# A Novel Feeder Link Handover Strategy for Backhaul in LEO Satellite Networks
## 3 System Model
### Backhaul Model  
In this article, we focus on solving the handover problem by jointly considering feeder links and ISLs to optimize the backhaul capacity. Then, we formulate the backhaul model related to the feeder link handover. The GSs and corresponding visible satellites are mainly considered since the backhaul traffic converges to these nodes. The visible satellite set of all GSs is denoted by \(\mathbf{V}(t)\). The visibility \(v_{n,m}(t)\) of \(S_{n}\) and \(G_{m}\) is defined as follows:  
\[v_{n,m}(t)= \begin{cases}1,&G_{m}\text{ is visible to }S_{n},\\ 0,&G_{m}\text{ is not visible to }S_{n}\end{cases}  \tag{1}\]  
Since the satellite may be visible to more than one GS simultaneously, \(\mathbf{V}(t)\) is expressed as  
\[\mathbf{V}(t)=\{S_{n}|\sum_{m=1}^{M}v_{n,m}(t)\geq 1\} \tag{2}\]  
During the whole communication process, GSs need to switch to a sequence of visible satellites to achieve seamless connection due to the moving of satellites. The connections between \(S_{n}\) and \(G_{m}\) at time slot \(t\) is denoted by \(x_{n,m}(t)\) as follows:  
\[x_{n,m}(t)= \begin{cases}1,&G_{m}\text{ is connecting to }S_{n},\\ 0,&G_{m}\text{ is not connecting to }S_{n}\end{cases}  \tag{3}\]  
In our proposed model, it is assumed that the satellites and GSs can establish multiple feeder links simultaneously. Therefore, the number of feeder links should meet the following constraints:  
\[\sum_{m=1}^{M}x_{n,m}(t)\leq L_{s},\quad\forall n\in N \tag{4}\]  
\[\sum_{n=1}^{N}x_{n,m}(t)\leq L_{g},\quad\forall m\in M \tag{5}\]  
Figure 1: LEO communication network for backhaul services. In our proposed model, the channel quality of the feeder link between \(S_{n}\) and \(G_{m}\) may be affected by the complex space environment. The channel gain \(g_{n,m}(t)\) between \(S_{n}\) and \(G_{m}\) mainly consists of the path loss, atmospheric fading, and Rician small-scale fading [29], are is expressed as follows:  
\[g_{n,m}(t)=\left(\frac{c}{4\pi vd_{n,m}(t)}\right)^{2}\cdot 10^{\frac{33 rd_{m,m}(t) }{10h}}\cdot Ri \tag{6}\]  
where \(\left(\frac{c}{4\pi vd_{n,m}(t)}\right)^{2}\) is path loss, \(10^{\frac{3 rd_{m,m}(t)}{10h}}\) is atmospheric fading, \(Ri\) is Rician small-scale fading. \(c\) and \(fc\) are the speed of light and the carrier frequency, \(\chi\) is attenuation through the clouds and rain in dB/km, and \(h\) is the altitude of satellites. The above four parameters and \(Ri\) can be approximated as constants. Therefore, \(d_{n,m}(t)\) is the only parameter varying with time, which represents the distance between \(S_{n}\) and \(G_{m}\). The available feeder link capacity from \(S_{n}\) to \(U_{m}\) at time \(t\) is expressed as follows:  
\[C_{n,m}^{g}(t)=B_{g}\log_{2}\left(1+\frac{g_{n,m}(t)P_{t}g_{t}g_{s}}{\sigma^{2 }}\right) \tag{7}\]  
where \(P_{t}\) is the transmission power of the satellite, \(g_{t}\) is the antenna gain of the transmitter, \(g_{s}\) is the antenna gain of the receiver, \(\sigma^{2}\) the noise power at GS, and \(B_{g}\) is the bandwidth allocated to the feeder link. The channels between satellites are regarded as ideal channels regardless of attenuation. Therefore, the available channel capacity of ISLs is assumed as a constant \(C^{s}\). The object of feeder link handover is to optimize the backhaul capacity, defined as the maximum backhaul flow in the graph. To formulate the traffic flow, a binary indicator \(l_{i,j}\) is used to indicate whether \(S_{i}\) links to \(S_{j}\) as follows:  
\[l_{i,j}= \begin{cases}1,&S_{i}\text{ links to }S_{j},i\
eq j,\\ 0,&S_{i}\text{ does not link to }S_{j},i\
eq j\end{cases}  \tag{8}\]  
The traffic flow from \(S_{i}\) to \(S_{j}\) is denoted as \(f_{i,j}^{s}(t)\). For each satellite \(S_{n}\) in the visible satellite set, it receives traffic from its adjacent satellites and connecting UEs, which is denoted by \(f_{n}^{in}(t)\) as follows:  
\[f_{n}^{in}(t)=\sum_{j=1,j\
eq n}^{N}l_{j,n}f_{j,n}^{\text{isl}}(t)+f_{n}^{in} (t),\qquad\forall S_{n}\in\mathbf{V}(t) \tag{9}\]  
where \(f_{n}^{in}(t)\) is the overall traffic flow of the ULs of \(S_{n}\).