# High Reliable Uplink Transmission Methods in GEO-LEO Heterogeneous Satellite Network
## 3 Performance Evaluation
### Performance with Ordered Channel Gain  
Consider that the large-scale fading of beam-edge user is usually much worse than that of the beam-center user. We can derive the following assumption. **Assumption 1**. : _In LEO NOMA scenario, the uplink channel condition of the beam-center user is considered to be superior compared with that of the beam-edge user, i.e., \(|\rho_{2,l}|^{2}<|\rho_{1,l}|^{2}\)._  
According to Assumption 1, the large-scale fading should be ordered first. The joint probability density function (pdf) of \(|\rho_{1,l}|^{2}\) and \(|\rho_{2,l}|^{2}\) can be expressed as [26]  
\[p_{ij}(y,z)=\frac{n!}{(i-1)!(j-i-1)!(n-j)!}[F(y)]^{i-1}[F(z)-F(y)]^{j-i-1}[1-F(z) ]^{n-j}f(y)f(z), \tag{20}\]  
where \(y<z\). \(n=2\) denotes the number of users; \(i=1\) and \(j=2\) represent the beam-edge and beam-center users. Then, the joint pdf is provided by  
\[p_{|\rho_{2,l}|^{2},|\rho_{1,l}|^{2}}(y,z)=2f_{|\rho_{2,l}|^{2}}(y)f_{|\rho_{1,l }|^{2}}(z). \tag{21}\]For the beam-center user, uplink outage satisfies \(|\rho_{2J}|^{2}<|\rho_{1J}|^{2}<\frac{T_{1}(|\rho_{2J}|^{2}K_{3}\alpha\beta+1)}{K_{ 1}(1-\alpha)\beta}\) after the channel ordering. Then, we have \(|\rho_{2J}|^{2}<\frac{T_{1}}{(K_{1}(1-\alpha)-T_{1}K_{2}\alpha)\beta}\). Let \(\chi=\frac{T_{1}}{(K_{1}(1-\alpha)-T_{1}K_{2}\alpha)\beta}\). (11) becomes  
\[ \begin{array}{ll}Pr_{1}&=\Pr(|\rho_{1J}|^{2}<\frac{T_{1}(|\rho_{2J}|^{2}K_{3} \alpha\beta+1)}{K_{1}(1-\alpha)\beta})=2\int_{0}^{\chi}\int_{y}^{\frac{T_{1}(yK _{2}\alpha\beta+1)}{K_{1}(1-\alpha)\beta}}f_{|\rho_{1J}|^{2}}(x)dxf_{|\rho_{2J}| ^{2}}(y)dy\\ &=2\int_{0}^{\chi}\Big{(}\epsilon_{1J}\sum_{k=0}^{\infty}\frac{(m_{1J})x_{1J}^ {k}}{(k!)^{2}\kappa_{1J}^{k+1}}\gamma(k+1,\frac{T_{1}(yK_{2}\alpha\beta+1)}{K _{1}(1-\alpha)\beta}\kappa_{1J})-\epsilon_{1J}\sum_{k=0}^{\infty}\frac{(m_{1J} )x_{1J}^{k+1}}{(k!)^{2}\kappa_{1J}^{k+1}}\gamma(k+1,y\kappa_{1J})\Big{)}f_{| \rho_{2J}|^{2}}(y)dy\\ &=2\int_{0}^{\chi}e^{-\kappa_{2J}y}\\ &\left(\epsilon_{1J}\sum_{k=0}^{\infty}\frac{(m_{1J})x_{1J}^{k}}{(k!)^{2} \kappa_{1J}^{k+1}}\gamma(k+1,\frac{T_{1}(yK_{2}\alpha\beta+1)}{K_{1}(1-\alpha )\beta}\kappa_{1J})-\epsilon_{1J}\sum_{k=0}^{\infty}\frac{(m_{1J})x_{1J}^{k}} {(k!)^{2}\kappa_{1J}^{k+1}}\gamma(k+1,y\kappa_{1J})\right)\epsilon_{2J1}F_{1}( m_{2J};1;\
u_{2J}y)\\ &\stackrel{{(c)}}{{=}}\frac{\chi^{\pi\pi}}{N}\sum_{n=1}^{N}\sqrt{1 -x_{n}^{2}}M(y_{n,1}),\end{array}  \tag{22}\]  
where \(x_{n}=\cos(\frac{2n-1}{2N}\pi)\) and \(y_{n,1}=\frac{(x_{n}+1)\chi}{2}\). \((c)\) follows Gauss-Chebyshev approximation [27]. The outage probability of the beam-edge user can be further expressed as  
\[\begin{array}{ll}Pr_{2}=\underbrace{\mathbb{P}(|h_{1}|^{2}<\frac{T_{1}(|h_{ 2}|^{2}\alpha\beta+1)}{(1-\alpha)\beta})}_{P_{1}}+\underbrace{\mathbb{P}(|h_{1 }|^{2}\geq\frac{T_{1}(|h_{2}|^{2}\alpha\beta+1)}{(1-\alpha)\beta},|h_{2}|^{2} <\frac{T_{2}}{\alpha\beta})}_{S1},|h_{2}|^{2}<\frac{T_{2}}{\alpha\beta}). \text{Since }Pr_{1}\text{ has been derived in (\ref{eq:p1}), we will focus on the deduction of }S_{1}\). Similarly, \(S_{1}\) can be rewritten as  
\[\begin{array}{ll}\mathbb{P}(|\rho_{1J}|^{2}\geq\frac{T_{1}K_{2}|\rho_{2J}|^{2 }\alpha\beta+T_{1}}{(1-\alpha)\beta K_{1}},|\rho_{2J}|^{2}<\frac{T_{2}}{ \alpha\beta K_{2}}).\text{ As }|\rho_{1J}|^{2}>|\rho_{2J}|^{2}\text{ is assumed in this section, we have }|\rho_{1J}|^{2}\geq\max\{\frac{T_{1}K_{2}|\rho_{2J}|^{2}\alpha\beta+T_{1}}{(1 -\alpha)\beta K_{1}},|\rho_{2J}|^{2}\}.\text{ Letting }\zeta=\max\{\frac{T_{1}K_{2}|\rho_{2J}|^{2}\alpha\beta+T_{1}}{(1-\alpha)\beta K_ {1}},|\rho_{2J}|^{2}\}\), \(S_{1}\) can be calculated as  
\[