# Performance of Dense Wireless Networks in 5G and beyond Using Stochastic Geometry
## 3 CF of the Aggregated Interference in LOS
### Numerical Validation on Aggregated Interference
Although the interference power of each one of them is highly attenuated, the summation of many far-away interferers is still too high. In this case, it is preferable to have a vertical beamforming component, which can filter out the effect of far-away UEs by creating a ring-shaped gain pattern around the AP. Furthermore, based on Figure 6, it can be seen that the increase in the path-loss exponent has an inverse effect on the two sets of curves. For the case of a small UE density, by increasing the path-loss exponent, the number of UEs served decreases, while in the case of a very dense network (\(\lambda=5\times 10^{-2}\)m\({}^{-2}\)), by increasing the path-loss exponent, the number of UEs served increases. This is because, in the case of dense networks, higher path-loss is beneficial by highly attenuating the interference coming from the large crowd of interferers that are located far away, while in the case of a small UE density, high path-loss attenuates to a larger extent than the signal coming from the target user. It can be noted that for \(2b=2\), the curve monotonously decreases, and it is the minimum for a UCA, while the other curves have a concave shape. This is due to the fact that \(2b=2\) is a singularity point when the area where the interferers is infinite (i.e., \(R_{max}\rightarrow\infty\)). This can be inferred by setting \(\alpha=1\) in relation (26), or from the literature on aggregated interference, e.g., [17; 48]. This means that for \(\alpha=1\), the average aggregated interference power diverges, and the AP does  
Figure 5: SIR vs. AP height \(h\), for a target UE located at \(R_{0}=\{20,30,40\}\) m equipped with a uniform circular array (UCA) with \(N_{\rm c}=128\) that has average power gain \(\bar{G}_{\rm c}^{2}=0.012\), for two sets of interfere density \(\lambda=\{10^{-4},10^{-3}\}\)m\({}^{-2}\). Parameters: path-loss exponent \(2b=2.4\), normalized amplitude \(|\bar{P}_{\rm o}|=1\), \(R_{max}\rightarrow\infty\). not serve any UE. In the example of Figure 6, the radius is \(R_{max}=400\) m, and thus, the service probability is not zero. Figure 7 illustrates the average number of the users served vs. the antenna ratio \(\log_{2}(N_{c}/N_{v})\), varying the SIR threshold \([T]_{dB}=\{0,5,10\}\) for small and large path-loss exponents \(2b=\{2,3.6\}\). It can be seen that in this scenario, by increasing the threshold \(T\), UCA performs worse. Note that in this setup, the UEs are very dense (\(\lambda=5\times 10^{-2}\) m\({}^{-2}\)). It can be shown that for smaller UE densities, UCA outperforms UcyIA. Figure 8 shows a similar example of the average number of users served vs. the antenna ratio \(\log_{2}(N_{c}/N_{v})\), fixing the threshold \([T]_{dB}=5\) and varying the AP height \(h=\{5,10,15,20\}\) m. Clearly, by increasing the AP height, the usage of UcyIA becomes more advantageous in terms of \(\bar{P}_{s}\), and it is more meaningful to use vertical beamforming, while for smaller heights, it is preferable to use a larger circular array. However, it can be observed that, in this scenario, where UEs are dense, the service probability, in general decreases by increasing the height. Thus, it is preferable to use an AP with a smaller height. Figure 9 shows the effect of UE density \(\lambda\). It is seen that for areas that are not very dense, a UCA is marginally more preferable. However for more dense areas, a UcyIA would be more preferable depending on different parameters. It can be shown that changing every one of the above-mentioned parameters affects the shape of the \(\bar{P}_{s}\) curves. Furthermore, the constraint of ground UEs is also responsible for the results. UcyIAs are favored when UEs have arbitrary heights (usually indoor UEs are at arbitrary heights, while when targeting high frquencies, penetration loss is too high).