## 4 Numerical Simulations and Analysis
Figure 6 illustrates the iteration graph of proposed method based on GA, where the Y-axis represents the value of objective function \(\min\sum_{i}^{K}|R_{i}-\hat{R}_{i}|^{2}\). From Figure 6, it can be seen that after 40 more iterations, the average value of objective function of population decreases and finally approaches the minimum value of objective function, which means that the algorithm has good convergence. The reason is that multi-action selection method can reduce computational complexity and the solution space can be adjusted adaptively and effectively reduced by GA [37]. As stated above, the design of BHTP and the corresponding algorithms mainly operate at system resource management unit of network control center. Therefore, powerful processing capabilities of network control center can ensure the strong engineering practicability of the genetic algorithm. As previously stated, time-varying rain attenuation brings changeable spectrum efficiency for each beam, which makes the quantity and arrangement of beam hopping  
Figure 5: Spectrum efficiency of each beam. Figure 6: GA optimization iteration graph. time slots become coupled. By employing GA as a global optimization method, the dynamic allocation algorithm is proposed to solve the nonlinear problem with multiple constrained conditions. It can obtain not only the number of time slots allocated to each beam, but also the arrangement of time slots in the beam hopping period to determine the order in which the beam is activated. The time slot allocation results of one period for each beam are shown in Figure 7, where the X-axis represents time slot index and the color-block represents the time slot allocated to corresponding beam. Moreover, the following parameters are introduced to evaluate the system performance. System actual throughput or capacity: it is the part of the allocated capacity not exceeding the requested traffic under the resource-constrained condition  
\[C_{A}=\sum_{i=1}^{K}\min\{\hat{R}_{i},R_{i}\} \tag{13}\]  
where \(\hat{R}_{i}\) is the request traffic of each beam and \(R_{i}\) is the system offered. Traffic matching ratio: it describes the satisfaction degree of the actual capacity with respect to the total requested traffic. It can be defined as:  
\[\rho_{s}=\frac{C_{A}}{\sum_{i=1}^{K}\hat{R}_{i}} \tag{14}\]  
Finally, the traffic matching ratio of the beam hopping system is shown in Figure 8. Here, BH method with constant rain attenuation represents the conventional beam hopping time slot allocation method which assumes constant rain attenuation as in [7] and [37]. Multi-beam with ACM represents a traditional fixed multi-beams HTS system employing ACM referring to DVB-S2X. Multi-beam without ACM represents the same but without utilizing ACM. BH method based on GA is our proposed algorithm under the effect of time-varying rain attenuation. For BH method with constant rain attenuation, it is obvious that the redundant rain loss margin will cause low efficiency utilization of spectrum resource, which leads to the unsatisfaction of demand traffic. For Multi-beam with ACM, although ACM is employed to deal with changeable communication link quality, the fixed resource allocation manner of multi-beam system could not manage the uneven traffic demand among beams. Figure 4.0 Beam hopping time slots allocation results. Fortunately, compared with conventional algorithms, the proposed BH method still obtains better performance, because the proposed method introduces GA to solve the two-dimensional time-space optimization problem of beam hopping under time-varying rain attenuation. The proposed method flexibly allocates and efficiently utilizes system resources to meet the traffic demand as much as possible. More importantly, different from conventional two-step strategy, through the proposed BH method, it obtains not only the number of time slots allocated to each beam, but also the arrangement of time slots in beam hopping period, which reduces the computational complexity.