# Load-Balancing Routing for LEO Satellite Network with Distributed Hops-Based Back-Pressure Strategy
## 3 System Model
### End-to-End Hops in LSN
#### 3.2.2 Intra-Plane Hops \(H_{v}\)  
According to the above analysis, inter-plane hops is determined by the longitude difference \(\Delta L_{0}\) between the ascending points of source-destination satellites. Correspondingly, intra-plane hops depend on the phase angle difference \(\Delta u\) between satellites. Since each intra-plane hop increases the phase angle by \(\Delta\Phi\) and each inter-plane hop increases the phase angle by \(\Delta f\), the phase angle of the destination satellite \(Sat_{2}\) can be expressed as:  
\[u_{2}=u_{1}+(H_{h}^{\rightarrow}\cdot\Delta f)+\underbrace{\left(H_{v}^{ \
earrow}\cdot\Delta\Phi\right)}_{\Delta\vec{u}}, \tag{3}\]  
where \(H_{h}^{\rightarrow}\) is the eastward inter-plane hops count and \(H_{v}^{\
earrow}\) is the eastward intra-plane hops count. In order to calculate intra-plane hops \(H_{v}\), the phase difference \(H_{h}^{\rightarrow}\cdot\Delta f\), resulting from \(H_{h}\), needs to be eliminated from overall phase difference \(\Delta u=u_{2}-u_{1}\). Similar to the calculation of the inter-plane hops count, the calculation of phase difference \(\Delta u\) distinguishes between east and west propagation directions can, respectively, be expressed by:  
\[\left\{\begin{array}{l}\Delta\vec{u}=\left(u_{2}-u_{1}-H_{h}^{\rightarrow} \cdot\Delta f\right)\mod\ 2\pi\\ \Delta\vec{u}=\left(u_{2}-u_{1}+H_{h}^{\leftarrow}\cdot\Delta f\right)\mod 2\pi,\end{array}\right. \tag{4}\]  
where \(\Delta f\) denotes the change of phase angle resulting from inter-plane hops. \(H_{h}^{\rightarrow}\) denotes the westward hops count and \(H_{h}^{\leftarrow}\) denotes the eastward hops count.  
Satellite orbits include ascending orbits (from southwest to northeast) and descending orbits (from northwest to southeast). The ascending and descending orbits can also be divided into forward hop and backward hop. In this work, we calculate inter-plane hops in four directions:  
\[\left\{\begin{array}{l}H_{v}^{\
earrow}=\left|\frac{\Delta\vec{u}}{\Delta \Phi}\right|\!,H_{v}^{\
earrow}=\left|\frac{\Delta\vec{u}}{\Delta\Phi}\right| \\ H_{v}^{\
earrow}=\left|\frac{2\pi-\Delta\vec{u}}{\Delta\Phi}\right|\!,H_{v}^{ \searrow}=\left|\frac{2\pi-\Delta\vec{u}}{\Delta\Phi}\right|\!,\end{array}\right. \tag{5}\]  
where \(\Delta\Phi\) denotes the phase difference between adjacent satellites in the same orbit; \(\Delta\vec{u}\) denotes the phase angle difference covered by westward intra-plane hops; \(\Delta\vec{u}\) demonstrates the phase angle difference covered by eastward intra-plane hops; \(H_{v}^{\
earrow}\) denotes intra-plane hops in the northwest direction; \(H_{v}^{\
earrow}\) denotes intra-plane hops in the northeast direction; \(H_{v}^{\
earrow}\) denotes intra-plane hops in the southwest direction; \(H_{v}^{\
earrow}\) denotes intra-plane hops in the southeast direction.  
The possible \(H_{h}\) and \(H_{v}\) are calculated, respectively. Therefore, the minimum end-to-end hops that count \(H\) in the inclined constellation can be obtained by minimizing the sum of \(H_{h}\) and \(H_{v}\).  
\[H=\text{min}\left\{\begin{array}{l}H_{h}^{\leftarrow}+H_{v}^{\
earrow}\\ H_{h}^{\leftarrow}+H_{v}^{\
earrow}\\ H_{h}^{\rightarrow}+H_{v}^{\
earrow}\\ H_{h}^{\rightarrow}+H_{v}^{\searrow}.\end{array}\right\} \tag{6}\]