# Performance of Dense Wireless Networks in 5G and beyond Using Stochastic Geometry
## 2 System Model
### Signal Model and Service Probability  
Let \(x\) be the transmitted signal; the signal received by the AP with beamforming pointing toward the UE of interest with angles \(\phi_{o}=0\) and radial distance \(R_{o}\) is  
\[y=\frac{\beta_{o}}{(R_{o}^{2}+h^{2})^{\frac{b}{2}}}G(\phi_{0},\theta_{0})x+ \iota+w, \tag{3}\]  
where \(w\sim CN(0,\sigma_{w}^{2}/N)\) is the additive Gaussian noise with power \(\sigma_{w}^{2}/N\) after the array gain, and  
\[\iota=\sum_{i=1}^{\infty}\frac{\beta_{i}}{\left(R_{i}^{2}+h^{2}\right)^{\frac{ b}{2}}}G(\phi_{i},\theta_{i})x_{i}. \tag{4}\]  
is the aggregated interference originated from PPP-distributed interfering UEs with density \(\lambda\); all signals generated by all UEs are \(x_{i}\sim CN(0,1)\). We assume that the aggregate interference power \(I=|\iota|^{2}\) is typically \(\mathbb{E}[I]\gg\sigma_{w}^{2}/N\) as in the macro-cell network [22]. This assumption is relaxed in Section 5, since in high frequencies, the noise is not negligible in small cells, especially in the presence of blockage [43]. The outage analysis depends on the PDF of the aggregated interference (4) for the PPP distribution of UEs, each having polar coordinates \((R_{i},\theta_{i})\). The PDF of \(I\) without any array-gain for interference mitigation (here is a specific case with \(G^{2}(\phi_{i},\theta_{i})=1\) for any \(i\)) and \(h=0\) has been extensively investigated in the literature (see, e.g., [11; 44; 45]). The scope here is to evaluate the PDF of aggregated interference \(I\) for UcylA and \(h\geq 0\). The \(N_{c}\times N_{v}\) UcylA is the most general case as UCA is when \(N_{v}=1\) and the single antenna is when \(N_{c}=N_{v}=1\), as shown in Table 1, with the corresponding references or sections for the analytic form of CF. Note that the density \(\lambda\) refers to the number of _active_ UEs per square meter coexisting on the same time-frequency and, depending on the specific radio resource allocation strategies, is likely to be meaningfully lower than the effective crowd density [46]. The signal-to-interference ratio (SIR) is defined as  
\[\text{SIR}=\frac{|\beta_{0}|^{2}G^{2}(\phi_{0},\theta_{0})}{I(R_{0}^{2}+h^{2} )^{b}}. \tag{5}\]  
The average probability of successful connection experienced by the UE of interest in \((\phi_{o},R_{o})\) depends on a certain threshold \(T\), on fading fluctuation \(|\beta_{o}|^{2}\), and on the overall interference \(I\). The service probability from the distribution of \(I\) is  
\[P_{s}(R_{o}|R_{max})=F_{I}\left(\frac{|\beta_{0}|^{2}G^{2}(\phi_{0},\theta_{0} )}{T(R_{o}^{2}+h^{2})^{b}}\right), \tag{6}\]  
where \(F_{I}(x)=\Pr(I<x)\) is the CDF. The service probability (6) is for the interference power \(I\), which accounts for the randomness of the position of the interferers according to the PPP model within a certain radius \(R_{max}\) and fading. The fluctuations of the interferers \(|\beta_{i}|\) are embodied in the CF derivations (Section 3). The reference user in \((R_{o},\phi_{o})\) is considered as deterministic for the computations of the (conditional) service probabilities, but whenever necessary for the unconditional probability, it can be assumed as PPP distributed in the same way as for the other interferers. Notice that the service probability \(P_{s}(R_{o}|R_{max})\) depends on the CDF of the aggregated interference \(I\) to be evaluated for the PPP distribution of active interferers, the array-type and its height \(h\), as evaluated in the next section based on CF analysis. For the unconditional service probability accounting for the fading of the user of interest, one should evaluate the expectation  
\[\tilde{P_{s}}(R_{o}|R_{max})=\mathbb{E}_{|\beta_{0}|^{2}}[P_{s}(R_{o})]. \tag{7}\]  
Differently from the interference analysis that aggregates multiple (and many) interfering contributions into the CF of \(I\), (7) depends on the specific PDF of \(\beta_{0}\) thus making the service