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TABLE 34.2 Results for Example 34.3 | ||||||||||||||
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Responce Time | Queue Lenths | |||||||||||||
Iteration | System | |||||||||||||
No. | CPU | Disk A | Disk B | System | Throughput | CPU | Disk A | Disk B | ||||||
1 | 0.92 | 2.20 | 1.47 | 44.00 | 0.42 | 6.11 | 9.17 | 3.06 | ||||||
2 | 0.85 | 2.91 | 0.78 | 46.64 | 0.39 | 5.38 | 11.50 | 1.54 | ||||||
3 | 0.76 | 3.58 | 0.49 | 50.46 | 0.37 | 4.49 | 13.14 | 0.90 | ||||||
4 | 0.66 | 4.05 | 0.37 | 52.83 | 0.35 | 3.70 | 14.24 | 0.65 | ||||||
5 | 0.56 | 4.36 | 0.32 | 54.23 | 0.34 | 3.10 | 14.97 | 0.56 | ||||||
6 | 0.49 | 4.57 | 0.31 | 55.08 | 0.34 | 2.67 | 15.45 | 0.52 | ||||||
7 | 0.44 | 4.70 | 0.30 | 55.62 | 0.34 | 2.37 | 15.78 | 0.50 | ||||||
8 | 0.41 | 4.80 | 0.30 | 55.97 | 0.33 | 2.17 | 16.00 | 0.49 | ||||||
9 | 0.38 | 4.86 | 0.29 | 56.20 | 0.33 | 2.04 | 16.15 | 0.49 | ||||||
10 | 0.37 | 4.90 | 0.29 | 56.35 | 0.33 | 1.94 | 16.25 | 0.48 | ||||||
11 | 0.36 | 4.93 | 0.29 | 56.45 | 0.33 | 1.88 | 16.31 | 0.48 | ||||||
12 | 0.35 | 4.95 | 0.29 | 56.52 | 0.33 | 1.84 | 16.35 | 0.48 | ||||||
13 | 0.34 | 4.96 | 0.29 | 56.57 | 0.33 | 1.82 | 16.38 | 0.48 | ||||||
14 | 0.34 | 4.97 | 0.29 | 56.59 | 0.33 | 1.80 | 16.40 | 0.48 | ||||||
15 | 0.34 | 4.97 | 0.29 | 56.61 | 0.33 | 1.79 | 16.41 | 0.48 | ||||||
16 | 0.34 | 4.98 | 0.29 | 56.63 | 0.33 | 1.78 | 16.42 | 0.48 | ||||||
The approximate analysis of queueing networks is an active area of research. The majority of new papers on queueing theory introduce approximations of one kind or another. Unfortunately, there are only a few approximation techniques that are used again by their inventors, and even fewer are used by people other than the inventors. Schweitzers approximation discussed in this section is one of those exceptional approximation techniques that has been used and discussed by many researchers.
One of the common mistakes we find in papers proposing approximate queueing techniques is that the authors compare the results obtained from their technique and those from a simulation to show that the error in throughput is small. The fact, on the contrary, is that a small error in throughput does not imply that the approximation is satisfactory. The same applies to device utilizations and the system response time. In spite of a small error in any of these, the error in the device queue lengths may be quite large. It is easy to see this from Table 34.2. There we see that the throughput reaches close to its final value within five iterations, while the response time reaches close to its final value within six iterations. The queue lengths take the longest to stabilize.
Another way to understand the relative magnitude of errors in throughput, response time, and queue lengths is to look at Figures 34.1 and 34.2, which show the system throughput and the system response times as obtained from the Schweitzer approximation. The number of users is varied from 1 to 20, and in each case we start the iterations with queue lengths initialized at N/M and stop arbitrarily after five iterations. Also shown in the figures are the throughput and response time as obtained using the exact MVA. Finally, Figure 34.3 shows the CPU queue length. Notice that for all values of N, the error in throughput is small, the error in response time is slightly larger, and the error in queue lengths is the largest. The error in queue lengths thus gives a better measure of the goodness of an approximation technique. For the same reason, a stopping criterion based on changes in queue lengths is better than one based on system throughput or response time.
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