## 5 Performance Evaluation  
For LSNs, the communication delay is a key optimization indicator. Compared to the link capacity, reducing the delay is more important in improving the network performance. Therefore, we assigned weight \(w_{1}=0.3\) to the delay, and weight \(w_{2}=0.7\) to the remaining capacity in the simulation, giving more importance to the delay. The performance of the proposed GAGR algorithm is assessed using several performance indicators, such as RTT, ISL utilization, the ratio of received/sent traffic, and the hop count, and compared to the VTSR algorithm [16] and FW routing algorithm [17]. VTSR is a DFS algorithm based on an SDN and virtual topology; the FW algorithm utilizes the idea of dynamic programming to calculate the shortest path between any two satellite nodes whenever the topology state changes. To evaluate its performance in terms of the link capacity and flow distribution within the network, we utilize the UDP protocol and TCP protocol. We selected 100 pairs of cities from the set of the 100 most populous cities to represent 100 commodities, and these three routing algorithms were simulated for 60 s in the Telesat simulation constellation provided by Hypatia v1. Telesat's satellites use laser ISLs to communicate with each other. A radio up-link from the source GS to the ingress satellite, some laser ISLs, and a radio down-link from the egress satellite to the destination GS form an end-to-end path between two GSs. In order to simulate a case where the maximum capacity is insufficient, we set the flow size of each commodity to about 100 Mbit, which is much higher than the maximum capacity of the link, 32 Mbit/s. In this way, we can observe the behavior and bottlenecks of the link under a high load. Its other attributes are shown in Table 2.  
The simulation environment is created using Hypatia on a personal computer configured as follows: Core i5-8300H 2.3 GHz CPU, 16 GB RAM, and Ubuntu 20.04. Hypatia converts the continuous motion of satellites and the consequent changes in their paths into discrete processes. Although the delay on the path is constantly updated, the forwarding status is only recalculated at a fixed time step. Moreover, we use commercial solvers such as Gurobi 10.0.3 to solve the linear relaxation problems. Firstly, we calculated the RTTs of all commodities under the TCP protocol under the three routing algorithms, as shown in Figure 3. The simulation time is within [0, 45] s; our proposed GAGR algorithm has an average RTT of 404.02 ms, while the FW algorithm has an average RTT of 469.57 ms, and the VTSR algorithm has an average RTT of 453.93 ms. The average RTT ratio of the GAGR algorithm to the FW algorithm is 0.86, and the average RTT ratio to VTSR is 0.89. The average RTT of our proposed routing algorithm is better after 45 s; our proposed GAGR algorithm has an average round-trip time of 334.93 ms, while the FW algorithm has an average round-trip time of 482.01 ms, and the VTSR algorithm has an average round-trip time of 438.84 s. Compared with the FW algorithm, the average RTT ratio of GAGR is 0.69, and, compared with the VTSR algorithm, it is 0.76. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Attribute** & **Value** \\ \hline \multirow{5}{*}{TeleSat-1015} & Alt. (km) & 1015 \\  & Inclin. & 98.98 \\ \cline{1-1}  & N. Sat & 351 \\ \cline{1-1}  & N. Orb & 27 \\ \cline{1-1}  & N. Sat. Orb & 13 \\ \cline{1-1}  & Cap. ISL. (Mbit/s) & 32 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Constellation parameters. Figure 3: Round-trip times under TCP protocol. In Figure 4, we show the average round-trip times of 100 commodities within the simulation time based on the UDP protocol and sort them in ascending order. During the entire simulation time, the average RTT of the GAGR algorithm is 378.51 ms, the average RTT of the FW algorithm is 472.64 ms, and the average RTT of the VTSR algorithm is 448.42 ms. In Figure 5, we show the average round-trip times of 100 commodities within the simulation time based on the TCP protocol and sort them in ascending order.