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distributional and independence assumptions listed above Eq. (12.5) are valid, then the GBAR process will be formed. Simulating the GBAR process only requires the ability to simulate independent and identically distributed gamma and beta random variables. This is easily done; for example, algorithms and Fortran programs are presented in Bratley et al. [3]. The GBAR process is used as a source model by generating noninteger values from Eq. (12.5) and then rounding to the nearest integer. It would be cleaner if a discrete process with negative-binomial marginals could be generated in the rst place. McKenzie describes such a process (his Eq. (3.6)). Unfortunately, that process requires much more computation to simulate, and the extra effort does not appear to be worthwhile. Validating the GBAR Model Ten sample paths of the GBAR process were generated and used as the arrival process (number of cells per frame with a xed interframe time) in a simulation of a service system with a nite buffer and a constant-rate server. The cell-loss rates from these paths were averaged to obtain a point estimate for the GBAR model. The traf c intensity is varied by changing the service rate. The points produced by the simulations are denoted by an asterisk. In Fig. 12.3, we see that cell-loss rates computed from the GBAR model are close to the cell-loss rates computed from the data. Note that for each traf c intensity, the decrease in the cell-loss rate as the buffer size increases is very slight, for both the model and the data. This con rms the prediction of Hwang and Li [19] of buffer ineffectiveness. Since the GBAR model has only short-range dependence, this effect is not caused by long-range dependence here.
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Fig. 12.3 Cell-loss rates for sequence A.
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Fig. 12.4
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Mean queue lengths for sequence A.
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Figure 12.4 shows that the mean queue lengths in an in nite buffer computed from ve GBAR paths are similar to the mean queue lengths computed using the data. In Fig. 12.4 the vertical axis on the left shows the mean queue length in cells. Video quality is poor when the cell delays are large; 100 ms is an upper bound on the acceptable delay at a node in a network that provides video services. Two buffer drain times are also shown; the practical region for the maximum is below and to the left of the 100 ms line. The range of the mean queue lengths shown exceeds the practical region for maximum queue length. In the practical region, the model and the trace give very similar mean delays. The differences between the mean queue length from the GBAR model and the mean queue lengths from the data would be even smaller if a nite buffer were imposed. (This is the truncating effects of nite buffers that is described in Section 12.4.2). The comparisons for sequences B, C, and D are qualitatively the same as for sequence A.
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Now we turn to more dynamic sequences, such as lms, news, sports, and entertainment television. Since the main purpose of the models is to aid in network performance evaluations, we are particularly interested in using the models to predict cell-loss rates.
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Broadcast VBR-coded video has different bit-rate characteristics than VBRcoded video conferences. Video conference sequences consist of head-and-shoulders pictures with little or no panning, while broadcast video is characterized by a succession of scenes. With interframe coding, it is clear that scene changes require more bits than intrascene frames, so broadcast video will differ from video conferences in at least this respect; this was demonstrated by Yasuda et al. [32]. There are some other differences too, as demonstrated by Verbiest et al. [31] and further ampli ed by Verbiest and Pinnoo [30]. In these papers, a DPCM-based coding algorithm is used. In the latter paper, it is shown that the number of bits per frame has a different autocorrelation function for broadcast video than for video conferences or video telephony. The autocorrelation functions for the last two are similar to each other and decay geometrically to zero. For broadcast video, the autocorrelation function does not decay to zero. Moreover, the rst frame after a scene change has signi cantly more bits than other frames in the scene. Ramamurthy and Sengupta [26] observe that the correlation function declines more rapidly at small lags than at large lags, and that the time series can be described by a semiMarkov process that has states identi ed by the bit rates for different types of scenes (and a state for scene changes). We build on this idea [12]; the simple DAR and GBAR models that described video conferences are not suf cient for broadcast video, although the DAR model is used as a building block in a more complex model. 12.3.1 Modeling Broadcast Video
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We obtained several data sets giving the number of bits per frame for sequences encoded by an intra eld=interframe DPCM coding scheme without use of DCT or motion compensation. We did not have access to the actual video sequences. Hence, a visual identi cation of scene-change frames was not possible. Our modeling strategy was to rst develop a way to identify scene changes, then construct models for the lengths of the scenes and the number of cells in a scene-change frame. Finally, models for the number of cells per frame for frames within scenes were developed. Preliminary Data Analysis Before describing the statistical models, we report some elementary statistics about the sequences we examined. Figure 12.5 shows the peak and mean bit rates, and their ratios. The peak-to-mean ratios vary from 1.3 to 2.4. By way of comparision, the peakto-mean ratio for video-conference sequence with this codec (sequence A) is 3.2. Note that the larger peak-to-mean ratios are associated with the lower mean bit rates. The sequences divers, lm, Isuara 1, and Isaura 2, which have a low mean rate and high peak-to-mean ratios, were different TV programs recorded from a Cable TV network (and designated as normal quality broadcast video). The sequences with low peak-to-mean ratios (such as football, sport, news, etc.) were taken directly from the TV studios (and designated high-quality broadcast video).
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