## 3 Outage Probability Estimation  
In order to evaluate the performance of the examined system, we estimate its OP, which is a crucial performance metric, in terms of reliability, for any communication system. This metric represents the probability that the instantaneous electrical SNR at the receiver falls below a critical threshold value, \(\gamma_{th}\), which corresponds to the receiver's sensitivity limit. Therefore, the OP inthe \(j\)th hop in the \(i\)th path is expressed as [10; 12; 15; 17]:  
\[P_{out,\{i,j\}}=\Pr\Bigl{(}I_{\{i,j\}}<I_{\{i,j\},th}\Bigr{)}=F_{I_{\{i,j\}}} \left(I_{\{i,j\},th}\right)=\int\limits_{0}^{I_{\{i,j\},th}}f_{I_{\{i,j\}}} \left(I_{\{i,j\}}\right)dI_{\{i,j\}} \tag{13}\]  
where \(\Pr(. )\) denotes probability, \(f_{I_{\{i,j\}}}\left(I_{\{i,j\}}\right)\) stands for the probability density function (PDF) either of the G-G or the NE turbulence distribution, and \(F_{I_{\{i,j\}}}\left(I_{\{i,j\},th}\right)\) represents the corresponding cumulative density function (CDF) of the G-G or the NE turbulence distribution, respectively. Assuming G-G modeled turbulence, from (10), (12) and (13) (Equation (07.34.17.0011.01) in [30], Equation (26) in [32]) and by setting then \(I_{\{i,j\}}=A_{\psi,th}\sqrt{\gamma_{\{i,j\},th}/\mu_{\{i,j\}}}\), [4], where \(\gamma_{\{i,j\},th}/\mu_{\{i,j\}}\) represents the normalized average electrical SNR, we conclude that (13) can be written as:  
\[ \begin{array}{ll}P_{GG,out,\{i,j\}}&=\frac{\psi_{\{i,j\}}^{2}}{\Gamma(\alpha )\Gamma(\beta)}\\ &\times G_{2A}^{3,1}\left(\frac{\alpha\beta\psi_{\{i,j\}}^{2}}{\psi_{\{i,j\}}^ {2}+1}\sqrt{\frac{\gamma_{th}}{\mu}}\begin{array}{c}1,\,\psi_{\{i,j\}}^{2} \\ \psi_{\{i,j\}}^{2},\,\alpha,\,\beta,\,0\end{array}\right)\end{array}  \tag{14}\]  
Alternatively, by performing once again the procedure above but by using (11) instead of (10), we conclude that for NE modeled turbulence, (13) gives the following expression:  
\[P_{NE,out,\{i,j\}}=\psi_{\{i,j\}}^{2}\,G_{2A}^{2,1}\left(\frac{\psi_{\{i,j\}}^ {2}}{\psi_{\{i,j\}}^{2}+1}\sqrt{\frac{\gamma_{th}}{\mu}} \begin{array}{c}1, \,\psi_{\{i,j\}}^{2}+1\\ \psi_{\{i,j\}}^{2},\,1,\,0\end{array} \right) \tag{15}\]  
Note that contrary to [4] that evaluates the average bit error rate (ABER) as a function of the average SNR, \(\mu\), in this work we evaluate through (14) and (15) the OP as a function of the normalized average electrical SNR \(\gamma_{\{i,j\},th}/\mu_{\{i,j\}}\) that is assumed to obtain the same value for each link of the examined FSO system, i.e., \(\gamma_{\{i,j\},th}/\mu_{\{i,j\}}=\gamma_{th}/\mu\), under G-G and NE modeled turbulence conditions, respectively. Moreover, (14) and (15) refer to mixed DF relay FSO configurations, and thus, contrary to the ABER expressions in [4] they introduce the indices \(i\) and \(j\). The OP of each multi-hop branch of the system, i.e., the OP of each parallel link of serially connected DF relays of the \(i\)th path is given as [10; 17]:  
\[P_{out,i}=1-\prod_{j=1}^{H}\Bigl{[}1-P_{out,\{i,j\}}\Bigr{]} \tag{16}\]  
and taking into account that \(N\) parallel links are used the total OP of the system is written as [10; 17]:  
\[P_{out}=\prod_{i=1}^{N}P_{out,i}=\prod_{i=1}^{N}\Biggl{\{}1-\prod_{j=1}^{H} \Bigl{[}1-P_{out,\{i,j\}}\Bigr{]}\Biggr{\}} \tag{17}\]  
from (12) and (14), the following closed-form mathematical expression for the OP of the total system under G-G modeled turbulence along with non-zero boresight pointing errors, is obtained:  
\[P_{GG,out}=\prod_{i=1}^{N}\Biggl{\{}1-\prod_{j=1}^{H}\Biggl{[}1-\frac{\psi_{\{ i,j\}}^{2}}{\Gamma(\alpha)\Gamma(\beta)}G_{2A}^{3,1}\Biggl{(}\sqrt{\frac{ \gamma_{\{i,j\},th}}{\mu_{\{i,j\}}}}\frac{\alpha\beta\psi_{\{i,j\}}^{2}}{\psi_{ \{i,j\}}^{2}+1} \begin{array}{c}1,\,\,\psi_{\{i,j\}}^{2}\\ \psi_{\{i,j\}}^{2},\,\alpha,\,\beta,\,0\end{array}