PREDICTABILITY OF SELF-SIMILAR TRAFFIC in Visual Studio .NET

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18.3 PREDICTABILITY OF SELF-SIMILAR TRAFFIC
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For the aggregate throughput traces with a 1:05 Figure 18.2 (top row) the three-dimensional (3D) conditional probability densities can be seen to be skewed diagonally from the lower left side toward the upper right side. This indicates that if the current traf c level L1 is low, say, L1 1, chances are that L2 will be low as well. That is, the probability mass of PrfL2 jL1 1g is concentrated toward 1. Conversely, the plots show that PrfL2 jL1 8g is concentrated toward 8. This is more clearly seen in Fig. 18.3(a), which shows two cross sections, that is, 2D projections, re ecting PrfL2 jL1 1g and PrfL2 jL1 8g. For the aggregate throughput traces with a 1:95 (Fig. 18.2 (bottom-row)), on the other hand, the shape of the distribution does not change as the conditioning variable L1 is varied. This is more clearly seen in the projections of PrfL2 jL1 1g and PrfL2 jL1 8g shown in Fig. 18.3(b). This indicates that for a 1:95 traf c observing the past (over the time scales considered) does not help much in predicting the future beyond the information conveyed by the xed a priori distribution. Given the de nition of Lk , the Gaussian shape of the marginal densities is consistent with short-range correlations, making the central limit theorem approximately applicable over larger time scales. In both cases a 1:05, 1.95), the shape of the distribution stays relatively constant across a wide range of time scales 500 ms to 20 s. For a 1:35, 1.65 the predictability structure lies ``in-between'' (not shown here).
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Predictability and Time Scale
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An important issue is how time scale affects predictability when traf c is long-range dependent. Going back to Fig. 18.2 (top row), one subtle effect that is not easily discernible is that as time scale is increased the conditional probability densities PrfL2 jL1 lg become more concentrated. Given that PrfL2 jL1 lg is a function of T1 , T2 , we would like to determine at what time scale predictability is maximized. One way to measure the ``information content'' that is, in the sense of randomness or unstructuredness in a probability distribution is to compute its
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Fig. 18.3 (a) Shifting effect of conditional probability densities P L2 jL1 1 and P L2 jL1 8 for a 1:05. (b) For a 1:95, the corresponding probabilities remain invariant.
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entropy. P a discrete probability density pi, its entropy S pi is de ned as For S pi i pi log 1=pi . In the case of our conditional density PrfL2 jL1 lg, Sl
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PrfL2 l H jL1 lg log PrfL2 l H jL1 lg:
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Thus, entropy is maximal when the distribution is uniform and it is minimal if the distribution is concentrated at a single point. Since we are given a set of eight conditional probability densities, one for each L1 1; 2; . . . ; 8, we de ne the  average entropy S as  S
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  The average entropy remains a function of T1 ; T2 : that is, S S T1 ; T2 .   Figure 18.4 plots S T1 ; T2 S T1 (recall that T1 T2 ) for the a 1:05 throughput series as a function of time scale or aggregation level T1 . Entropy is highest for small time scales in the range $250 ms, and it drops monotonically as T1  is increased. Eventually, S T1 begins to atten out near the 3 5 second mark, reaching saturation, and stays so as time scale is further increased. From our analysis of various long-range dependent traf c traces, we nd that the ``knee'' of the entropy curve is in the range of 1 5 seconds. Note that increasing T1 further and further to gain small decreases in entropy brings forth with it an important problem, namely, if prediction is done over a ``too long'' time interval, then the information may not be effectively exploitable by various congestion control strategies. In the next section,
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Aggregation Level (seconds)
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