## 5 Numerical Results  
In this section we present the numerical results concerning both the OP and MOD performance of the examined relay-assisted FSO system, for weak to saturated turbulence along with different amounts of pointing mismatch, over a wide SNR range. Specifically, by using (18) and (19) we obtain the numerical results concerning the OP of the system for gamma-gamma modeled and negative exponential modeled turbulence conditions, respectively, along with the presence of non-zero boresight pointing errors. Similarly, by using (21) and (22) we get the numerical results for the corresponding MOD values of the system, i.e., for G-G and NE turbulence, respectively, along with non-zero boresight misalignment-induced fading. The relay-aided FSO system under investigation is assumed to operate at \(\lambda\) = 1.55 \(\mu\)m and receiver's aperture diameter, \(D\) = 0.1 m. Additionally, each hop length of the system, \(L\), is 1.5 km, while the system employs \(N\) = 1, 2 or 5 parallel DF relays and \(H\) = 2 or 4 hops. For G-G modeled, weak turbulence conditions, the parameter \({C_{n}}^{2}\) value is fixed at 2 \(\times\) 10\({}^{-14}\) m\({}^{-2/3}\), while for NE modeled, saturated turbulent channels, the parameter \({C_{n}}^{2}\) obtains its maximum value, i.e., \({C_{n}}^{2}\) = 10\({}^{-13}\) m\({}^{-2/3}\). Concerning the point errors effect, the boresight is zero only for \(N=1\) due to precise positioning, [4]. In that case it holds that \(\left(\mu_{x}/r_{a},\mu_{y}/r_{a}\right)\) = (0,0) and \(\sigma_{x}\) = \(\sigma_{y}\), while \(\psi_{1}\) = 5 for \(\left(r_{a},w_{z}/r_{a},\mu_{x}/r_{a},\mu_{y}/r_{a},\sigma_{x}/r_{a},\sigma_{y}/ r_{a}\right)\) = \(\left(5\text{ cm},10,0,1,1\right)\). When \(N=3\) relays are considered, we assume for the remaining paths that \(\left(r_{a},w_{z}/r_{a},\mu_{x}/r_{a},\mu_{y}/r_{a}\right)\) = \(\left(5\text{ cm},10,1,2\right)\), while for weak to strong non-zero boresight pointing errors, the parameter \(\psi_{2}\) takes the value 2.3 or 1.3 for \(\left(\sigma_{x}/r_{a},\sigma_{y}/r_{a}\right)\) = (2.1,1.5) or \(\left(\sigma_{x}/r_{a},\sigma_{y}/r_{a}\right)\) = (4,3), respectively. Finally, for \(N=5\), the non-zero boresight pointing errors effect is stronger and even more stronger, with \(\psi_{3}\) = 0.9 or \(\psi_{3}\) = 0.6, for \(\left(\sigma_{x}/r_{a},\sigma_{y}/r_{a}\right)\) = (5.8,4.8) or \(\left(\sigma_{x}/r_{a},\sigma_{y}/r_{a}\right)\) = (9,7), respectively. Figure 2, illustrates the evolution of OP performance over a wide normalized average electrical SNR range, obtained by using (19), for the total relay-assisted examined FSO system when it employs \(N=1\), 3 or 5 dual-hop paths, i.e., \(H_{j}=2\), under saturated turbulence conditions, modeled through the NE distribution, along with zero boresight pointing errors for the single path configuration, i.e., for \(N=1\), and additional weak to strong non-zero boresight pointing errors for multiple path configurations, i.e., for \(N=3\) or 5. It is clearly shown that even in saturated turbulence conditions, the OP performance is significantly improved by using parallel relaying method with dual-hop paths, and especially, by increasing the number of paths of the total system. Additionally, further OP performance enhancements are depicted as the normalized average electrical SNR obtains larger values and as the non-zero boresight misalignment-induced fading gets weaker, i.e., for larger \(\psi_{2}\) or \(\psi_{3}\) parameter values that refer to triple-path (\(N=3\)) and quintuple-path (\(N=5\)) dual-hop implementations, respectively. Moreover, it is worth mentioning that the performance comparison between the corresponding curves of Figure 2 and those which appear in Ref. [17] obtained by the analytical expression Equation (15) in [17], demonstrates significantly larger OP corresponding values due to the fact that Figure 2 refers to saturated NE modeled turbulence channels, while the results of [17] have been obtained for weak turbulence conditions with a different model, i.e. the Gamma distribution.