## 4 Performance Evaluation
### Performance Comparison  
Cost function adopted for the optimization is Equation (1), which aims to minimize the difference between the traffic demand and the offered capacity and at the same time to minimize the resources used in the satellite while complying with all the constraints of the system presented in Equations (2)-(5). Therefore, to evaluate the performance of different ML models, two KPI (Key Performance Indicator) are proposed with which an important trade-off can be observed [33]. In this sense, the first KPI is defined as the average capacity gap.  
\[KPI_{1}=\frac{1}{B}\sum_{b=1}^{B}|C_{t}^{b}-R_{t}^{b}| \tag{16}\]  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Parameter** & **Value** \\ \hline Satellite Orbit & \(9^{\mathrm{o}}\,\mathrm{E}\) \\ \hline Number of Beams & 82 \\ \hline Noise power density & \(-204\) dBW/MHz \\ \hline Max. beam gai & 51.8 dBi \\ \hline User antenna gai & 39.8 dBi \\ \hline Maximum bandwidth per beam & 500 MHz \\ \hline Maximum power per beam & 100 W \\ \hline Total available transmit power & 1000 W \\ \hline \hline \end{tabular}
\end{table}
Table 4: System parameters.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**ML Technique** & **Model** & **Training** & **Implementation** \\ \hline \multirow{2}{*}{Supervised Learning} & DNN & \multirow{2}{*}{offline} & \multirow{2}{*}{On-board} \\ \cline{2-3} \cline{5-5}  & CNN & & \\ \hline \multirow{2}{*}{Reinforcement Learning} & QL & \multirow{2}{*}{online} & \multirow{2}{*}{On-ground} \\ \cline{2-3} \cline{5-5}  & DQL & & \\ \cline{2-3} \cline{5-5}  & DDQL & & \\ \hline \hline \end{tabular}
\end{table}
Table 3: Features of the ML models evaluated.  
The second KPI is the power saving operation, which is defined as follows:  
\[KPI_{2}=\frac{P_{Total,LIPA}}{P_{Total,model}} \tag{17}\]  
where \(P_{Total,LIPA}\) is the total payload power when using uniform power allocation (UPA), and \(P_{Total,model}\) is the total payload power when using power allocation and the proposed ML model.  
The ML model will try to minimize \(KPI_{1}\) in (16) and maximize \(KPI_{2}\) in (17). In this sense, the performance of the model used for RRM is evaluated by exploiting a joint KPI and is defined as follows:  
\[KPI_{3}=\frac{A_{1}}{\rho}KPI_{1}+\frac{A_{2}}{KPI_{2}} \tag{18}\]  
where \(\rho\) is the normalization parameter, and the normalization parameter allows comparing the performance of the different ML models when used in different traffic demand scenarios. \(A_{1}\) and \(A_{2}\) are weighting parameters that provide additional importance to the two KPIs. If one algorithm performs better, \(KPI_{3}\) is lower.  
Taking \(KPI_{1}\) and \(KPI_{2}\) to be of equal importance, we set both \(A_{1}\) and \(A_{2}\) to 0.5. We evaluated the five ML models presented in Table 3 and used a normalization parameter \(\rho=R_{max}\), where \(R_{max}\) represents the maximum required capacity per beam for each scenario in which the model was evaluated. For this evaluation, it is not taken into account whether the model is trained online or offline. The results obtained are presented in Figure 9.  
The normalized \(KPI_{1}\) is represented on the left axis and \(KPI_{2}\) is on the right axis, while the gray bars represent the value obtained from the computation represented by \(KPI_{3}\). In the case of the DNN-based model (\(KPI_{3}=0.48\)), it is shown that it is not a suitable approach. As the availability of possible resources to be optimized increases, DNN loses efficiency, which is explained in [33]. The models that obtain the smallest normalized average capacity gap, i.e., \(\frac{1}{\rho}KPI_{1}\), are the DQL and DDQL models, while they also obtain the highest \(KPI_{2}\). Therefore, they are the most efficient models, reaching a \(KPI_{3}\) of 0.33. However, it is observed that CNN obtains quite competitive results, reaching a \(KPI_{3}\) of 0.35.  
Figure 9: Performance comparison of different ML models for RRM.