## 3 System Model
### Satellite Coverage and Transmission Model  
#### 3.3.1 LEO Satellite Coverage Model  
LEO satellite can only cover users in a certain time and space range to complete the transmission of information. The geometric relationship between LEO satellite and user is shown in Figure 3.  
In Figure 3, O is the geocentric, \(R_{e}\) is the earth radius, \(h\) is the LEO satellite orbit altitude, and \(P\) represents the ground user; at \(t_{0}\) time, the maximum elevation of the ground user is \(\theta_{\max}\), and the LEO satellite position and the sub-satellite points are \(S^{\prime}\) and \(M\). At \(t\) time, the LEO satellite position and satellite sub-satellite points are \(S\) and \(N\). Furthermore, \(\gamma(t_{0})\), \(\gamma(t)\) and \(\psi(t)\) represent the corresponding geocentric angles between \(P\) and \(M\), \(P\) and \(N\), and \(M\) and \(N\), respectively. In addition, \(\theta(t)\) represents  
Figure 2: Network control architecture of the hierarchical and domain-controlled SIN architecture. the elevation of the ground user at time \(t\); \(\theta(t)\) is the minimum elevation of the ground user, and the corresponding maximum geocentric angle at time \(t\) is \(\gamma_{\max}\). According to the spherical triangle _PMN_ and the triangle _OPS_ shown in Figure 3, we can obtain the following:  
\[\cos\gamma(t)=\cos\psi(t)\cos\gamma(t_{0}); \tag{1}\]  
\[\gamma(t)=arc\ cos\bigg{(}\frac{R_{e}}{R_{e}+h}\cos\theta(t)\bigg{)}-\theta(t). \tag{2}\]  
The effective coverage time \(t_{c}\) of LEO satellite to ground users is  
\[t_{c}=\frac{2}{\omega}\psi(t)=\frac{2}{\omega}arc\ cos\bigg{(}\frac{\cos\gamma _{\max}}{\cos\gamma(t_{0})}\bigg{)}, \tag{3}\]  
where \(\omega=\omega_{s}-\omega_{c}i_{0}\) is the angular velocity of a satellite in the Earth-centered, Earth-fixed, (ECEF) coordinate system, \(\omega_{s}\) is the angular velocity of a satellite in the Earth-centered inertial (ECI) coordinate system, \(\omega_{e}\) is the angular velocity of the earth's rotation under ECI, and \(i_{0}\) is the orbital inclination angle. Ground users are randomly distributed. We assume that the distance from the ground user to the sub-satellite point obeys a uniform distribution. Therefore, when the LEO satellite covers ground users, \(\gamma(t_{0})\) satisfies the \(U(0\sim\gamma_{\max})\) uniform distribution. The probability density function \(f_{\gamma(t_{0})}(\gamma(t_{0}))\) of \(\gamma(t_{0})\) is  
\[f_{\gamma(t_{0})}(\gamma(t_{0}))=\left\{ \begin{array}{c}1/\gamma_{\max},0 \leq\gamma(t_{0})<\gamma_{\max}\\ 0,\text{ Other}\end{array} \right.. \tag{4}\]  
According to Equations (3) and (4), the cumulative distribution function of coverage time \(t_{c}\) is  
\[ \begin{array}{l}F_{T_{c}}(t_{c})=P\Big{(}\frac{2}{\omega}arc\ cos\Big{(} \frac{\cos\gamma_{\max}}{\cos\gamma(t_{0})}\Big{)}\leq t_{c}\Big{)}=1-\frac{1 }{\gamma_{\max}}arc\ cos\Big{(}\cos\gamma_{\max}/\cos\frac{\omega t_{c}}{2} \Big{)},\\ 0<t_{c}\leq T_{m}\end{array}  \tag{5}\]  
where \(T_{m}\) represents the maximum effective coverage time of satellite to ground users. When \(\gamma(t_{0})=0\), according to Equation (3), we can get  
\[T_{m}=\max(t_{c})=2\gamma_{\max}/\omega. \tag{6}\]  
Figure 3: Geometric diagram of low Earth orbit (LEO) satellite and user. According to Equation (5), the probability density function of coverage time \(f_{t_{c}}(t_{c})\) is  
\[f_{t_{c}}(t_{c})=\left\{ \begin{array}{c}\frac{\omega cosg_{max}tan(\omega t_{c}/ 2)}{2\gamma_{\max}\sqrt{\cos^{2}(\omega t_{c}/2)-\cos^{2}\gamma_{\max}}},0<t_{ c}\leq\frac{2\gamma_{\max}}{\omega}\\ 0\text{ Other}\end{array}