## 3 LEO Satellite Network Model  
### System Architecture  
The figure shows the polar-orbit Walker constellation configuration. Except for the first orbit and the \(N\)th orbit, each satellite communicates with four satellites in the front, rear, left and right through four ISLs. There are two seams between the first orbit and the \(N\)th orbit. There is no link between the seams, so the satellites in the first orbit and the \(N\)th orbit are only connected to three satellites. As the satellite passes through the polar region, it constantly adjusts the position and pointing of the inter-plane laser terminal to ensure that the connection status of the inter-plane ISL remains unchanged. From the perspective of network routing, the propagation delay of the inter-plane ISL changes, while the connection relationship of the satellite laser terminal interface remains unchanged. In the polar-orbit Walker constellation, the length of intra-plane ISLs remains constant, while the length of inter-plane ISLs continues to change. The symbols for the constellation parameters are as follows: \(R\) represents the orbit radius, \(N\) represents the number of orbits, \(M\) represents the number of satellites in each orbit, \(\varphi\) represents the phase difference of satellites in adjacent orbits and \(RI\) represents the right ascension difference of the ascending nodes of adjacent orbits. The length of the inter-plane intersatellite link can be expressed as  
\[L_{\flat}=\sqrt{2}R\sqrt{1-\cos\left(\frac{360^{\circ}}{M}\right)}. \tag{1}\]  
The length of the inter-plane intersatellite link is related to the latitude. As shown in Figure 1, take a point \(A^{\prime}\) on the orbit where satellite \(B\) is located, and take a point \(B^{\prime}\) on the orbit where satellite \(A\) is located, so that \(AA^{\prime}\) and \(BB^{\prime}\) are parallel to the equatorial plane. Then, \(AA^{\prime}\ BB^{\prime}\) is an isosceles trapezoid, and its side length calculation formula is as follows:  
\[AA^{\prime}=\sqrt{2}R\sqrt{1-\cos(RI)}\cos(latA), \tag{2}\]  
\[BB^{\prime}=\sqrt{2}R\sqrt{1-\cos(RI)}\cos(latB), \tag{3}\]  
\[AB^{\prime}=\sqrt{2}R\sqrt{1-\cos(\varphi)}, \tag{4}\]  
where \(latA\) represents the latitude of satellite \(A\) and \(latB\) represents the latitude of satellite \(B\). If satellites \(A\) and \(B\) are located in the southern or northern hemisphere at the same time, \(|latA-latB|=\varphi\); if satellites \(A\) and \(B\) are located in different hemispheres, \(|latA|+|latB|=\varphi\)[7]. From the isosceles trapezoid waist length formula, we can know that the length of the inter-plane intersatellite link is [7]  
\[L_{h}=\sqrt{AB^{\prime 2}+AA^{\prime}\cdot BB^{\prime}}=\sqrt{2}R\sqrt{1-\cos \left(\varphi\right)\ +(1-\cos(RI))\cos(latA)\cos(latB)}. \tag{5}\]  
The data links between a satellite and ground terminals, ground stations and other satellites are called user links, feeder links and inter-satellite links, respectively. As shown in Figure 2, according to the minimum pitch angle for satellite link establishment and the scale of the satellite constellation, a ground gateway station can establish links with multiple satellites. Ground stations, satellites, and the data links connecting them together form an LEO satellite transmission network. Due to the existence of inter-satellite links, there are different paths in the transmission network that can be used for data transmission for specific services, so routing problems arise. The traffic carried in the user link is the traffic input between satellites and between satellites and ground base stations. Due to the movement of satellites, user links are constantly switched, and the business requirements in the transmission network continue to change.