## 6 Evaluation and Analysis  
To evaluate the performance of the GF-TAR, LF-TAR, and MS-TAR algorithms, we perform the simulations with four satellite network topologies. All the topologies consist of 25 ground stations randomly chosen from the major ground stations around the world, as shown in Figure 3. Three GEOs in the Walker constellation with one orbital plane are also included in all the topologies. In the six-plane Star topology and seven-plane Star topology, we construct a Walker Star constellation with six and seven orbital planes, respectively. In the six-plane Delta topology and seven-plane Delta topology, we construct a Walker Delta constellation with six and seven orbital planes, respectively. The Walker Star and Walker Delta are two typical constellations [25; 31]. Each of the LEO orbital planes has 11 LEO satellites. Figure 3 shows the distribution of ground stations and satellites on the six-plane Star topology as an example of topologies. The LEO satellite has an orbit altitude of 500 km, an inclination of 90 degrees, and an orbital period of 5677 s. The GEO satellite has an orbit altitude of 35788.1 km, an inclination of 0 degrees, and an orbital period of 86170.5 s (approximately equal to 24 h). Due to the ground coverage time of LEO being commonly up to 5 min, we set the duration of the physical discrete topology to 1 min (i.e., \(\Delta\) is equal to 1 min). We set the simulation time \(T\) to 48 h and then obtain 2881 physical topologies for different moments. We jointly use the STK 11.6 and MATLAB 2018b to obtain the related access and distance information of the satellites as well as process the dates to obtain the physical topology matrixes. Then, we use the MATLAB for the simulations of the proposed algorithms and the benchmark algorithms. The simulations are performed on a computer with 2.3 GHz Intel Core i7-10875H CPU and 16 GB RAM.  
In the simulation, each LEO can create quantum links to four ground stations (i.e., \(l_{\text{g}}\) is equal to four) and each LEO can establish quantum links with four LEOs (i.e., \(l_{\text{b}}\) is equal to four). Each GEO can establish quantum links to eight LEOs or ground stations (i.e., \(l_{\text{h}}\) is equal to eight) and each LEO can establish the quantum link with one GEO. The keys required for the request \(k_{r}\) are randomly chosen from the set \(K_{\text{s}}=\{0.8\sigma,\,\varnothing,\,1.2\sigma\}\). The QKD key rates are set according to the key rate data in [11; 22]. Given that there is little research on the hybrid GEO/LEO quantum satellite network, we use four simple benchmark algorithms in this paper. The GEO-free benchmark with non-adaptive topology (Benchmark-GFU) and GEO-free benchmark with topology updating (Benchmark-GFU) use only LEOs for QKD. The LEO-first benchmark with non-adaptive topology (Benchmark-LFN) and LEO-first benchmark with topology updating (Benchmark-LFU) use both GEOs and LEOs for QKD, where the GEO-based QKD is used as an alternative plan for the LEO-based QKD. The Benchmark-GFN and Benchmark-LFN route on the physical topology without updating according to the rest of the time resources on the link. The Benchmark-GFU and Benchmark-LFU route on the physical topology with updating according to the rest of the time resources on the link. In Section 6.1, we will analyze the success probability and the max traffic supported on different topologies. In Section 6.2, we will compare the average number of relaying hops for the successful requests. The analysis of the total number of secret keys produced is conducted in Section 6.3.