## 4 Results
### Experimental Setup  
We created a SEC network simulation environment to carry out experiments. Six Walker Delta constellations were developed, each with different elements in Table 2, including Orbcomm [57], Globalstar [58], and Starlink [59]. The simulation started on [1 June 2022 00:00:00.000 UTCG], and the required orbit elements were obtained using the software STK [60]. We conducted extensive experiments using three groups of testing instances: _small_, _medium_, and _large_. Table 3 shows the constellations (\(\kappa\)) and the number of applications (\(v\)) for each instance type. In total, there are \(3\times 2\times 3\) = 18 combinations and we generated 10 testing instances for each combination. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Constellation (\(\kappa\))** & **Altitude (km)** & **Inclination (deg)** & **Planes** & **Satellites (\(n\))** \\ \hline A [57] & 825 & 45 & 4 & 32 \\ B [58] & 1414 & 52 & 6 & 48 \\ C & 825 & 45 & 8 & 160 \\ D & 1110 & 53.8 & 12 & 300 \\ E & 1175 & 60 & 18 & 864 \\ F [59] & 1110 & 53.8 & 32 & 1600 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Satellite constellations used in simulation. In each texting instance, the edge server's properties such as computing capacity \(f_{k}\), processing energy consumption coefficient \(g_{w}\), standby power \(g_{s}\), and unit rental cost \(h_{e}^{k}\) for edge server \(u_{k}\in\) U are generated randomly from the ranges [5, 10], [\(1\times 10^{-28},2\times 10^{-28}\)], [0.1, 0.2], and [1, 2], respectively. The assumed values for transmission power \(g_{c}\), unit data transmission cost \(h_{d}\), and unit network rental cost \(h_{o}\) are 30, 0.02, and 0.1, respectively. The evaluation of the algorithm was performed using five real-world applications [61], and their task structures are shown in Figure 10. A testing instance is composed of a random selection of these applications. For any task \(m_{p}\), the amount of data \(d_{i,p}\) and workload \(e_{p}\) are randomly generated from [5, 10] and [1, 2], respectively. Table 4 provides a summary of the main parameter settings used in the simulation. We conducted a comparative analysis between our proposed algorithm and three existing algorithms, namely IMOPSOQ [27], MCHO [15], and NSGA-II [41] (using Al \begin{table}
\begin{tabular}{l c} \hline \hline
**Parameters** & **Value** \\ \hline Number of edge servers & \(n\) \\ The computing capacity \(f_{k}\) & [5, 10] Cycles/s \\ The processing energy consumption coefficient & [\(1\times 10^{-28},2\times 10^{-28}\)] \\ \(g_{w}\) & \\ The standby power \(g_{s}\) & [0.1, 0.2] W/s \\ The unit rental cost \(h_{e}^{k}\) & [1, 2] 5/s \\ The transmission power \(g_{c}\) & 30 W \\ The unit data transmission cost \(h_{d}\) & 0.02 5/MByte \\ The unit network rental cost \(h_{o}\) & 0.1 5/s \\ The communication capability of ISL \(r\) & 1 Gbps \\ The amount of data volume \(d_{i,p}\) & [5, 10] MByte \\ The workload \(e_{p}\) of a task \(m_{p}\) & [1, 2] Kcycles/Byte \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Simulation parameters. Figure 10: The structure of computation-intensive applications. (**a**) Montage; (**b**) CyberShake; (**c**) Epigenomics; (**d**) LIGO; (**e**) SIPHT. gorithm AM to meet the precedence constraints). The performance evaluation of these algorithms was based on three indicators:  
1. _Number of non-dominated solutions_ (_NS_) indicates the average quantity of nondominated solutions in each experiment. 2. _Hypervolume_ (_HV_) [62] represents the hypercube's size enclosed by individuals and a reference point in the target space. The reference point \(\mathbf{r}\) is normalized to a unit vector, and the enclosed volume of solution \(\mathbf{x}\) is calculated as follows: \[V(\mathbf{x})=\{\mathbf{y}\in\mathbf{O}|\mathbf{x}\prec\mathbf{y}\wedge\mathbf{y}\prec\mathbf{r}\}\] (31) where \(\mathbf{O}\) is the objective space; then, the hypervolume of \(\Psi_{nd}\) can be obtained by (32). \[HV(\Psi_{nd})=\underset{\mathbf{x}\in\Psi_{nd}}{\cup}V(\mathbf{x})\] (32)
3.