## 5 Simulation Results
This may be attributed to a mismatch between the scaling factor \(\rho\) in Algorithm 1 and the number of GPFs. Validation confirms that adjusting the scaling factor \(\rho\) can improve algorithm performance. The algorithm achieves its maximum performance when the number of GPFs equals the sum of the maximum connectable devices for each AP. As the number of GPFs gradually increases, the total system energy efficiency shows a decreasing trend. This is due to intensified competition for connections among GPFs in each time slot as the number of GPFs increases, leading to some GPFs being unable to establish connections with APs. Improvement can be achieved by increasing the number of APs or enhancing the number of connectable devices for each AP. It is evident that, under most scenarios of device numbers, the performance of the proposed algorithm is superior to the other three algorithms. The curve of the energy efficiency of the proposed algorithm changing with the number of ground devices is improved by 7.13%, 31.15%, and 4.23%, respectively, compared with the RP algorithm, EP algorithm, and RS algorithm. Figure 8: Total system energy efficiency with respect to throughput load rate. In the table below, we list the quantitative indicators of the proposed algorithm's improvement in energy efficiency. It can be seen that the proposed algorithm has a significant improvement compared with the EP algorithm, but the optimization effect is not obvious compared with the other two algorithms. This may be because the number of APs in the simulation scenario is small and the users are relatively close to each other, so the effect of equipment selection and path optimization is not obvious enough. However, in future work, we will study the larger number and distribution of users, and it is expected that more ideal results can be obtained. In addition to the analysis of simulation performance, we also analyze the limitations and computational complexity of the algorithm. First of all, in terms of algorithm limitations, compared with the RP algorithm, the proposed algorithm considers the path optimization of the AP and will perform better in scenarios where the AP needs to move in a large range. Compared with the AP algorithm, the proposed algorithm can optimize the transmit power of GPFs, adapt to the different communication needs of each GPF, and greatly improve the energy efficiency of the system. Compared with the RS algorithm, the proposed algorithm can optimize the device selection of the GPF and ensure that the system can perform better in scenarios where a large number of GPFs are connected to the network. The proposed algorithm breaks through the limitations of GPF's large-scale distribution, multi-service communication, and large-scale access. Therefore, the proposed algorithm has strong applicability. However, when dealing with larger-scale state space and action space problems in the future, the proposed algorithm needs to perform a certain discretization of the state space and action space, and the training time is long, and sensitive to parameter selection. This is also the limitation of the proposed algorithm. In this article, Algorithm 1 is used to optimize the flight path of APs, and its computational complexity is \(O(KTM)\), where \(K\) represents the number of iterations, \(T\) represents the number of task slots, and \(M\) represents the number of APs. Algorithm 2 is used to optimize the device selection and power control of GPFs, and its computational complexity is \(O(KTN)\), where \(N\) represents the number of GPFs. The computational complexity of the overall algorithm flow is \(O(KT(M+N))\). As shown in Table 4, except for the slightly lower complexity of the RP algorithm, the complexity of the other two benchmark algorithms is consistent with the proposed algorithm. Figure 9: Total system energy efficiency with respect to the number of ground devices.