## 6 Simulation Results  
In order to verify the effectiveness of the proposed matching algorithm, preliminary simulations are conducted. The main simulation parameters are listed in Table 1. We compare the proposed approach with some state-of-the-art algorithms, including deep deterministic policy gradient (DDPG), deep Q-network (DQN), and two greedy methods. * For the first greedy method (abbreviated as Greedy 1), each LEO satellite will choose the \(L\) HAP drones with the highest channel gains within its visible range to establish connections. * For the second greedy method (abbreviated as Greedy 2), each LEO satellite will choose the \(L\) HAP drones with the longest remaining visible time within its visible range to establish connections. * For both Greedy 1 and Greedy 2, each HAP drone that has established a connection with an LEO satellite will choose the traffic type with the largest transmitted traffic value for transmission. Figure 3 illustrates the transmitted traffic values of the proposed matching approach in one time slot under episode lengths of 500, 1000, 1500, 2000, 2500, and 3000, respectively. When the length of the episode does not exceed 2000, the transmitted traffic values in one time slot increase significantly with the increase of the episode length. However, when the length of the episode exceeds 2000, the transmitted traffic values in one time slot are basically the same for various episode lengths. Thus, the length of the episode is set to 2000 in subsequent simulations, thereby saving computational resources while ensuring performance. Furthermore, it can be observed that for any episode length, the transmitted traffic value will first increase and then remain essentially stable, which validates the convergence of the proposed matching algorithm. Figure 4 illustrates the variation of the relative mean square error of Q-values obtained by the target network and the main network under learning rates of 0.15, 0.1, 0.08, and 0.05, respectively. As the learning rate \(\beta\) increases from 0.05 to 0.1, the rate of decrease in relative mean square error accelerates. Nevertheless, as the learning rate continues to increase from 0.1 to 0.15, the rate of decrease in relative mean square error remains almost unchanged, but its fluctuations will increase. Therefore, in order to balance the convergence speed and stability, we set the learning rate \(\beta\) to 0.1 in subsequent simulations. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Description** & **Notation** & **Value** \\ \hline Self-transition probability from on to on & \(p_{1}\) & 0.9 \\ Self-transition probability from off to off & \(p_{2}\) & 0.95 \\ Total number of traffic types & \(Q\) & 4 \\ Value factors & \(\mu_{q}\), \(q=1,2,3,4\) & \(1,2,3,4\) \\ Number of LEO satellites & \(J\) & 5 \\ Number of HAP drones & \(I\) & 20 \\ Number of beams for each LEO satellite & \(L\) & 4 \\ Beam bandwidth & \(W\) & 10 MHz \\ Number of time slots & \(T\) & 1000 \\ Length of time slot & \(T_{0}\) & 0.1 s \\ HAP drone altitude & \(h_{1}\) & 20 km \\ LEO altitude & \(h_{2}\) & 600 km \\ Carrier frequency & \(f_{c}\) & 10 GHz \\ Number of signaling & \(\kappa\) & 8 \\ Discount rate & \(\gamma\) & 0.98 \\ Learning rate & \(\beta\) & 0.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: System parameters. Figure 3: Transmitted traffic values in one time slot with different episode lengths. Figure 5 illustrates the total transmitted traffic values of different algorithms under varying HAP drone transmission powers. It can be seen that with the increase of the transmission power, the total transmitted traffic values of all algorithms increase. This is because, according to (2), increasing the transmission power of HAP drones can improve the traffic transmission rates, thereby increasing the total transmitted traffic value of the space-air-ground integrated network. From Figure 5 we can see that the proposed multi-agent deep RL algorithm is the best.