# Performance Modeling Framework for IoT-over-Satellite Using Shared Radio Spectrum
## 4 SINR Model Approximation and Optimal Parameter of IoT-over-Satellite
### IoT-over-Satellite Duty Cycle and Optimal Beamwidth Modeling  
To establish a proper uplink channel, the minimum SINR level is at least \(-18\) dB to provide a PER at \(10^{-7}\) according to Table 4, so we define the available system time interval as the interval that SINR is larger than the threshold (\(\gamma_{0}\)) in decibel. The link-availability is equal to the available period divided by the accessible period. The accessible period is the time interval that the satellite has the LoS to the IoT devices, and it is equal to the time interval that elevation angle is higher than the horizon angle of the satellite (\(\theta_{h}\)). Figure 8 shows the SINR level of a measurement point in a single satellite pass. The period when SINR is above the threshold (grey-line) is the available period, and we assume the accessible interval to be the whole period. The effect of satellite antenna beamwidth has been taken into consideration. The mathematical expression for the link availability is shown in the following:  
\[F\left(\theta,\gamma_{0}\right)=\frac{T\left(\overline{\gamma}>\gamma_{0} \right)}{T\left(\theta_{e}>\theta_{h}\right)}. \tag{17}\]  
The link-availability is calculated, and the relationship between the duty cycle, SINR threshold, and antenna beamwidth is shown in Figures 9 and 10. The approximation is made by using two variables fitting where beamwidth (\(\theta\)), SINR threshold (\(\gamma_{0}\)) are the variables. The polynomial approximation can be written as the following:  
\[F\left(\theta,\gamma_{0}\right)=\sum_{j=0}^{4}\sum_{i=0}^{4-j}C_{ij}\theta^{i} \gamma_{0}^{j}, \tag{18}\]  
where \(C_{ij}\) is the polynomial coefficients and is given in Tables 5-7 for classes 1, 2 and 3, respectively.  
Figure 7: Average Uplink SINR of Australia over a week.  
Figure 8: Uplink-SINR during the accessible period with different antenna beamwidths of the satellite. The grey line represents the SINR threshold, and the period when SINR is above the threshold represents the available period for communication.  
Figure 10: The optimal satellite beamwidth with different SINR thresholds. It shows that the optimal antenna beamwidth for the satellite is from \(22^{\circ}\) to \(28^{\circ}\) based on the simulation.  
\begin{table}
\begin{tabular}{l c c c c c} \hline \hline \(C_{ij}\) & \(i\) & \(0\) & \(1\) & \(2\) & \(3\) & \(4\) \\ \hline j & & & & & \\
0 & 3.296 & \(-0.3102\) & \(0.01652\) & \(-0.0003975\) & \(3.57\times 10^{-6}\) \\
1 & \(-0.2542\) & \(0.01051\) & \(-0.0001986\) & \(9.556\times 10^{-7}\) & - \\
2 & \(0.01015\) & \(-0.0003013\) & \(3.305\times 10^{-6}\) & - & - \\
3 & \(-0.0001688\) & \(2.166\times 10^{-6}\) & - & - & - \\
4 & \(1.115\times 10^{-6}\) & - & - & - & - \\ \hline \hline \end{tabular}
\end{table}
Table 5: Surface Polynomial Coefficients for Class 1.  
Figure 9: An example of the availability percentage varies with the antenna beamwidth and the SINR threshold in Class 1. The dots indicate the simulation data and the heat map is the surface fitting using (18).  
Figure 10: The optimal satellite beamwidth with different SINR thresholds. It shows that the optimal antenna beamwidth for the satellite is from \(22^{\circ}\) to \(28^{\circ}\) based on the simulation.