## 5 Numerical Results and Discussion  
In this section, based on the previously presented theoretical analysis, numerical evaluated results are presented and discussed. If not otherwise stated, the values of the parameters considered in these results can be found in Table 1 and are mainly based on previous relevant studies, e.g., [20]. Moreover, for comparison purposes, we have also investigated the performance of a scheme in which in the second phase of communication, the link with the highest SNR, between the RF and the FSO, is always selected. This selection policy is also adopted in [18]. The numerical evaluation of the analytical expressions has been performed using the Mathematica software package. In particular, Mathematica supports all necessary functions for obtaining these results, such as the \(\texttt{NIntegrate}[]\) (for numerical integration) and \(\texttt{MeijerG}[]\) (for implementation of the Meijer's G function). \begin{table}
\begin{tabular}{|l l l|} \hline
**Parameter** & **Definition** & **Value** \\ \hline \(\lambda_{f}\) & FSO wavelength & 1550 nm \\ \hline \(h_{s}\) & Satellite height & 620 km \\ \hline \(h_{h}\) & HAPS height & 20 km \\ \hline \(h_{p}\) & Ground station height & 10 m \\ \hline \(G_{Tf}\) & Transmit telescope gain & 5 dB \\ \hline \(P_{f}\) & FSO transmit power & 5 dBm \\ \hline \(G_{Rf}\) & Receive telescope gain & 10 dB \\ \hline \end{tabular}
\end{table} 
Table 1: Communication parameter definitions and simulation values. In Figure 2, the performance of both schemes, i.e., the one introduced in this paper, labeled as \"Proposed Scheme\", and the one that always selects the highest SNR, labeled as \"Maximum SNR\", is evaluated using the criteria of OP (using (25)), the NLE (using (28)), and the SP (using (29)). The performance of these criteria is evaluated as a function of switching threshold \(\gamma_{\text{th}}\), assuming that \(\frac{\gamma_{T}}{\gamma_{k}}=-10\) dB. It is shown that as \(\gamma_{\text{th}}\) increases, the OP performance of the proposed scheme approaches the one of maximum SNR. What is very important to note is that the proposed scheme offers a considerable improvement in overhead estimation criteria that have been examined in this paper, namely, the NLE and PS. For example, for \(\gamma_{\text{th}}=5\) dB, the OP is equal for both schemes; SP is 10-14% lower for the proposed scheme, while the NLE is more than 70% lower. Therefore, based on the results of this figure, it can be concluded that in the proposed scheme, an excellent compromise between performance improvement and overhead reduction can be achieved by setting \(\gamma_{\text{th}}=\gamma_{T}\). It is noted that by setting \(\gamma_{\text{th}}=\gamma_{T}\) in (17), it can be easily mathematically proved that the CDF expressions of the two policies coincide. Nevertheless, for the numerical results that follow, equal values for these two thresholds have been considered. In Figure 3, an effort to depict the impact of the wind velocity \(w\) on the OP has been made. More specifically, the OP is plotted as a function of the average SNR (assuming \(\bar{\gamma}_{1}=\bar{\gamma}_{2}\)). In this figure, it is shown that the OP improves as the wind velocity increases, with the highest improvement being noticed when \(w\) decreases from 51 m/s to 31 m/s. In the same figure and using the corresponding high SNR expression for the CDF, based on (26) and (27), excellent tightness between the exact and the asymptotic results is proved. In Figure 4, the impact of the elevation angle \(\theta\) on the OP of the proposed scheme has been evaluated. More specifically, the OP is plotted as a function of the average SNR for various values of \(\theta\). It is shown that the performance improves as \(\theta\) decreases. Moreover, an excellent tightness is also observed between the exact and the asymptotic results.