## 5 Performance Evaluation
### Performance Comparisons  
We compared the performance of the proposed method with those of three alternatives. The first alternative is EVEN, which determines \(\alpha_{i}(t)=1/|N|\). The second alternative determines \(\alpha_{i}(t)\) in proportion to the queue-length threshold \(\delta_{i}\). Specifically, if we denote \(\eta_{i}=1/\delta_{i}\), the second alternative determines \(\alpha_{i}(t)=\eta_{i}/\sum_{j\in N}\eta_{j}\). Henceforth, we refer to this as TPRA. The third alternative, WoPred, is similar to our method in that it uses a Lyapunov optimization framework. However, WoPred uses only the observed queue length without considering the input workload dynamics when determining \(\alpha_{i}(t)\)s at the start of each time slot. We also show the results obtained when our method uses \(w_{i}(t)\)s instead of \(\widehat{w}_{i}(t)\)s, which indicate the upper performance bound of our method. First of all, we analyzed the time complexity of these methods. Algorithm 1 describes our online resource allocation method. The method is composed of two main parts--the prediction of the workload to be imposed on the queue \(i\) during time slot \(t\) and the determination of the time fraction that a MES allocates to each queue during a time slot \(t\). To analyze the time complexity of the workload prediction, we considered the time consumed by the LSTM deep learning model to predict the workload. Since the deep learning technique is used for prediction after going through the training phase, the required time for the prediction is \(O(1)\). Next, the time complexity of the time fraction determination is \(O(N)\), which is proportional to the set of queues managed by a MES. Therefore, the time complexity of our method is \(O(N)\). Meanwhile, in the case of the ideal method, the time complexity is \(O(N)\) because it has only the time fraction determination process. In _EVEN_, _TPRA_, and _WoPred_, only the time fraction determination process is included. However, the difference among these three methods is in how they determine the fraction of the time that a MES allocates to each queue during a time slot \(t\) (\(\alpha_{i}^{*}(t)\)). Since _EVEN_ and _TPRA_ allocate \(\alpha_{i}^{*}(t)\)s statically, their time complexity is \(O(1)\). In contrast, since _WoPred_ dynamically allocates service-time at each time slot by calculating \(\alpha_{i}^{*}(t)\)s, its time complexity is \(O(N)\). In Figure 3, we show how each method changes \(Q_{i}(t)\)s over time. We can see in Figure 3a that WoPred makes \(Q_{1}(t)\) fluctuate widely and does not make \(Q_{1}(t)<\delta_{1}\) most of thetime. We also observe similar behaviors in Figure 3b,c. This is because WoPred uses only the current queue length when it allocates service-time for each queue. When \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Class 1**} & \multicolumn{2}{c}{**Class 2**} & \multicolumn{2}{c}{**Class 3**} \\ \cline{2-7}  & **Avg.** & **Std. Dev.** & **Avg.** & **Std. Dev.** & **Avg.** & **Std. Dev.** \\ \hline \(h=3\) & 0.1022 & 0.0875 & 0.1182 & 0.1069 & 0.0738 & 0.0662 \\ \(h=4\) & 0.0957 & 0.0767 & 0.1152 & 0.1016 & 0.0889 & 0.0795 \\ \(h=5\) & 0.1057 & 0.0836 & 0.1249 & 0.0948 & 0.0931 & 0.0709 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Performance comparison of LSTM prediction in terms of RMS. Figure 2: Comparison between actual and predicted values. EVEN and TPRA are used, \(Q_{1}(t)\) is maintained under \(\delta_{1}\). However, \(Q_{2}(t)\) and \(Q_{3}(t)\) are not always maintained below \(\delta_{2}\) and \(\delta_{3}\). This is attributed to the fact that they statically allocate service-time. If the workloads imposed on the Class 2 queue and Class 3 queue are temporarily larger than the service-time allocated to them, \(Q_{2}(t)\) and \(Q_{3}(t)\) rise sharply. In this figure, we can observe that our method makes \(Q_{i}(t)<\delta_{i}\) for all \(i\) all the time by dynamically allocating service-time to each queue according to the current queue length and the expected amount of workload.