## 4 Results
### Sensitivity Analysis  
1. Impact of the number of AEOSs and tasks  
Figure 9 shows the average total profit and profit acquisition rate of TPHO-MRC under different numbers of tasks and AEOSs. The profit acquisition rate is the ratio of the total profit of scheduled tasks to the total profit of all tasks, reflecting the degree of task overload. Figure 9a shows that the total profit of TPHO-MRC increases with an increasing number of AEOSs and tasks. In the 3-300 and 4-300 scenarios, the profit acquisition rates are equal to 1, implying that all tasks are scheduled. As the number of tasks increases, the total profit increases, and the profit acquisition rate decreases. This is because excessive tasks exceed the resource ability of the AEOSs, but the TPHO-MRC can make good selection decisions from excess tasks to improve the total profit. Figure 8: Average runtimes of algorithms. The runtime curves of the four TPHO-based algorithms nearly overlap. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{1}{c}{**Number of**} & \multicolumn{5}{c}{**Runtime(s)**} \\ \cline{2-7}
**AEOSs and Tasks** & **TPHO-MRC** & **TPHO-Linear** & **TPHO-Profit** & **TPHO-EFT** & **A-ALNS** & **SA** \\ \hline
3-600 & 17.058 & 15.171 & 16.965 & 14.994 & 276.333 & 46.696 \\
3-900 & 32.482 & 31.029 & 27.686 & 28.140 & 390.706 & 106.019 \\
3-1200 & 54.714 & 52.903 & 46.446 & 45.561 & 2587.387 & 189.259 \\
4-300 & 6.540 & 5.429 & 6.520 & 6.382 & 0.193 & 12.666 \\
4-600 & 21.270 & 19.693 & 21.954 & 18.828 & 370.466 & 52.161 \\
4-900 & 43.123 & 40.969 & 36.348 & 36.058 & 515.719 & 120.106 \\
4-1200 & 70.073 & 69.514 & 69.203 & 58.393 & 3391.777 & 210.437 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: _Cont._  
Figure 9: The trends of average total profit and profit acquisition rate. (**a**) Average total profit of TPHO-MRC under different numbers of tasks and AEOSs. (**b**) Average profit acquisition rate of TPHO-MRC under different numbers of tasks and AEOSs. Impact of satellite memory capacity  
In the proposed TPHO-MRC, the designed reward function and heuristic rule are related to the memory capacity constraint. To evaluate the performance of algorithms under looser memory capacity constraints, we compared TPHO-MRC with TPHO-Profit and TPHO-EFT in scenarios in which the memory capacities of the AEOSs are set to 450 GB (increased by 100). Table 9 shows the detailed average total profit and profit acquisition rate for the three algorithms and that TPHO-MRC still obtains the best results among them. A significant test was also performed for the pairs of the three algorithms. TPHO-MRC significantly outperforms TPHO-Profit (\(p=0.003\)) and TPHO-EFT (\(p=0.003\)) in scenarios with relaxed memory capacity constraints. The total profit increases with a rise in memory capacity. Figure 10 shows the rate of increase in total profit for the three algorithms. The rise in total profit increases as the task number increases and decreases as the AEOS number increases, aligning with the normal relationship between task load and capacity demand. The TPHO-Profit result increases the most and that of TPHO-EFT increases the least, except in 4-600 scenarios in which all tasks are executed by TPHO-Profit and TPHO-MTC. Notably, TPHO-EFT outperforms TPHO-Profit in 2-AEOS and 3-AEOS scenarios, but the opposite conclusion can be drawn in 4-AEOS scenarios, as shown in Tables 6 and 9. This indicates that the MRC rule shows more prominent effectiveness in scenarios with more AEOSs. The methods and experimental conclusions are discussed further in Section 5. Figure 10: Increase in total profit for TPHO-MRC, TPHO-Profit, and TPHO-EFT.