# The Optimal Deployment Strategy of Mega-Constellation Based on Markov Decision Process
## 4 Case Study
### Parameters of Simulation Model  
Given the collision probability \(p_{t}\), the Markov transition matrix was computed by space orbital debris flux, as shown in Table 4.  
Based on historical data, the probability of aunch vehicle failure was denoted by \(P\left(e_{of}\right)=0.03\). Let the collision probability \(P\left(e_{sf}|e_{c}\right)=0.95\) because the consequencse would be severe when collisions occur during operation periods. Therefore, the probability due to a subsystem malfunction is relatively small, \(P\left(e_{sf}|e_{c}^{-1}\right)=0.03\). Given the three realizations of the probability distribution, let \(T=10\); then the probability is shown as in Table 5.  
Based on the collision probability within the time steps, the probability at different states was calculated, as shown in Table 6. \begin{table}
\begin{tabular}{c c c c} \hline \hline \(p_{t}\) & \(P(p_{t+1}=0|p_{t})\) & \(P(p_{t+1}=1|p_{t})\) & \(P(p_{t+1}=2|p_{t})\) \\ \hline
0 & 0.4431 & 0.1053 & 0.4516 \\
1 & 0.2483 & 0.0156 & 0.7361 \\
2 & 0.0672 & 0.1693 & 0.7635 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The transition matrix of collision probability. \begin{table}
\begin{tabular}{c c c} \hline \hline \(p_{t}\) & \(P(p_{t+1}=0|p_{t})\) & \(P(p_{t+1}=1|p_{t})\) & \(P(p_{t+1}=2|p_{t})\) \\ \hline
0 & 0.4431 & 0.1053 & 0.4516 \\
1 & 0.2483 & 0.0156 & 0.7361 \\
2 & 0.0672 & 0.1693 & 0.7635 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The transition matrix of collision probability. \begin{table}
\begin{tabular}{c c} \hline \hline
**The Solving Process of Net Profit Function** \\ \hline
1: Set \(NP_{0}(x)=0\) \\
2: For \(t=1:T\), \(NP_{t+1}(x)=\max\limits_{u_{t}}E[NP(n_{t},p_{t};u_{t})+NP_{t}(x)]\) \\
3: Find \(u_{t}^{*}\) and finish solving \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Optimal deployment strategy flow of finding. Assuming that the mega-constellation needs at least 30 satellites to operate normally in order to provide service for customers in the target region, the constellation consisted of 60 satellites and the upper limit of the launch capability was 10 satellites. Therefore, \(u_{t}\in\{0,1,\cdots,10\}\), \(n_{t}\in\{10,11,\cdots,60\}\), and the size of state space was 153, as shown in Table 7.  
There is an assumption that at least 30 satellites need to be operated on orbit to assure the system normal performance, so the boundary condition was \(\varepsilon=30\). Given the capability of launching additional satellites, the optimal deployment strategies are shown as Figure 7, considering different state variables \(x_{t}\). \begin{table}
\begin{tabular}{c c c} \hline \hline
**State** & **Collision Probability** & **Value** \\ \hline decreased & \(P(\varepsilon_{c}|p_{t}=0)\) & \(1.7608\times 10^{-4}\) \\ maintained & \(P(\varepsilon_{c}|p_{t}=1)\) & \(1.8471\times 10^{-4}\) \\ increased & \(P(\varepsilon_{c}|p_{t}=2)\) & \(1.8723\times 10^{-4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Collision probability during different states. \begin{table}
\begin{tabular}{c c} \hline \hline
**State** & **Collision Probability** \\ \hline \(x_{t}=0\) & \((10,0)\) \\ \(x_{t}=1\) & \((10,1)\) \\ \(x_{t}=2\) & \((10,2)\) \\ \(x_{t}=3\) & \((11,0)\) \\ \(\vdots\) & \(\vdots\) \\ \(x_{t}=151\) & \((60,1)\) \\ \(x_{t}=152\) & \((60,2)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: State space of [PERSON] decision process. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Value** & **Parameter** & **Value** \\ \hline \(C_{ex}\) & 50,000 & \(E_{op}\) & 50,000 \\ \(\alpha_{coo}\) & 600 & \(\alpha_{lan}\) & 350 \\ \(\alpha_{ord}\) & 600 & \(\alpha_{op}\) & 30 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: The data of simulation parameters. Figure 7: The optimal deployment decision of state variables. According to the simulation parameters, the optimal decision actions which are denoted as \(u^{*}\) are shown as Figure 8. Four scenarios could be divided with different deployment stages. 1. 10\(\sim\)19 satellites on-orbit operation.