Results and Findings

For the purpose of performance analysis, we define baseline condition which consists of the dedicated assignment, sizing as described in Section 2.3.2 and round-robin routing

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(b) Pattern A - different SCOV

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(d) Pattern B - different SCOV

Figure 2.3: Comparing average total number in system across all classes over 1 cycle:

dedicated vs pooled

with traffic intensity ρ = 0.9. We will evaluate our approach by using baseline condition in following sections.

2.7.2.1 Performance Comparison Between Assignment Strategies

First, we compare the performance of assignment strategies to verify our insight de-scribed in Section 2.4. As dede-scribed in Theorem 1, we use Bernoulli routing for the ded-icated assignment and pooled assignment. Note that the total number of servers powered on at any time period is the same for both assignment strategies, we can make a fair com-parison between assignment strategies. Since, as described in Section 2.5.2.1, the pooled

assignment results in a non-homogeneous system, it would not be possible to use “dummy"

requests for pooled assignment cases. Therefore, we compare the dedicated assignment case without using “dummy" requests. We compare the average total number of requests in the system by plotting it across time (also averaged across all classes) with constant SCOV in Figure 2.3a for arrival rate pattern A and in Figure 2.3c for pattern B. From the results, we can verify that dedicated assignment is better than pooled assignment. More-over, we try to compare assignments with different SCOV defined in the Section 2.7.1 to check our conjecture that our insight can be extended to more general cases where SCOV is not constant and each class has a different SCOV value. Figures 2.3b and 2.3d show that dedicated assignmentis also better than pooled assignment with a different SCOV value for each arrival rate pattern. Since cases with different SCOV values are regarded as more general, we will consider only different (and high) SCOV for further analysis.

2.7.2.2 Analysis of Time-Stability

Next we analyze the time-stability of our suggested approach. As described in Section 2.5, our approach stabilizes the distributions of the queue lengths as well as the sojourn times (see Section 2.6.1). Based on both time-stable distributions, first we show that the mean and standard deviation of queue lengths for 5 classes are time-stable for round-robin (baseline) in Figure 2.4. Note that both round-round-robin and Bernoulli routing result in time-stable performance as mentioned in Section 2.5.2.1, but round-robin routing indeed results in better performance than Bernoulli routing as described in Section 2.3.3. For this reason we analyze time-stable performance of baseline (which use round-robin routing) for further analysis. Also based on Figures 2.4 and 2.5, it is worthwhile to indicate that our time-stable performance does not depend on arrival rate patterns which verifies the discussion in Section 2.5.2.1. In addition we check that distribution of sojourn times is also stabilized described in Section 2.6.1. As we indicated, Figure 2.5 show that the mean

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Figure 2.4: Mean and standard deviation of queue length for the 5 classes across a cycle with round-robin routing

and standard deviation of sojourn times for 5 classes are time-stable.

2.7.2.3 Bounded Performance

In Section 2.5.3 we mentioned that our time-stable performance measures would be an upper bound on actual performance without using dummies. Figure 2.6 compares the actual time-varying performance obtained our approach without dummies as opposed to time-stable performance bound. As we already mentioned, if dummies are not used then the mean queue lengths are time-varying across time intervals (due to empty queue of newly powered-on servers), but they are strictly bounded by the time-stable mean queue

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Figure 2.5: Mean and standard deviation of sojourn times for the 5 classes across a cycle with round-robin routing

lengths obtained by adding dummies. In addition, we have claimed that the gap between actual performance and bound would be affected by both variance of workload (i.e. SCOV of workload distribution) and utilization (which can be controlled by the desired traffic intensity ρ in our approach), but it is not dependent on the time interval length. Figure 2.7 compares the differences between actual performance and bound for application class 3 (which shows the largest variation without dummies) according to the different conditions of SCOV, utilization and time interval length. As we expected, the performance gap would be bigger with bigger SCOV and higher utilization, but the same with smaller time interval length. Although the performance gap seems to be large for bigger SCOV of workload

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Figure 2.6: Performance of the mean queue length for the 5 classes across 24 60-minutes time intervals

distribution and higher utilization, considering that it is also difficult to analyze dynamics of time-varying and transient system for both cases, we believe that our suggest approach still provide significant benefits based on time-stability.

2.7.2.4 The Effect of Time Interval Length

As we discussed in Section 2.6.2, in order to check whether our suggested approach performs well for the case with smaller time interval length, we run simulation for 288 5-minutes time intervals by decomposing 24 60-minutes interval into smaller ones with the same daily pattern. Recall that we used data set which has the mean service times of 5 classes as 38.46, 29.034, 48.0769, 33.8826, and 40.3846 seconds, and thus we believe that 5 minutes time intervals are appropriate to check the case of smaller time interval length.

Figure 2.8 shows the bound and actual performance of the mean queue lengths for 288 5-minutes time intervals, and based on the comparison with Figure 2.6 we can conclude that time-stability obtained by our suggested approach is robust to time interval length.

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(a) 60-minutes, SCOV=2.2, ρ = 0.9

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Figure 2.7: Analysis of the gap between bound and actual performance of class 3 for different conditions

2.7.2.5 The Effect of System Size

As we mentioned in Section 2.6.3, our suggested approach has a limitation that perfor-mance would not be stabilized with smaller size N for round-robin routing. To analyze the limitation of our suggested approach, we have run simulation by scaling with size N = 100 instead of N = 1000, summarized the results as shown in Figure 2.9. As shown in Figure 2.9, performance by using Bernoulli routing is stabilized with smaller size N = 100 (but as we mentioned performance is wore than round-robin), however round-robin does not yield time-stable performance for the case of size N = 100.

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Figure 2.8: Bounds and actual performance of the mean queue length for the 5 classes across 288 5-minutes time intervals

(a) Round-robin with N = 100

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(b) Bernoulli with N = 100

Figure 2.9: Bounds on the mean queue length with N = 100 for both routing policies:

Round-robin and Bernoulli

2.7.2.6 Dummy Traffic Analysis

In Section 2.5.2.2, we claimed that the amount of dummy traffic to adjust arrival rates of each application a is insignificant. We show percentile gap between the actual arrival rates and adjusted arrival rates in Table 2.1, and the additional dummy traffic is reasonably negligible to the actual arrivals.

Time Class Indices

Intervals class 1 class 2 class 3 class 4 class 5 1 0.6200 0.3275 0.4073 1.6294 0.2857 2 0.2857 0.9125 1.0880 1.1895 0.1143 3 1.4000 0.7370 1.0880 0.6569 0.6984 4 0.5333 0.3275 0.3392 1.6294 0.0755 5 1.0880 0.1000 1.6229 0.3462 0.5042 6 1.7391 1.0540 0.1520 1.9990 0.8163 7 2.1091 1.0100 1.0880 0.5058 0.8722 8 0.4488 1.4226 0.3392 0.7533 1.6229 9 0.4073 0.4062 0.4073 0.5570 0.2857 10 0.7390 0.4250 0.5531 0.0717 1.6229 11 0.3935 0.3774 0.3711 0.8464 0.2857 12 0.1915 1.2682 0.0947 0.6986 2.2521 13 0.0552 0.4062 0.0800 0.3462 0.2857 14 0.5702 1.4226 0.3109 1.2685 1.2987 15 0.5568 1.1706 0.2631 0.7151 0.8722 16 0.3663 0.7084 0.0446 0.4384 0.2857 17 0.2857 0.5750 0.0350 0.8281 0.4746 18 0.3737 0.2857 0.2426 0.1776 0.1143 19 0.5049 1.7351 0.0552 0.3462 0.6234 20 0.2857 1.3512 0.4073 1.9990 0.0902 21 0.6200 0.0350 0.4488 0.9365 0.0583 22 0.2857 0.5136 0.5531 1.9990 1.2987 23 0.6909 0.3275 0.8800 1.4194 0.4883 24 1.1484 1.0540 0.3392 0.1192 0.2857 Overall 0.5318 0.7370 0.3752 0.7847 0.5165

Table 2.1: Percentage gap between the actual arrival rates (A) and adjusted arrival rates (B): ^(B−A)_A × 100(%)