# Performance Modeling Framework for IoT-over-Satellite Using Shared Radio Spectrum
## 5 Optimal Operating Parameters
### Optimal Beamwidth of LEO Satellite Antenna  
In order to maximize the link-availability, we need to elongate the duration at which the SINR is above the minimum threshold. Having a narrower beamwidth results in a higher gain, thus higher a SINR; however, the duration of such an increase in SINR will be reduced. These two competition factors can result in a maximum link-availability at a particular beamwidth. The dynamic of these two factors is illustrated in Figure 8. As it can be noted from Figures 9 and 11, the optimum beamwidth is specific to a certain SINR threshold. And the optimal antenna beamwidth of a satellite occurs when maximum average link-availability is achieved. Hence, the optimal satellite beamwidth can be derived from the differentiating of (17) because it is a concave function, as follows,  
\[\theta^{*} =\operatorname*{argmax}_{\theta}\left[F\left(\theta,\gamma_{0} \right)\right]\] \[\text{where}\ \left[\ \frac{\partial}{\partial\theta}\left[F \left(\theta^{*},\gamma_{0}\right)\right]=0\ \right] \tag{19}\]  
We depict in Figure 10 the optimum beamwidth as a function of the SINR threshold in the three different classes. The result of the duty cycle percentage when adopting the \(\theta^{*}\) is shown in Figure 12, and the solution of (19) is shown in Figure 10. We substitute the optimal beamwidth into (18), and the result of maximum link-availability is shown in Figure 12. The sea area has the highest link-availability, and rural area has the lowest among these groups, as shown in Figure 12. It shows that the communication started to be available when the beamwidth of the antenna is smaller than \(35^{\circ}\), and having the optimal beamwidth at around \(22\)-\(27^{\circ}\). If we decrease the beamwidth of the antenna, it shows better link-availability because narrow-band antenna decreases the area of the visible horizon. So it decreases the number of terrestrial interference sources.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(C_{ij}\) & \(\mathbf{i}\) & \(\mathbf{0}\) & \(\mathbf{1}\) & \(\mathbf{2}\) & \(\mathbf{3}\) & \(\mathbf{4}\) \\ \hline j & & & & & \\
0 & 5.193 & \(-\)0.7377 & 0.04943 & \(-\)0.001439 & 1.566 \(\times\)\(10^{-5}\) \\
1 & \(-\)0.2536 & 0.01297 & \(-\)0.0003978 & 2.702 \(\times\)\(10^{-6}\) & - \\
2 & 0.009473 & \(-\)0.0003127 & 6.509 \(\times\)\(10^{-6}\) & - & - \\
3 & \(-\)0.0001397 & 9.906 \(\times\)\(10^{-7}\) & - & - & - \\
4 & \(9.301\times 10^{-7}\) & - & - & - & - \\ \hline \hline \end{tabular}
\end{table}
Table 6: Surface Polynomial Coefficients for Class 2.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(C_{ij}\) & \(\mathbf{i}\) & \(\mathbf{0}\) & \(\mathbf{1}\) & \(\mathbf{2}\) & \(\mathbf{3}\) & \(\mathbf{4}\) \\ \hline j & & & & & \\
0 & 1.006 & \(-\)0.1815 & 0.01493 & \(-\)0.0004503 & 4.597 \(\times\)\(10^{-6}\) \\
1 & 0.002753 & \(-\)0.003322 & \(-\)2.033 \(\times\)\(10^{-5}\) & 1.53 \(\times\)\(10^{-6}\) & - \\
2 & 0.0004753 & 0.0001392 & \(-\)1.35 \(\times\)\(10^{-6}\) & - & - \\
3 & \(-\)3.165 \(\times\)\(10^{-5}\) & \(-\)1.09 \(\times\)\(10^{-6}\) & - & - & - \\
4 & 3.75 \(\times\)\(10^{-7}\) & - & - & - & - \\ \hline \hline \end{tabular}
\end{table}
Table 7: Surface Polynomial Coefficients for Class 3.