## 4 Numerical Simulations and Analysis  
The satellite system simulation parameters are listed in Table 1, where the radiation pattern of the multi-beam antenna refers to ITU-S.672 [39] and other parameters conform to the DVB-S2X interface specification [16]. Moreover, in order to fully verify the proposed BH algorithm, the maximum rain attenuations of each beam are chosen as [35.8, 19, 11.65, 6.43, 23] dB to simulate light or heavy rains. The uneven traffic demand of each beam is set as [70, 530, 550, 600, 320] Mbps. The genetic algorithm parameters are shown in Table 2. For \(K\) = 5 beams in the system, as previously stated, chromosome length can be set as \(log_{2}K=3\). Other parameters can be appropriately set as the classic case of GA [25] to minimize the computation complexity while guaranteeing that the optimal solution can be found [37]. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Label** & **Value** \\ \hline Transmission standard & - & DVB-S2X \\ Beams number & \(K\) & 5 \\ Total bandwith & \(B_{W}\) & 400 Mhz \\ Beam hopping time slot & T & 100 ms \\ Beam hopping period & W & 256 \\ Transmitter power & P & 100 W \\ Carrier frequency & \(F\) & 20 GHz \\ Roll-off & \(\alpha\) & 0.05 \\ Altitude of satellite orbit & H & 35,786 km \\ Transmit antenna gain & \(G_{T}\) & 54 dB \\ Receive antenna gain & \(G_{R}\) & 38.5 dB \\ Propagation loss & \(L_{SL}\) & 210 dB \\ Traffic demand of each beam & - & [70, 530, 550, 600, 320] Mbps \\ Maximum rain attenuation of each beam & - & [35.8, 19, 11.65, 6.43, 23] dB \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Satellite system simulation parameters. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Crossover probability & 0.6 \\ Mutation probability & 0.02 \\ Chromosome length & 3 \\ Population & 100 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Genetic algorithm parameters. Firstly, long-term rain attenuation time series based on Dirac log-normal distribution is illustrated in Figure 4. To simplify calculation while ensuring sufficient validation of algorithm, the duration of one rainfall can be chosen from the long-term rain attenuation time series. It is shown in top right corner of Figure 4. Applying the above method to each beam, attenuation time series of one rainfall for each beam can be obtained with different maximum attenuation values. The rain attenuation time series are changed every one second. The reason is for both lower computation complexity and to satisfy beam hopping time slot \(<\)\(\frac{1}{\text{the rainy environment changing frequency}}<\) beam hopping period. On the basis of time-varying rain attenuation, SNR of each beam can be obtained by bringing Equation (3) into Equation (7), which is also time-varying. As previously stated, the interface specification of BH system conform to DVB-S2X, and the spectrum efficiency \(\eta_{ij}=f_{\text{DVB-S2X}}\left(SNR_{ij}\right)\) is a piecewise function defined in [16], where \(SNR_{ij}\) represents signal-to-noise ratio of \(i\)-th beam at \(j\)-th time slot. With this function, time-varying SNR leads to different spectrum efficiency. Then, spectrum efficiency of each beam is also a time-varying piecewise curve, as shown in Figure 5.  
Due to the finer gradation and extension number of MOCODs in DVB-S2X, when SNR of each beam changes during transmission under the effect of the time-varying attenuation, appropriate MOCOD can be chosen to obtain optima spectrum efficiency. Note that, in this simulation condition, the short-term rain attenuation time series based on the 'Event-on-Demand' model [40] can also be introduced to validate the proposed beam hopping method. Nevertheless, it is limited to the short term. Thus, without loss of generality, the Dirac log-normal distribution model is recommended (as in Section 2.3) for its time series including rain and no rain, long-term and short-term situations. Figure 4: Rain attenuation time series.