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where r is the lag-one autocorrelation coef cient. I is the identity matrix, and each row of Q consists of the steady-state probabilities. In our case the steady-state probabilities are the negative-binomial probabilities described above, truncated at some convenient value at least as large as the peak rate (the missing probability is added to the last probability kept). Equation (12.1) is convenient for analytical work, but it masks the simplicity of the DAR model. Let Xn be the size (in bits, bytes, or cells as appropriate) of the nth frame and r be as above; the DAR model is  Xn Xn 1 XH with probability r; with probability 1 r; 12:2
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where X H is a sample from the marginal distribution (negative-binomial in our case). From Eq. (12.2) we see that the X -process maintains a constant value (cell rate) for a geometrically distributed number of steps (frames) with mean 1= 1 r , and then another value (possibly the same as the old value) is chosen. When r is close to one (it is about 0.98 in our examples, see Table 12.1), the mean time between cell rate changes is large (about 50 frames in our examples). This means that the sample paths are constant for long intervals. The data trace doesn't have this property, which is the reason the GBAR model described in Section 12.2.3 was introduced. This difference between the sample paths of the model and the data trace is mitigated H when several sources are multplexed. The probability that Xn Xn 1 is small enough to be ignored in the following calculation. When k sources are multiplexed, Xn Xn 1 with probability rk, so the mean time between potential cell rate changes with r 0:98 and k 16 is 3.6. Consequently, sample paths of the multiplexed cell streams from 16 sources are not constant for long intervals. 12.2.2.1 Validating the DAR Model We validate the DAR model by looking at performance models for multplexing gain and connection admission control. To estimate statistical multiplexing gain, we use cell-loss probabilities [16] from a simple model of a switch. The source model is a FIFO buffer that is drained at 45 Mb=s. The length of the buffer is expressed as the time to drain a full buffer; this is the maximum possible delay. The results of ten simulations of the DAR model for sequence C are given by 95% con dence intervals and are shown in Table 12.2. The
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TABLE 12.2 Cell-Loss Rates for Trace and 95% Con dence Intervals for DAR Model of Sequence C Buffer Size (ms) Source 1 2 3 4 5
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Probability of Loss 10 6 for Various Buffer Sizes Trace DAR model 2070.0 (1738, 2762) 527.0 (433, 775) 141.0 (107.4, 212.6) 33.3 (15.1, 54.1) 2.88 (2.26, 9.34)
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12.2 VIDEO CONFERENCES
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results of these simulations show that the DAR model does a good job of estimating the cell-loss rate when 16 sources are multiplexed. Similar results were obtained for the other sequences [18]. Now we consider connection admission control (CAC). Since the DAR model is a Markov chain model of the source, it conforms to one of the sets of conditions a source model must have for the effective bandwidth (EBW) theory of Elwalid and Mitra [8]. Moreover, the DAR model is a reversible Markov chain, and so it inspired a powerful extension of the EBW method, called the Chernoff-dominated eigenvalue (CCE) method [7]. Suppose we have a switch that can process at rate C (Mb=s) and has a buffer of size B (ms). We want to nd the maximum number of statistically homogeneous sources that can be admitted while keeping the cell-loss rate no larger than 10 6 . The CDE method gives an approximate analytic solution with known error bounds; this solution is denoted by KCDE. Another way to obtain the solution is to test candidate values by evaluating the cell-loss rate by simulation; we treat this as the exact solution and denote it by Ksim. Table 12.3 compares the results of the CDE method to the CAC found from simulations. The number admitted by the CDE method is a very close approximation to the ``true'' value obtained by simulation. This implies that the DAR model captures enough of the statistical properties of the trace to produce good admission decisions.
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