# 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  
Previous studies and measurement campaigns [54] have shown that the NLOS clusters of rays are present in mmW communications where they form sparse multipath faded channels. These paths can increase the amount of interference, but at the same time would lead to a more useful signal being received in the case of the coherent reception of a signal. On the other hand, diversity is an efficient way to compensate for the blockage and to increase the reliability of communication systems. ### NLOS Paths  
Let \(L\) be the total number of paths that the signal arrives from the user of interest to the same AP; the \(L\times 1\) set of signals \(\mathbf{y}=[y_{1},y_{2},\ldots,y_{L}]^{T}\) after the multi-beam beamforming to each of the paths from the user of interest transmitting \(x\) is  
\[\mathbf{y}=\mathbf{h}x+\mathbf{t}+\mathbf{w}, \tag{42}\]  
where \([\mathbf{h}]_{\ell}=h_{\ell}=\beta_{o,\ell}/D_{\ell}^{b}\) for distance \(D_{\ell}=(R_{o,\ell}^{2}+h^{2})^{1/2}\) corresponding to the \(\ell\)th paths of arrival in case \(\ell=1\) it is the direct LOS link and \(R_{o,1}=R_{o}\) is the geometric distance between the user of interest, while the distances for NLOS links are modeled later but \(R_{o,\ell}\geq R_{o}\). The ensemble of the aggregated interference amplitudes from the PPP-distributed interferers is \(\boldsymbol{\iota}=[t_{1},t_{2},\ldots,t_{L}]^{T}\), which are independent and identically distributed (iid) random variables obtained from a set of \(L\) beamforming toward the distinct angles \(\phi_{o,1},\phi_{o,2},\ldots,\phi_{o,L}\) for LOS (\(\phi_{o,1}\)) and NLOS (\(\phi_{o,2},\ldots,\phi_{o,L}\)) of the user of interest, so that adapting (4) to this case with multipath for interference is:  
Figure 13: Comparison of the CDF of the aggregated interference power, within \(R_{max}=400\) m using a UCA of size \(N_{c}=16\) positioned at AP height \(h=30\) m, for different interferer density \(\lambda=\{1,5,10\}\times 10^{-3}\) m\({}^{-2}\), path-loss exponent \(2b=2.6\), transmit power \([P_{Tx}]_{dBm}=20\) dBm, central frequency \(F_{c}=28\) GHz and \(|\beta_{o}|=\sqrt{P_{Tx}}\,4\pi F_{c}/c\), for three cases: (a) true CF model described in (19); (b) approximate mixture model described in (40); and (c) \(\alpha\)-stable distribution. \[\iota=\sum_{i=1}^{\infty}\sum_{\ell=1}^{L}\frac{\beta_{i,\ell}}{\left(R_{i,\ell}^{ 2}+h^{2}\right)^{\frac{k}{2}}}G_{c}(\phi_{i,\ell})x_{i,\ell}. \tag{43}\]  
This assumption is justified by the interfering ray-paths on every beamforming that has different attenuations and phase shifts, and is thus independent. \(\mathbf{w}=[w_{1},w_{2},\ldots,w_{L}]^{T}\) is the collection of noise amplitudes. The LOS/NLOS links are shown in Figure 14 and using the Weyl model (similar to Saleh-Valenzuela [55] adapted for mmW [56; 57]), where the indirect NLOS paths from the transmitter are reflected from a secondary point, which is uniformly distributed around a circle with a radius \(d\) around each of the transmitter's location. Usually, at high frequencies, there are not many NLOS paths, which are typically \(L=2-3\)[54; 58; 59]. The receiver for \(L\) paths, possibly with different (and likely delay-resolvable for mmW and sub-THz system with large bandwidth) delays, is expected to combine to maximize the service probability.