## 5 Experimental Results
### Experimental Results for BCC Algorithm
#### 5.2.4 Accuracy of Inter-Consolidation Time Modeling  
In this section, the inter-consolidation time for the BCC algorithm is analyzed and the accuracy of the mathematical estimation of inter-consolidation time is evaluated. The expected inter consolidation cost estimated from (15) is shown in Figure 10 for a data center consisting of 800 servers as discussed in Section 5.2.1.  
Figure 9: Average throughput for different data center architecture with NC, CES, GRD and BCC consolidation normalized with flow injection rate.  
To verify the mathematical model for cost of consolidation (15), we ran a Monte Carlo simulation for each \(\rho\) 300 times and calculated the value of the cost function. For these cases, the values for \(\eta_{1}\), \(\
u\), \(\kappa\) considered here were 200, 1000, and 1 respectively. The average size of migration \(\
u\) was considered as 1000 Megabytes which represents a practical value. For \(\eta_{1}\), we ran the server consolidation simulation for different random initial conditions for thousands of times and used the average number of idle servers in PowerNap mode after the first cycle of server migration. We considered \(\kappa=1\) to put equal emphasis on power saving and network performance on the cost. Nevertheless, these values needed not necessarily be exactly the same across all the data centers. Depending on the capacity, performance requirements and physical limitations, these values could vary for different HPC data centers.  
The average of the simulated values from the different run for each \(\rho\) are shown in Figure 10. The cost estimates in this method relied on many repetitive simulations and were highly computationally expensive as each of simulation at a particular \(\rho\) and \(t\) was repeated at least 1000 times to find the expected cost using Monte-Carlo method. On the contrary, using (15) the cost and optimal inter-consolidation time could be approximated much faster. The optimal inter-consolidation time for different \(\rho\) identified from both methods is shown in Figure 11. It was observed that for lower values of \(\rho\) (\(\rho\leq 0.55\)) the optimal inter-consolidation time estimated from the mathematical equation closely approximated the measured value from the Monte Carlo simulation. On the contrary, for higher values of \(\rho\), the optimal inter-consolidation time estimated from the mathematical analysis deviated from the value measured through simulations. However, at higher \(\rho\), the absolute value of the cost was less sensitive to the inter-consolidation time. This shows that although at higher \(\rho\) the inter-consolidation time suggested by the model may deviate from the actual optimal interval, the actual cost incurred at this non-optimal interval was not much different compared to that at the optimal interval. Hence the optimal inter-consolidation time could be estimated reasonably accurately from the mathematical form.  
We used MATLAB R2018b on a system having Intel Core i7 with 16 GB memory to calculate the inter consolidation time with both Monte-Carlo simulation and mathematical model. On average, computation time required for the Monte-Carlo simulation took 1039.4 s to complete the calculation while the mathematical model took only about 0.798 s on average. Hence there was more than 1000\(\times\) speedup in calculation time using the mathematical model.  
Figure 10: Consolidation cost estimated from mathematical analysis (dashed line) and from Monte Carlo simulation (solid line) with respect to inter-consolidation time.