PREDICTABILITY OF SELF-SIMILAR TRAFFIC in Visual Studio .NET Drawer Code-39 in Visual Studio .NET PREDICTABILITY OF SELF-SIMILAR TRAFFIC 18.3 PREDICTABILITY OF SELF-SIMILAR TRAFFICScanning Code 39 Full ASCII In Visual Studio .NETUsing Barcode Control SDK for Visual Studio .NET Control to generate, create, read, scan barcode image in VS .NET applications.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).Drawing Code 39 In .NETUsing Barcode creator for .NET framework Control to generate, create Code-39 image in Visual Studio .NET applications.Predictability and Time Scale Code 3/9 Reader In VS .NETUsing Barcode decoder for Visual Studio .NET Control to read, scan read, scan image in Visual Studio .NET applications.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 itsBar Code Generator In .NET FrameworkUsing Barcode maker for .NET framework Control to generate, create bar code image in Visual Studio .NET applications.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.Scan Barcode In Visual Studio .NETUsing Barcode recognizer for Visual Studio .NET Control to read, scan read, scan image in .NET framework applications.CONGESTION CONTROL FOR SELF-SIMILAR NETWORK TRAFFIC Code 39 Full ASCII Creator In C#Using Barcode creation for Visual Studio .NET Control to generate, create Code39 image in VS .NET applications.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 Code-39 Drawer In VS .NETUsing Barcode printer for ASP.NET Control to generate, create USS Code 39 image in ASP.NET applications.8 P l H 1 Code 39 Extended Generation In Visual Basic .NETUsing Barcode generation for .NET Control to generate, create Code-39 image in VS .NET applications.PrfL2 l H jL1 lg log PrfL2 l H jL1 lg:Generate Bar Code In .NET FrameworkUsing Barcode drawer for .NET framework Control to generate, create bar code image in .NET framework applications.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 Making Data Matrix 2d Barcode In .NETUsing Barcode drawer for .NET framework Control to generate, create Data Matrix 2d barcode image in Visual Studio .NET applications.8 P l 1 UPC A Creator In .NET FrameworkUsing Barcode creation for Visual Studio .NET Control to generate, create UPC Symbol image in VS .NET applications.Sl =8:Identcode Maker In .NETUsing Barcode encoder for .NET Control to generate, create Identcode image in VS .NET applications.  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. 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