20.3 THE SMALL-TIME SCALING BEHAVIOR OF NETWORK TRAFFIC

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The aim of multifractal analysis (MFA) is to provide information about these singularity exponents in a given signal and to come up with a compact description of the overall singularity structure of signals in geometrical or in statistical terms. Before describing in more detail some of the commonly used MFA methods, we note that since wavelet decompositions contain information about the degree of local irregularity of a signal, it should come as no surprise that the singularity exponent t is related to the decay of wavelet coef cients wj;k Y s j;k s ds around the point t, where is a bandpass wavelet function and where j;k s : 2 j=2 2 j s k (e.g., in the case of the well-known Haar wavelet, s equals 1 for 0 s 1; 1 for 1 s 2, and 0 for all other s; for a general overview of wavelets, we refer to Daubechies [5]). Indeed, assuming only that s ds 0 one can show as in Jaffard [18] that 2n=2 w n;kn C 2 n t ; as kn 2 n 3 t: 20:9

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Moreover, it is known that under some regularity conditions (for a precise statement see Jaffard [18] or Daubechies [5, Theorem 9.2]), relation (20.9) characterizes the ~ degree of local irregularity of the signal at the point t. This suggests to de ne t as ~ in Eq. (20.8) but with n t replaced by n t , where ~ ~k n t : nn : 1 log 2n=2 jw n;kn j : n log 2 20:10

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In general, this may give a different but nevertheless useful description of the singularity structure of Y , particularly for nonmonotonous processes (for an example, see Gilbert et al. [13]). Using wavelets may also have numerical ~ advantages. The remainder of this section remains true if t is replaced by t and Eq. (20.8) by (20.10), that is increments by normalized wavelet coef cients. Conceptually, the geometrical formulation of MFA in the time domain is the most obvious one. Its objective is to quantify what values of the limiting scaling exponent t appear in a signal and how often one will encounter the different values. In other words, the focus here is on the ``size'' of the sets of the form K ft: t g: 20:11

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To illustrate, since for FGN there exists only one scaling exponent (i.e., t H , the set K is either the whole line (if H) or empty, and FGN is therefore said to be ``monofractal.'' Similarly, for the concatenation of several FGNs with Hurst parameters H i in the interval I i i; i 1 , we have KH i I i . In general, however, the sets K are highly interwoven and each of them lies dense on the line. Consequently, the right notion of ``size'' is that of the fractal Hausdorff dimension dim K , which is, unfortunately, impossible to estimate in practice and severely limits the usefulness of this geometrical approach to MFA. Therefore, we will focus below on different statistical descriptions of the multifractal structure of a given signal.

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NETWORK TRAFFIC DYNAMICS

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One such description involves the notion of the coarse Holder exponents (20.8). To illustrate, x a path of Y and consider a histogram of the n k 0; . . . 2n 1 k taken at some nite level n. It will show a nontrivial distribution of values but is bound to concentrate more and more around the expected value as a result of the law of large numbers (LLN): values other than the expected value must occur less and less often. To quantify the frequency with which values other than the mean value occur, we make extensive use of the theory of large deviations. Generalizing the Chernoff Cramer bound, the large deviation principle (LDP) states that probabilities of rare events (e.g., the occurrence of values that deviate from the mean) decay exponentially fast. To make this more precise consider a sequence of independent, identically distributed (i.i.d.) random variables W , W1 , W2 ; . . . and set Vn : W1 Wn . Using Chebyshev's inequality and the independence, we nd, for any q > 0, P 1=n Vn ! a P 2qVn ! 2nqa E2qVn E 2qW 2 qa n : 2nqa 20:12

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Since q > 0 is arbitrary, we can replace the right-hand side in Eq. (20.12) by its in mum over q > 0. A symmetry argument shows that P b ! 1=n Vn E 2qW 2 qb n , for all q < 0. Combining all this yields the following two upper bounds: 1 log P b ! 1=n Vn ! a n 2 inf q>0 flog2 E 2qW qag; inf q<0 flog2 E 2qW qbg: 20:13

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For a discussion of this simple result, let L q E 2q W a . Since log is a monotone function, nding the in mum of L is the same as nding the in mum of log L . We note rst that LHH q > 0, for all q P R, hence L is a strictly convex function and must have a unique in mum for q P R. From L 0 1 we conclude that this in mum must be less than or equal to 1. Focusing now on q > 0, we infer from LH 0 log 2 CE W a that inf q>0 L q is assumed in q 0 and equals 1 if and only if E W ! a. On the other hand, inf q>0 L q < 1 if E W < a. An analogous result holds for the second bound. In summary, if b > EW > a then the bounds on the right-hand side (RHS) in Eq. (20.13) are both zero and thus re ect the LLN, which says that 1=n Vn 3 E W almost surely. On the other hand, if E W is not contained in a; b and when P b ! 1=n Vn ! a is the probability of 1=n Vn deviating far from its expected value, then exactly one of the bounds will be negative, proving (at least) exponential decay of this probability. LDP theorems extend this result to a more general class of random sequences Vn and establish conditions under which the bound in Eq. (20.13) is attained in the limit n 3 I [6, 7]. To apply the LDP approach to our situation, we x a realization of Y and consider the location t, encoded by kn via t P kn 2 n , kn 1 2 n , as the only randomness relevant for the LDP. Since kn can take only 2n different values, which we will

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