## 2 System Model and Problem Formulation
### Signal Model  
As shown in Figure 1, when multiple satellites share the same frequency bands, the UT will receive communication and interfering signals simultaneously. Moreover, satellite-to-ground communication undergoes several stages of propagation and attenuation. Free space path loss, atmospheric absorption, and rain attenuation are mainly considered in path loss (PL). Then, PL can be given as follows:  
\[PL^{S}=PL_{b}(d^{S},f_{c})+PL_{g}(el^{S},f_{c})+PL_{r}(el^{S}). \tag{3}\]  
where \(PL_{b}(d^{S},f_{c})=32.45+20\log_{10}(f_{c})+20\log_{10}(d^{S})\) is the free space path loss [23], \(PL_{g}(el^{S},f_{c})=\frac{A_{\text{menth}}(f_{c})}{\sin(el^{S})}\) is the atmospheric absorption [23]. Specifically, \(d^{S}\) is the distance between the satellite and the UT, \(f_{c}\) is the carrier frequency, \(A_{\text{zenith}}(f_{c})\) is the zenith attenuation, and \(el^{S}\) is the elevation angle. Moreover, rain attenuation is the most dominant cause of signal degradation for the Ka-band, and it can be represented as follows: \(PL_{r}(el^{S})=A_{0.01}\frac{p-(0.655+0.033\ln(p)-0.045\ln(A_{0.01})-\beta(1-p )\sin(el^{S}))}{0.01}\)[24], in which \(A_{0.01}\) is the predicted attenuation exceeded for 0.01% of an average year, \(p\) is the exceedance probability, \(el^{S}\) is the elevation angle, and \(\beta\) is given as follows:  
\[\beta=\left\{\begin{array}{cc}0,&p>1\%\text{ or }|lat_{U}|\geq 36^{\circ}\\ -0.005(|lat_{U}|-36),&p<1\%\text{ and }|lat_{S}|<36^{\circ}\text{ and }el^{S}\geq 25^{\circ}\\ -0.005(|lat_{U}|-36)+1.8-4.25\sin(el^{S}),&\text{other}\end{array}\right. \tag{4}\]  
and \(lat_{U}\) is the latitude of the UT. The above rain attenuation model has been used in the existing research [25; 26]. Moreover, the effect of rain attenuation during radio propagation and the new rain attenuation model have been studied in [27; 28; 29; 30; 31]; however, this is not the focus of this paper.  
Figure 1: The dense LEO satellite communication system.  
The received signal strength is given as follows:  
\[P^{\mathcal{S}}=p_{t}+G_{t}^{\mathcal{S}}+G_{r}^{\mathcal{S}}-PL^{\mathcal{S}}, \tag{5}\]  
where the subscripts \(t\) and \(r\) denote the transmitter and receiver. \(\mathcal{S}\) is the set of visible satellites, which is composed of the serving satellite \(S\) and the interfering satellites \(I\). \(p_{t}\) denotes the satellite antenna transmit power. \(G_{t}^{\mathcal{S}}\) and \(G_{r}^{\mathcal{S}}\) are the antenna transmitting and receiving gain according to (1) and (2). Moreover, the received signal strength of serving signal and interfering signal are obtained by (5), and can be represented as \(P^{\mathcal{S}}\) and \(P^{I}\).  
Thus, the SINR at the receiver is written as follows:  
\[\mathrm{SINR(n)}=-10\log_{10}(10^{-0.1(P^{\mathcal{S}}(n)-KTB_{w})}+10^{-0.1(P ^{\mathcal{S}}(n)-P^{I}(n))}), \tag{6}\]  
where \(P^{I}=\sum_{q=1}^{Q}\sum_{d=1}^{D}P^{l_{q,d}}\), \(Q\) is the total number of interfering satellites at the time index \(n\), \(D\) is the total number of interfering links generated by an interfering satellite, which is equal to the number of beams that use the same frequency bands. \(l_{g,d}\) denotes the \(d\)th interfering link of the \(q\)th interfering satellite. \(K\) is the Boltzmann's constant and, given as \(-228.6\) dBW/K/Hz, \(T\) represents the noise temperature, and \(B_{w}\) means the bandwidth of the LEO satellite communication system.