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Simulation Results
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Figure 12.12 shows the simulation results evaluating the accuracy (as a predictor of cell losses) of the DAR model for the lm sequence. This sequence has a mean rate of 16.7 Mb=s and a peak rate of 37.31 Mb=s. The multiplexer simulations were done with 8 sources being multiplexed. The output rate of the multiplexer was set to 155 Mb=s. The link utilization is 0.849. Simulations were done for maximum delays in the multiplexer equal to 0.5 ms, 1 ms, 2 ms, 3 ms, and 4 ms. The peak input rate from the source to the multiplexer was set to 43 Mb=s. This value only determines the cell transmission time to the multiplexer. The intercell spacing within a frame is equal and in an interframe interval the cell arrival rate is equal to the number of cells in the most recent frame divided by the interframe interval. We generated 10 sample paths using the model and these are used in 10 runs of the simulation. In Fig. 12.12, the loss rates of the 10 sample paths for different buffer sizes are represented by their mean loss rate. The values corresponding to the mean plus one standard deviation and the mean minus one standard deviation are also plotted. For the actual traf c, the simulation was run not only using the trace of actual traf c but also with the rates in the actual traces scaled up and down by 1%. These are also shown in Fig. 12.12. The 1% scaling was used to provide a range of values against which the loss probabilities generated by the model may be compared. Since source rates may not be known very accurately, it is reasonable to compare the model's losses to actual losses with a 1% uncertainty in the source mean rate [9].
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Fig. 12.12
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Cell-loss probabilities for lm.
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As can be seen from Fig. 12.12, for all multiplexer buffer sizes studied, the mean loss rates of the 10 sample paths fall within the range of losses obtained for actual 1%. Also, the actual loss rate falls within the range of mean losses for the model 1 standard deviation. Hence, the DAR model is a suf ciently accurate model for the lm sequence. Frater et al. [9] successfully described this source with a simpler model. Their scenic model is a generalization of the basic DAR model of Eq. (12.1) that permits the times between changes of state to have any distribution on the positive integers. Figure 12.13 shows the simulation results for the Isaura 1 sequence. This sequence has a mean rate of 14.9 Mb=s and a peak rate of 33 Mb=s. The simulation was done with 8 sources being multiplexed. The buffer sizes and the multiplexer input and output rates are the same as those used for lm. The output link utilization was 0.77. The losses resulting from use of the DAR and the Markov chain models are compared to the actual losses. As in Fig. 12.12, the actual losses and the losses resulting from a scaling of the mean rate of the actual sequence by 1% is shown in the gure. The losses due to the models are represented using the mean losses from 10 sample paths and the mean 1 standard deviation values. From the gure, it can be seen that the DAR model overestimates cell losses and is not an acceptable model for this sequence unless conservative estimates are suf cient. However, as can be seen in Fig. 12.13, the Markov chain model is suf ciently accurate since the mean loss probabilities obtained using the Markov chain model are always lower than the losses obtained for actual rate plus 1% and always higher than the losses obtained for
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