## 5 Performance Evaluation
### Scalability  
In Figures 8 and 9, we show the average queue lengths of the different methods when the number of task classes (\(N\)) is 6 and 9, respectively. When \(N=6\), the average queue lengths obtained by TPRA are much larger than those achieved by other methods in Classes 4-6. For example, the average queue lengths obtained by TPRA are 11-25-times larger than those achieved by EVEN or WoPred. Moreover, in Class 6, the average queue length obtained by EVEN is much larger than those achieved by other methods. Therefore, in these subfigures, we omit the cases when TPRA and EVEN are applied to show the differences among the other methods visually. When \(N=9\), the average queue lengths obtained by TPRA are much larger than those achieved by other methods in Classes 7-9. Thus, in Classes 7-9, we omit the cases when TPRA is applied to show the differences among the other methods visually. In these figures, we also observe that the average queue lengths obtained by our method look identical to those obtained by the ideal method that knows \(w_{i}(t)\)s in advance. This is because they achieve similar average queue lengths for all \(V\)s. For example, when \(N=6\) and \(log_{10}V=16\) (i.e., \(V=10^{16}\)), the differences in the average queue length between our method and the ideal method ranges from 0.0053 to 0.097. When \(N=9\) and \(V=10^{16}\), the differences range from 0.0027 to 0.073. From Figures 8 and 9, we can observe that unlike the other methods, our method guarantees the average queue length below a given threshold for all the queues.  
Figure 7: Average queue length with different V. The average length of the Class 3 queue obtained by TPRA is more than 2100 megacycles, which is much larger than those achieved by other methods. Therefore, in Figure 7c, we omit the case when TPRA is applied to show the differences among the other methods visually. (**a**) Class 1, (**b**) Class 2, and (**c**) Class 3.  
Figure 8: Average queue length with different Vs when the number of classes is 6. In Classes 4–6, the average queue lengths obtained by TPRA are much larger than those achieved by other methods. Therefore, in Classes 4–6, we omit the case when TPRA is applied and adjust the range of the x-axis to show the differences among the other methods visually. We also note that since the results obtained by the proposed method are almost identical to those obtained by the ideal case, they look the same in these figures. (**a**) Classes 1 and 2, (**b**) Classes 3 and 4, and (**c**) Classes 5 and 6.  
In Figure 10, we show the average energy consumption of a MES when \(N=6\) (Figure 10a) and \(N=9\) (Figure 10b) with different \(V\)s. We can see that the energy consumed by our method is comparable to that consumed by the ideal method. Moreover, we can observe that our method consumes the least energy. We can also observe that there is a trade-off between the average energy consumption and the average queue length, which we can control by using the parameter \(V\).  
Figure 10: Average energy consumption of a MES with different Vs when \(N=6\) and \(N=9\). (**a**) \(N=6\) classes, and (**b**) \(N=9\) classes.  
Figure 9: Average queue length with different Vs when the number of classes is 9. In Classes 7–9, the average queue lengths obtained by TPRA are much larger than those achieved by other methods. Therefore, in Classes 7–9, we omit the case when TPRA is applied to show the differences among the other methods visually. We also note that since the results obtained by the proposed method are almost identical to those obtained by the ideal case, they looks the same in these figures. (**a**) Classes 1, 2, and 3, (**b**) Classes 4, 5, and 6, and (**c**) Classes 7, 8, and 9.