# Performance of Dense Wireless Networks in 5G and beyond Using Stochastic Geometry
## 5 Outage Analysis in the Presence of NLOS, Noise and Blockage
### NLOS Paths
If using the Maximal Ratio Combining (MRC) combiner for the \(L\) paths related to the user of interest affected by the interference powers \(I_{\ell}=|\iota_{\ell}|^{2}\), one obtains the following service probability analysis (see Appendix C for derivation and specific MRC notation):  
\[P_{MRC}=\mathbb{P}_{I}\left(\sum_{\ell=1}^{L}\alpha_{\ell}I_{\ell}<\frac{1}{T }\left(\sum_{\ell=1}^{L}\frac{|\beta_{\ell}|^{2}}{\hat{I}_{\ell}^{2}\,D_{ \ell}^{2b}}\right)^{2}\right). \tag{44}\]  
where \(|\beta_{\ell}|^{2}\) is the fluctuation for the signal from the user of interest from the \(\ell\)th path, and the distribution of total interference \(I_{MRC}=\sum_{\ell=1}^{L}\alpha_{\ell}I_{\ell}\) follows on from the CF for iid interferers over the L-beamformers:  
\[\Psi_{I_{MRC}}(\omega)=\prod_{\ell=1}^{L}\Psi_{I}(a_{\ell}\omega). \tag{45}\]  
The NLOS distances \(D_{\ell}=(R_{\alpha,\ell}^{2}+h^{2})^{1/2}\), and \(R_{\alpha,\ell}\) for \(\ell>1\) are affected by the random angular position \(\eta\) of the NLOS reflections around the radius \(d\), as depicted in Figure 14:  
\[R_{\alpha,\ell}=d+\sqrt{R_{\alpha,1}^{2}+d^{2}-2\,d\,R_{\alpha,1}\cos(\eta)} \,\,\forall\,\,\ell>1. \tag{46}\]  
Pragmatically, the NLOS distance for the served user is dependent on the specific multipath model assumed here, and a convenient way to incorporate the NLOS attenuation into the additional path \(R_{\alpha,\ell}-R_{\alpha,1}\) is to approximate this term is by considering the mean distance for NLOS \(\bar{R}_{\alpha,\ell}=E_{\eta}[R_{\alpha,\ell}]\). The distance \(\bar{D}_{\ell}=(R_{\alpha,\ell}^{2}+h^{2})^{1/2}\) is for the NLOS \(\ell>1\)  
Figure 14: NLOS model: every user has a few NLOS links in addition to an LOS link. LOS links are shown with thick red lines, and NLOS links are shown with dashed lines reflected back from the perimeter of a circle around the UE. Here, the \(UE_{o}\) is the user of interest and and \(UE_{i}\) is an interferer. and thus the relationship (44) is somewhat simplified by constant distances. Recalling that LOS/NLOS models hold for interferers' \(i_{\ell}\), the power is augmented by the NLOS components, and the service probability is reduced accordingly compared to \(L=1\) (LOS-only). The CF for \(L>1\) is derived in Appendix D, accounting for LOS and NLOS. Figures 15 and 16 show the average number of users served \(M_{s}\) within a \(\Delta\times\Delta\) square-shaped area (from (32) it is \(M_{\rm s}=\lambda\Delta^{2}\bar{P}_{\rm s}\)) when the AP with UCA is located at the center. The multipath increases the interference and the number of NLOS paths from 1 to 3 (or \(L=1,2,3,4\)). In Figure 15, the performance degrades as the density of the interferers is quite high (here \(\lambda=10^{-2}\,\rm{m}^{-2}\)); in contrast, in Figure 16, the aggregated interference is lower due to the lower density (\(\lambda=10^{-3}\,\rm{m}^{-2}\)), and thus, there is a clear benefit arising from the multipath that vanishes for a large height (here \(h>25\) m). The performance of MRC shown in Figures 15 and 16 is compared to the selection combining (SC), which select the path with the largest SIR ([60], Ch. 7). The derivation of SC is straightforward (not shown here). As expected, the MRC outperforms the SC, but one might observe that degradation becomes more severe for large multipath. In the remainder of the paper, we consider the multipath condition with both LOS and NLOS links.