact that there are two different regimes, each with its own invariance To understand this difference, let us imagine that we de ned, not a second order logscale diagram, on the basis of quantities |dX (j, k)|2 , but, in a similar manner, a logscale diagram at q th order based on |dX (j, k)|q , from which we estimated, over the same range of scales, an exponent q In the case of fractional Gaussian noise, though each of the { q } (here positive real q are taken) are different, they are connected in a simple, linear way; essentially there is only one true exponent On the other hand, in the multifractal case the { q } enjoy considerable freedom The invariance is fundamentally different for each different moment, and therefore it is necessary to know the entire spectrum of exponents, instead of just one, to fully understand the nature of the invariance present A detailed treatment of multifractals is beyond the scope of this chapter (see [RIE 95, RIE 99], 1, 3 and 4 for a more in-depth analysis) but it is nevertheless relevant to describe a simple example: a deterministic multiplicative cascade We begin with a unit mass, uniformly spread out over [0, 1] We then distribute a fraction p (0, 1) of the mass on the rst half [0, 1 ] and the 2 remainder on ( 1 , 1] The total mass is thus preserved and we say that the cascade 2 is conservative Now, let us repeat this procedure to obtain masses {p2 , p(1 p), p(1 p), (1 p)2 } on the quarters of the interval, then {p3 , p2 (1 p), p2 (1 p), p(1 p)2 , p2 (1 p), p(1 p)2 , p(1 p)2 , (1 p)3 } on the eighths, and so on This procedure, repeated inde nitely, creates a mass distribution, that is a measure, which is singular: it is multifractal Less rigid and random variants can be easily de ned and taken to be traf c models by normalizing [0, 1] to the duration [0, t], and by regulating the number of iterations which controls the time resolution The singularity and positivity of these measures are apt to describe the astonishing variability observed in small scale traf c, where a Gaussian hypothesis is far from reasonable Thus, multifractal modeling offers the hope of venturing into the dif cult

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