# A Beam Hopping Scheme Based on Adaptive Beam Radius for LEO Satellites
## 3 Model
### Transport Model  
Channel coefficients:  
\[h_{k,n}=\frac{\sqrt{G_{R}G_{k,n}}}{4\pi\frac{d_{k}}{\lambda}\sqrt{K_{B}TB}}, \tag{1}\]  
\(h_{k,n}\) is the channel coefficient of the link between \(user_{k}\) and beam \(n\). \(G_{R}\) refers to the receiving antenna gain of the receiving terminal, \(G_{k,n}\) refers to the transmitting antenna gain between \(user_{k}\) and beam \(n\), \(d_{k}\) refers to the distance between the user and the satellite, \(T\) refers to the noise temperature of the terminal, \(B\) refers to the bandwidth, and \(K_{B}\) refers to the Boltzmann constant. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Symbol** & **Parameter** & **Symbol** \\ \hline Beam number & \(N_{beam}\) & The leaving time of the data packet whose destination user is \(i\) & \(pck^{user}\).end\({}_{time}\) \\ \hline Cell number & \(N_{cell}\) & The delay of one packet. & \(D\) \\ \hline delay sensitivity factor & \(\rho\) & The total delay of \(cell_{i}\). & \(D_{cell_{i}}\) \\ \hline Time slot length & \(T_{slot}\) & The total delay of one period. & \(Delay\) \\ \hline Time slot number & \(N_{T}\) & The length of the data packet whose destination user is \(j\). & \(pck^{user}\).len \\ \hline Hopping window & \(T\) & User demand & \(D_{user_{j}k}\) \\ \hline The power that is allocated to \(cell_{i}\) in \(t\). & \(P_{cell_{i},t}\) & User Rate & \(R_{user_{j}k}\) \\ \hline The arrival time of the data packet whose destination user is \(i\) & \(pck^{user_{j}}\).start\({}_{time}\) & Virtual demand traffic of \(cell_{i}\) & \(VirD_{t}^{cell_{i}}\) \\ \hline The set of service cell & \(\mathcal{C}\) & Demand traffic of \(cell_{i}\) & \(D_{t}^{cell_{i}}\) \\ \hline The frequency that is allocated to \(cell_{i}\) in \(t\) & \(B_{cell_{i},t}\) & offered throughput & \(R_{t}^{cell_{i}}\) \\ \hline Actual throughput & \(Throughput\) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The set of symbols and their meaning. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Values** \\ \hline Orbit Height & 550 km \\ \hline Minimum User Elevation & 40\({}^{\circ}\) (40–90\({}^{\circ}\)) \\ \hline \(P_{max}\) & 1000 w \\ \hline \(G_{R}\) & 21.8 dBi \\ \hline \(N_{B}\) & 16 \\ \hline Frequency & 2G Hz (19 GHz) \\ \hline Free Space Loss & 20 lg(f) + 20 lg(r) + 32.4 = 169.941664 dB \\ \hline Noise Temperature & 290 K \\ \hline Boltzmann constant & \(1.38\times 10^{-23}\) J/m \\ \hline Noise Power & \(\text{KTB}=-110.9669292079\) (dB) \\ \hline Delay Sensitivity Factor & 2,5 and 10 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The set of parameters and their values. A model with signals that have undergone large-scale fading channels and have been added with Gaussian white noise:  
\[y_{k,n}=\mathbf{h}_{k,n}^{T}\mathbf{x}+n_{k,n}, \tag{2}\]  
\(\mathbf{x}\) is the transmitted signal vector, and \(n_{k,n}\) is the additive Gaussian white noise. \(SINR_{k}\):  
\[\text{SINR}_{k}=\frac{\left|h_{k,n}\right|\cdot P_{n}}{\sigma^{2}+\sum_{m\
eq n,j\
eq k}\left|h_{j,m}\right|\cdot P_{m}} \tag{3}\]  
\(SINR_{k}\) is the signal- to- interference plus noise ratio suffered by \(user_{k}\). The above formula represents the SINR equation in the manuscript. In this equation, there is one term in the numerator and two terms in the denominator. The numerator term is the product of the magnitude of the complex channel coefficient and the power allocated to \(user_{k}\). This term represents the energy of the useful signal. Let us now consider the denominator. \(\sigma^{2}\) is the noise signal energy. As long as the temperature of an object is above absolute zero, thermal noise will be present. \(\sum_{m\
eq n,j\
eq k}\left|h_{j,m}\right|\cdot P_{m}\) represents the interference signals energy. From the expression of this term, it appears to be the summation of the energy of many useful signals. However, since these useful signals are intended for users other than \(user_{k}\), they consequently interfere with \(user_{k}\).