## 5 Computational Experiments
### Experimental Setup  
We establish an SEC simulation environment using MATLAB, as in [1; 21; 27], adopting six Walker Delta constellations A-F, listed in Table 3. The simulation is set to start at [20 Mar 2024 00:00:00.000 UTCG], with satellite coordinates obtained from the Satellite Tool Kit (STK). We conduct the experiments under three types of instances: small, medium, and large, resulting in 48 combinations (3 \(\times\) 16), detailed in Table 4. Each combination consists of 10 different testing instances. The altitudes of observation satellites are set to 500 km, with an average distance between satellites of 1000 km and 100 km for low-density and high-density instances, respectively. The task deadlines \(\sigma_{i}\) are randomly generated from [15; 25] for emergency cases and [15; 31] for normal cases. Additional parameters are detailed in Table 5.  
The proposed PITO is compared with three state-of-the-art algorithms: CNP [24], DETS [26], and ADMM [27]. The brief explanations of these algorithms are as follows.  
CNP is a distributed coordination mechanism based on market-like agreements, where agents bid on tasks announced by a manager agent, with the most suitable agent winning the contract.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Constellation** & **Altitude (km)** & **Inclination (deg)** & **Planes** & **Satellites (m)** \\ \hline A & 5000 & 97.4 & 2 & 6 \\ B & 5000 & 53.8 & 3 & 9 \\ C & 3000 & 60 & 4 & 16 \\ D & 480 & 97.4 & 3 & 24 \\ E & 550 & 60 & 6 & 36 \\ F & 780 & 86.4 & 6 & 66 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Satellite Constellations.  
DETS is a centerless method based on the directed diffusion algorithm, using a novel computing gradient that jointly considers communication and computation resources.  
ADMM is a distributed mechanism that uses binary variable relaxation to convert the original nonconvex problem into a linear programming problem. After solving it with distributed convex optimization, a variable recovery algorithm is applied to obtain a discrete solution.  
Then, we use the following three performance indicators:  
(1) \(RV\): the relative value of \(F(\theta)\) for an algorithm compared with others; we have  
\[RV=\frac{F(\theta)-F_{all}(\theta)}{F_{all}(\theta)} \tag{29}\]  
where \(F(\theta)\) is the objective value of an algorithm for a testing instance, and \(F_{all}(\theta)\) is the optimal objective value obtained by all algorithms for the same instance. A lower \(RV\) value means better performance. It helps to reduce variation among instances in the same combination.  
(2) \(CT\): the communication times, indicating the communication burden between satellites during task offloading.  
(3) \(RT\): the running time required for the task offloading process.  
To eliminate randomness, each algorithm is independently run 10 times for each instance. We then calculate average \(RV\) (\(aRV\)), average \(CT\) (\(aCT\)), and average \(RT\) (\(aRT\)), to evaluate their performance. All these algorithms are coded in MATLAB and run on a PC with an Intel Core i7-14700k CPU @ 5.6 GHz and 64 GB of RAM in the 64-bit Windows 11 operation system.