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2 The factors 2n appearing in f and  q are for convenience. The sign of  q is chosen such as to render Eqs. (20.20) and (20.19) symmetrical. The signs of q in (20.13) and (20.20) are opposite to each other.
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20.3 THE SMALL-TIME SCALING BEHAVIOR OF NETWORK TRAFFIC
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For the example considered earlier where we concatenated a number of FGNs, we nd  q mink qHk 1 , which is again consistent with t taking the values Hk on sets of dimension 1 (compare Eq. (20.33), see also Levy Vehel and Riedi [36] for more details). This example shows also how noncavity in  q can result in loss of information:  q and its Legendre transform re ect only the minimal and the maximal of the Hk . In contrast, truly concave behavior of  q indicates that there is a whole interval of -values present in the signal and not just a few (hence the term multifractal). 20.3.2 Multiplicatively Generated Multifractals or Cascades
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A construction that fragments a given set into smaller and smaller pieces according to some geometric rule and, at the same time, divides the measure of these pieces according to some other (deterministic or random) rule is called a multiplicative process or cascade (e.g., see Evertsz and Mandelbrot [9]). The limiting object generated by such a multiplicative process de nes, in general, a singular measure or multifractal and describes the highly irregular way the mass of the initial set gets redistributed during this simple fragmentation procedure. The generator of the cascade speci es the mass fragmentation rule, and we consider in the following the class of conservative cascades, introduced by Mandelrot [23] characterized by a generator that preserves the total mass of the initial set at every stage of the construction (i.e., mass conservation). To illustrate, we will construct a binomial conservative cascade or measure  on the interval I : 0; 1 . More precisely, we will construct its distribution function Y t  0; t and since the underlying generator will be random, Y will de ne a stochastic process. By construction it will have positive increments and Y 0 0 almost surely. This iterative construction starts with a uniform distribution on the unit interval of total mass M 0 and then ``redistributes'' this mass by splitting it among the two 1 1 1 1 subintervals of half-size in the ratio M0 to M1 , where M0 M1 1. Proceeding iteratively one obtains after n steps a distribution that is uniform on intervals n n Ikn : kn 2 n , kn 1 2 n . The mass lying in Ikn is redistributed among its two n 1 n 1 n 1 n 1 dyadic subintervals I2kn and I2kn 1 in the proportions M2kn and M2kn 1, where n 1 n 1 M2kn M2kn 1 1 almost surely. To summarize, for any n let us choose a sequence k1 ; k2 ; . . . ; kn such that the i interval Ikll lies in Iki whenever i < l. In other words, the ki are the n rst binary digits n of any point t P Ikn . We call this a nested sequence, and it is uniquely de ned by the value of kn . Then we have
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The various Mli , which collectively de ne the generator of the conservative cascade, may have distributions that depend on i and l and that are arbitrary, as long as they are positive and provided that for all i and all m,
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