# Efficient Resource Allocation for Beam-Hopping-Based Multi-Satellite Communication Systems
## 5 Numerical Result  
In this section, we conduct performance simulations for the proposed offline and online algorithms. We consider a multi-satellite system consisting of \(J=3\) satellites, and each serves \(M=20\) cells on the Earth. The total available system bandwidth is \(B_{tot}=210\) MHz, with each satellite using \(N=7\) subcarriers of bandwidth \(B_{sc}=30\) MHz. We set the length of a time segment as \(T_{seg}=1\) s, and the length of a time-slot as \(T_{s}=100\) ms. Meanwhile, we assume that the traffic demand of each cell follows the Poisson distribution. To compare with traditional non-beam-hopping systems, we propose an average resource allocation method. Average resource allocation means that satellites do not consider the current channel conditions and traffic requirements of different cells, and we allocate time-frequency resources and power resources equally to each cell. Due to the absence of beam-hopping, the value of all elements \(a^{j}_{m,nk}\) in \(\mathbf{A}^{j}_{m}\) is 1, and \(p^{j}_{m,nk}\) in \(\mathbf{P}^{j}_{m}\) is equal to \(P_{max}/NK\) for satellite \(j\).  
Figure 1 shows the time-frequency resource allocation result of a certain satellite in the system. In Figure 1, five random cells are selected from the twenty cells it serves, and each is represented by a pattern. And when the time-frequency resource corresponding to a grid is allocated to one of the five cells, the grid is filled with the corresponding pattern. From Figure 1, we can see that some cells will be allocated more time-frequency resource grids, but other cells will be allocated fewer time-frequency resource grids. This is due to the different traffic demands and channel conditions of cells. And this reflects the advantage of on-demand allocation in the beam-hopping system.  
Figure 2 shows the convergence performance of power allocation using the SCA algorithm under different iteration steps \(\eta\). It is observed that, although the iteration step will not affect the final convergence result, it will have an impact on the convergence speed of the algorithm. Therefore, in the subsequent simulation, we set the iteration step size to 0.9 and the maximum number of iterations to 15.  
Figure 3 shows the system throughput performance of the proposed offline algorithm and online algorithm under different satellite transmission powers. We can find that, compared with the average resource allocation with no beam-hopping, both proposed algorithms achieve 1.5 \(\sim\) 1.8 times performance gain. From Figure 3, we can also see that,  
compared with the upper bound obtained by the offline algorithm, the online algorithm has a performance gap of about 20%.  
In Figure 4, we randomly select ten satellite beams in the system, count their respective traffic demand, and compare them with the single-beam throughput achieved by applying the proposed algorithms. It is observed that, when the average resource allocation is applied, the throughput of some beams is lower than the traffic demand, such as beam 8 and beam 9. However, when the offline algorithm or online algorithm is applied, because relevant constraints are added to the optimization, we can ensure that the traffic demand of all beams is met.  
Figure 1: Time-frequency resource allocation pattern.  
Figure 2: Convergence of the SCA Algorithm in power allocation.