# A Novel Feeder Link Handover Strategy for Backhaul in LEO Satellite Networks
## 4 Problem Analysis and Strategy Design
### Problem Analysis
When completing a handover decision, the available satellite capacity is expressed as follows:  
\[C_{i}^{s}(t)=4C^{\mathrm{isl}}+C^{u}-\sum_{m}^{M}(\sum_{n\
eq i}^{N}x_{n,m}(t )\frac{C_{n,m}^{g}(t)}{4h_{n,j}}+x_{i,m}(t)C_{i,m}^{g}(t)) \tag{16}\]  
where \(4C^{\mathrm{isl}}+C^{u}\) is the limit of the available capacity of the satellite, and the remaining is the overall influence of all feeder links on the available satellite capacity. Next, the available satellite capacity must be normalized to the available satellite capacity ratio \(r^{s}n,m(t)\). When \(C_{i}^{s}(t)\) is large, \(r^{s}n,m(t)\) should keep close to one since the feeder link is more likely to be the bottleneck. When \(C_{i}^{s}(t)\) is small, the ISLs are more likely to be the bottleneck so \(r^{s}n,m(t)\) must accurately reflect the subtle changes in \(C^{s}_{i}(t)\). Therefore, a sigmoid function is used for normalization as follows:  
\[r^{s}_{n,m}(t)=\frac{1}{1+e^{-\alpha(\frac{C^{s}_{i}(t)}{C^{s}_{\text{max}}}- \beta)}} \tag{17}\]  
where \(\beta\) is the offset and \(\alpha\) represents the gradient. This normalization method makes \(r^{s}_{n,m}(t)\) close to one when the available satellite capacity exceeds the available feeder link capacity and close to zero when the available satellite capacity is small. \(r^{s}_{n,m}(t)\) would change significantly if the available satellite capacity ranged from zero to the upper limits of the available feeder link capacity. Considering that the backhaul capacity depends on the status of the ISLs and the feeder links, the available backhaul capacity ratio is expressed as follows:  
\[r^{b}_{n,m}(t)=r^{s}_{n,m}(t)r^{g}_{n,m}(t) \tag{18}\]  
The available backhaul capacity ratio can estimate the maximum flow performance achieved by the feeder link handover and improve the backhaul capacity performance in the optimization. To reduce the handover frequency, we introduce the service time factor and handover control factor. The service time factor facilitates the feeder links switching to a satellite with a larger visible time, which can be calculated based on the ephemeris and the locations. The service time factor is expressed as follows:  
\[r^{t}_{n,m}(t)=\frac{D_{n,m}(t)}{D_{\text{max}}} \tag{19}\]  
where \(D_{n,m}(t)\) is the remaining time that \(S_{n}\) is visible to \(G_{m}\). \(D_{\text{max}}\) is the upper limit of the remaining service time. If a UE aims at maximizing the instantaneous throughput at each time slot, this UE may change its NT-BS selection at every time slot, which results in frequent handovers. The handover control factor is introduced to avoid frequent handovers in a greedy strategy. It is expressed as follows:  
\[r^{h}_{n,m}(t)=1-h_{n,m}(t) \tag{20}\]  
Figure 2: Schematic diagram of updating the available satellite capacity. Based on the proposed three handover factors, we propose a handover utility function for handover decisions expressed as follows:  
\[W_{n,m}(t)=w_{1}r_{n,m}^{b}(t)+w_{2}r_{n,m}^{t}(t)+w_{3}r_{n,m}^{h}(t) \tag{21}\]  
where \(w_{1}\), \(w_{2}\), and \(w_{3}\) represent the weights of each handover factor with the following constraint:  
\[w_{1}+w_{2}+w_{3}=1 \tag{22}\]