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The second term in Eq. (19.13) considers only the case when the total arrivals of all connections in a frame is less than the total capacity. It is possible to not drop any cell from connection k, even if the total number of arrivals is greater than the total capacity. Therefore, the equivalent bandwidth computation above is an upper bound.
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ALGORITHM ALTERNATIVE 2 1. Initialize Ci in accordance with Eq. (19.8). Let D be a (small) quantum of capacity that may be subtracted from Ci i 1; 2 . Let Ci0 Ci i 1; 2 , and let the iteration step n 1. 2. Set Cin Cin 1 D i 1; 2 . Set boolean variable tryagain to TRUE. 3. The probability that no cells will be dropped from class 1 is the probability that the arrivals from class 1, X1 , is less than C1 OR X1 is greater than C1 but the total arrivals in the frame, X1 X2 C1 C2 . Therefore, we need to check if
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19:11 If inequality (19.11) is not satis ed, mark tryagain FALSE. 4. Analogous to Eq. (19.11), check if the following is satis ed: C1 n
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19:12 If Eq. (19.12) is not satis ed, mark tryagain FALSE. 5. If tryagain is TRUE go to Step 2. Else go to Step 6. n 1 n 1 6. A tight upper bound on the equivalent bandwidth is C1 C2 . 19.4.1.3 Numerical Results and Simulation Equivalent Bandwidth Calculation Alternatives 1 and 2 were compared using the MPEG-I video traces mentioned earlier. Table 19.2 shows the equivalent bandwidths needed for three of the traces (Terminator, Gold nger, and Soccer) with one VC per connection when all of them belonged to the same jitter class (same frame size) and needed different cell-loss probabilities. The marginal distribution of each trace appeared to be a Gamma distribution, so a tted Gamma was used in the equivalent bandwidth computations. The numerical integration was evaluated using Mathematical Software package. Alternative 2 resulted in approximately 40% savings for
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19.4 EQUIVALENT BANDWIDTH
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Comparison of Alternatives 1 and 2a Alternative 1 Capacity Alternative 2 Capacity Bandwidth (kbits=frame) (kbits=frame) Reduction 256 388 460 150 220 280 0.41% 0.43% 0.39%
Cell-Loss Probabilitiesa E1 ; E2 ; E3 1:5 10 2 ; 2 10 2 ; 10 2 2 10 3 ; 3 10 3 ; 1 10 3 10 2 ; 10 3 ; 1 10 4
E1 ; E2 , and E3 represent the cell-loss probabilities for Terminator, Gold nger, and Soccer, respectively.
these experiments. Note that without framing and active cell-discard, developing a solution for heterogeneous E1 ; E2 ; E3 is nontrivial. Effectiveness of the Firewall fCi g and Dropping Policy The effectiveness of guaranteeing a minimum capacity Ci per frame for class i and the dropping policy is evaluated next to the context of heterogeneous E values. This is achieved through trace-driven simulation. The scenario was as follows.  MPEG-I video traces shown in Table 19.3 were used as input to the simulations.  Their Ci were calculated using Alternative 2.  The dispatcher was either the pseudo-earliest-due-date dispatcher described in Section 19.2.3 or LIFO for comparison purposes. In the experiments, each class had one connection. Also, the traces were phaseshifted randomly to ensure that they were mutually unsynchronized otherwise, one would have to consider correlated streams. Recall from Section 19.2.3, that when channel_image[] was full, the cell dispatcher gave preference to classes fjg that had not yet received at least Cj cell allocations in the current frame, and among those that did receive Cj , to those with smaller current estimates of the ratio Sj . The performance of the algorithm is shown in Table 19.3. Results for a simple last-in- rst-out (LIFO) dropping policy is also shown for comparison purposes. In
TABLE 19.3 Performance of the Proposed Cell Dispatcher for Heterogeneous Cell-Loss Probabilities, fEi g Experiment Number I II Video Trace Terminator Gold nger Soccer Terminator Gold nger Soccer Ei Desired 0.01 0.001 0.0001 0.002 0.003 0.001 Ei Delivered (LIFO) 0.0008 0.0012 0.0011 0.0014 0.0020 0.0018 Ei Delivered (Proposed Dispatcher) 0.0039 0.00096 0.00009 0.0019 0.0029 0.0007