## 4 UAV Blockage Analysis
### Geometric Structure of LoS Zones  
We start by characterizing the LoS visibility zone induced by BR BS located along the street, see Figure 2a. As one may observe, this zone is of rectangular shape with sides that depend on (i) heights of buildings, \(H_{B,1}\) and \(H_{B,2}\), (ii) maximum coverage of BS, \(r\), and (iii) UAV altitude \(h_{R}\).  
Observing Figure 3a, the length of the LoS visibility zone is  
\[D=2\sqrt{r^{2}-(h_{R}-h_{T})^{2}}, \tag{1}\]  
where \(r\) is the maximum communications distance,  
\[r=\sqrt{10^{\frac{P_{A}+C_{R}+C_{T}-N_{0}-S_{T}-32.4-20\log_{10}C_{C}}{2}}-[h_ {R}-h_{T}]^{2}}, \tag{2}\]  
where \(S_{T}\) is the SNR threshold, \(G_{T}\) and \(G_{R}\) are the transmit and receive antenna gains, \(P_{A}\) is the emitted power at mmWave BS, \(N_{0}\) is the thermal noise, \(F_{C}\) is the carrier frequency.  
The width of the LoS visibility zone, \(L\), is an RV that is determined by building heights, \(H_{B,1}\) and \(H_{B,2}\), where both have the same pdf \(f_{H_{B}}(x)\), see Figure 3b. Observe that angles \(\alpha_{1}\) and \(\alpha_{2}\) are given by  
\[\alpha_{i}=\tan^{-1}\left(\frac{H_{B,i}-h_{T}}{l_{A,i}}\right),\;i=1,2, \tag{3}\]  
where \(l_{1}\), \(l_{2}\) are the distances to the buildings, see Figure 3b.  
Figure 2: Two types of feasible LoS visibility zones in the considered scenario. (**a**) BS located along the street; (**b**) BS located at the crossroad.  
Further, using \(\tan\beta_{i}=L_{i}/(h_{R}-h_{T})\), \(i=1,2\) and observing that angles \(\beta_{i}\) are related to \(\alpha_{i}\) as \(\beta_{i}=\pi/2-\alpha_{i}\) we arrive at the following expressions for RVs \(L_{1}\) and \(L_{2}\)  
\[L_{i} = (h_{R}-h_{T})\tan\biggl{(}\frac{\pi}{2}-\tan^{-1}\biggl{[}\frac{ H_{B,i}-h_{T}}{l_{A,i}}\biggr{]}\biggr{)}= \tag{4}\] \[= \frac{l_{A,i}(h_{R}-h_{T})}{H_{B,i}-h_{T}},\,i=1,2.\]  
One may now determine the mean area of the LoS visibility zone as  
\[E[S_{B}] = D\int\limits_{0}^{\infty}\int\limits_{0}^{\infty}f_{H_{B}}(x)f_{ H_{B}}(y)[L_{1}(x)+L_{2}(y)]dxdy= \tag{5}\] \[= 2\sqrt{r^{2}-(h_{R}-h_{T})^{2}}E[L_{B}],\]  
where the mean length of the LoS visibility zone, \(E[L_{B}]\), is provided by  
\[E[L_{B}] = \int_{0}^{\infty}\int_{0}^{\infty}\biggl{(}\frac{l_{A,i}(h_{R}-h_ {T})}{x-h_{T}}\biggr{)}\times \tag{6}\] \[\times \biggl{(}\frac{l_{A,i}(h_{R}-h_{T})}{y-h_{T}}\biggr{)}f_{H_{B}}(x) f_{H_{B}}(y)dxdy,\]  
that can be evaluated in closed-form for a given distribution of the building height.  
A simple yet reliable approximation for (5) can be obtained by assuming the same random height of both buildings on the street, as it is usually the case in practice. In this case, the width of the blockage zone becomes  
\[L_{B}=L_{1}+L_{2}=\frac{(h_{T}-h_{R})(l_{A,1}+l_{A,2})}{x-h_{T}}, \tag{7}\]  
Figure 3: Geometrical illustration of the sides of LoS visibility zone. (**a**) Length of the LoS visibility zone; (**b**) Width of the LoS visibility zone.  
implying that (6) can be written as  
\[E[L_{B}]=\int_{0}^{\infty}f_{H_{B}}(x)\frac{(h_{T}-h_{R})(l_{A,1}+l_{A,2})}{x- hT}dx. \tag{8}\]  
For example, for \(H_{B}\) having uniform distribution in \((A,B)\) we have  
\[E[L_{B}]=\frac{(h_{T}-h_{R})(l_{A,1}+l_{A,2})(\log[1-\frac{A}{h_{T}}]-\log[1- \frac{B}{h_{T}}])}{B-A}. \tag{9}\]  
Similarly, the mean perimeter of the LoS visibility zone \(B\) is \(E[L_{B}]=2D+2E[L]\). The LoS visibility zones induced by BS deployments on the crossroads can be found similarly. Indeed, as one may observe in Figure 2b, they consist of two overlapping LoS visibility zones forming a \"cross\". Individually, parameters of these two zones can be estimated as shown above.