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Proof Both Eqs (434) and (435) are obvious from the de nition of blw (n) and tlw (n) For Eq (434), due to precedence constraints, all descendant nodes of n must be executed after n (Condition 42) Their execution in precedence order takes at least blw (n) time, including the execution time of n For Eq (435), due to precedence constraints, all ancestors of n must be executed before n (Condition 42) Their execution in precedence order takes at least tlw (n) time Note that Lemma 46 relies on the computation levels, to re ect the best possible case A lemma based on the top and bottom levels needs the additional restriction that all communications among the ancestor and descendent nodes are remote Lemma 46, however, is always true The top path ptl(n) and the bottom path pbl(n) of a node n G together form a path ptb(n) = ptl(n) pbl(n) , (436)



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Fig. 14.2 TCP run. Throughput as a function of le size distribution and three aggregation levels. File size distributions constitute Pareto with a 1:05, 1.35, 1.95, and exponential.





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weak dependency structure in the underlying time series. As a approaches 1, however, burstiness is preserved even at large time scales, indicating that the 10 ms time series possesses long-range dependence. The last column depicts time series obtained by employing an exponential le size distribution at the application layer with the mean normalized so as to equal that of the Pareto distributions. We observe that the aggregated time series between exponential and Pareto with a 1:95 are qualitatively indistinguishable. A quantitative measure of self-similarity is obtained by using the Hurst parameter H, which expresses the speed of decay of a time series' autocorrelation function. A time series with long-range dependence has an autocorrelation function of the form r k $ k b as k 3 I;

We have discussed a number of ideas for improving the performance of our code. This section brings several of these together in an example. The following code is a slightly extreme example of how not to do it. The code goes through a company database extracting information about each employee:

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where 0 < b < 1 The Hurst parameter is related to b via b H 1 : 2 Hence, for long-range dependent time series, 1 < H < 1 As H 3 1, the degree of 2 long-range dependence increases A test for long-range dependence in a time series can be reduced to the question of determining whether H is sign cantly different from 1 2 In this chapter, we use two methods for testing self-similarity2 These methods are described more fully in Beran [5] and Taqqu et al [23], and are the same methods used in Leland et al [12] The rst method, the variance time plot, is based on the slowly decaying variance of a self-similar time series.

which is a longest path of G through n, starting in a source node and ending in a sink node This path is called a level path of n and its length is len( ptb(n) ) = tl(n) + bl(n) (437)

The second method, the R=S plot, uses the fact that for a self-similar data set, the rescaled range or R=S statistic grows according to a power law with exponent H as a function of the number of points included Thus the plot of R=S against this number on a log log scale has a slope that is an estimate of H Figure 143 shows H-estimates based on variance time and R=S methods for three different network con gurations Each plot shows H as a function of the Pareto distribution parameter for a 1:05, 115, 125, 135, 165, and 195 Figure 143(a) shows the results for the baseline TCP Reno case in which network bandwidth and buffer capacity are both limited (15 Mb=s and 6 kB), resulting in an 4% packet drop rate for the most bursty case a 1:05 .

The plot shows that the Hurst parameter estimates vary with le size distribution in a roughly linear manner The H 3 a =2 line shows the values of H that would be predicted by the on=off model in an idealized case corresponding to a fractional Gaussian noise process Although their overall trends are similar (nearly coinciding at a 1:65), the slope of the simulated system with resource limitations and reliable transport layer running TCP Reno's congestion control is consistently less than 1, with an offset below the.

2 A third method based on the periodgram was also used. However, this method is believed to be sensitive to low-frequency components in the series, which led in our case to a wide spread in the estimates; it is omittted here.

Equation (437) holds for every node ni ptb(n) , ni V For the nodes of a critical path, this lemma follows Lemma 47 (Critical Path Length and Node Levels) Let G = (V, E, w, c) be a task graph For any node ncp,i of a critical path cp len(cp) = tl(ncp,i ) + bl(ncp,i ) (438)

Fig. 14.3 Hurst parameter estimates (TCP run): R=S and variance time for a 1:05, 1.35, 1.65 and 1.95. (a) Base run, (b) large bandwidth=large buffer, and (c) large buffer.

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