## 3 System Model and Problem Formulation
### Channel Model  
In LEO mega-satellite communication networks, transmission channels can be divided into two main categories:  
* Inter-satellite link channel model: In this paper, the inter-satellite link channel model we used is the free-space path loss model. The path loss can be expressed as follows: \[FSPL=(\frac{4\pi Rf}{c})^{2}\] (4) where \(R\) is the communication distance, \(c\) is the speed of light (in m/s), and \(f\) is the communications center frequency (in Hz) of inter-satellite links. The received power \(P_{R}\) can be calculated using the following equation: \[P_{R}=P_{Tx}\cdot G_{T}\cdot G_{R}\cdot\frac{c^{2}}{(4\pi Rf)^{2}L}\] (5) where \(P_{Tx}\) is the transmit power, and \(G_{T}\) and \(G_{R}\) are the transmit and receive antenna gains, respectively. \(L\) is the additional system loss. The data rate (in bps) of inter-satellite links can be expressed as follows: \[R_{ISL}=B\cdot log_{2}(1+\frac{P_{R}}{N_{0}B})\] (6) here, \(N_{0}\) is the noise power spectral density, and \(B\) is the channel bandwidth. * Satellite-ground link channel model: This includes the transmission channel between user terminals and satellites as well as between ground stations and satellites. From the ground node to the satellite node, the wireless signal propagation experiences the ground, the atmosphere, and the space environment, and it needs to consider the influence of various factors such as the ground shadow effect, the multipath effect, the atmospheric propagation loss, the free space propagation loss, etc. In this paper, we refer to the relevant parameters of 3 GPP TR 38.811 [24] to establish the satellite-ground link channel model. Specifically, the parameter set of the urban scenario in Table 6.7.2 is used. The Path Loss (PL) is composed of the following components: \begin{table}
\begin{tabular}{p{113.8 pt} p{284.5 pt}} \hline \hline
**Term** & **Definition** \\ \hline Non-Terrestrial Network k(NTN) & Communication networks that include satellites or other non-terrestrial nodes. \\
3 GPP & 3 rd Generation Partnership Project; develops mobile communication standards. \\  & Satellites: Orbiting communication nodes. \\ Node Types & Ground Stations: Fixed communication points on Earth. \\  & User Terminals: Devices that connect to the network. \\  & Inter-Satellite Links (ISLs): connections between satellites. \\ Links & Ground-to-satellite links: Connections between ground stations and satellites. \\  & User-to-satellite links: Connections between user terminals and satellites. \\ Network topology & The arrangement of nodes and links in the network. \\  & Link capacity: The maximum amount of data that a communication link can transmit per unit of time. \\ Key constraints & Note capacity: Maximum amount of data that can be processed per unit of time by a satellite or ground station. \\ Graph Neural Network (GNN) & A type of neural network designed to operate on graph structures. \\ Deep Reinforcement Learning (DRL) & A machine learning technique that enables agents to learn optimal actions through trial and error. \\ Service Function Chaining (SFC) & A networking concept where a set of network functions is applied to a traffic flow in a specific order. \\ \hline \hline \end{tabular}
\end{table} 
Table 2: List of terms. \[PL=PL_{b}+PL_{g}+PL_{s}+PL_{e} \tag{7}\]  
where \(PL\) is the total path loss (in dB), \(PL_{b}\) is the basic path loss (in dB), \(PL_{g}\) is the attenuation due to atmospheric gasses (in dB), \(PL_{s}\) is the attenuation due to either ionospheric or tropospheric scintillation in dB, and \(PL_{e}\) is the building entry loss (in dB). The basic path loss in units of dB is modeled as follows:  
\[PL_{b}=FSPL(s,f_{c})+SF+CL(\alpha,f_{c}) \tag{8}\]  
where \(FSPL(s,f_{c})\) is the free space path loss, \(CL(\alpha,f_{c})\) is the clutter loss, and \(SF\) is the shadow fading loss represented by a random number generated by the normal distribution, i.e., \(SF\sim N(0,\sigma^{2})\).