## 4 UAV Blockage Analysis
### Blockage Probability with Grounded Infrastructure  
We are now in a position to evaluate blockage probability, \(p_{B}\), with ground-mounted BSs. The input parameters are the number of LoS visibility zones characterized by their mean areas and perimeters, \(E[S_{B}]\) and \(E[L_{B}]\), in \(\Re^{2}\) plane positioned at the UAV flying altitude \(h_{R}\). To provide simple yet accurate expression for blockage probability, we will rely upon the tools of integral geometry. Further, we need two fundamental notions of integral geometry. A curious reader is referred to [29] for a basic account of information and to [30] for modern developments in the field. **Definition 1** (Kinematic density, [29]). : _Let \(K\) denote the group of motions of a set \(A\) in the plane. The kinematic density \(dA\) for the group of motions \(K\) in the plane for the set \(A\) is_  
\[dA=dx\wedge dy\wedge d\phi, \tag{10}\]  
_where \(\wedge\) is the exterior product [31], \(x\) and \(y\) are Cartesian coordinates, \(\phi\) is the rotation angle of \(A\) with respect to \(OX\)._  
**Definition 2** (Kinematic measure, [29]). : _The kinematic measure \(m\) of a set of group motions \(K\) on the plane is defined as the integral of the kinematic density \(dA\) over \(K\), that is,_  
\[m_{A}=\int_{K}dA=\int_{K}dx\wedge dy\wedge d\phi. \tag{11}\]  
Consider first a single mmWave BS in the area of interest \(A\) and let \(B\) define a LoS visibility zone. We are first interested in the probability \(p_{C}\) that UAV, located at a randomly chosen point \(P\) in \(A\), is in coverage of this BS, that is, it is located in \(B\). Using conditional probability we may write  
\[p_{C}=\frac{Pr\{P\in A\cap B\}}{Pr\{A\cap B\
eq 0\}}, \tag{12}\]  
where the probability that UAV location \(P\) belongs to the intersection area of two sets, \(A\) and \(B\), is in the nominator, while the probability that these sets do intersect is in the denominator. Using the notion of kinematic measure, we get [29]  
\[Pr\{P\in A\cap B\} =m(A:P\in A\cap B),\] \[Pr\{A\cap B\
eq 0\} =m(A:A\cap A\
eq 0), \tag{13}\]where the first expression is the kinematic measure of the set of motions of \(A\) such that \(P\in A\), while the second one provides the measure of all motions of \(A\), for which the intersection between \(A\) and \(B\) is non-zero. Following [29], the first measure is  
\[m_{j}(P\in A\cap B)=\int_{P\in B}f(x,y)dx\wedge dy\wedge d\phi, \tag{14}\]  
where \(f(x,y)\) is the density of LoS visibility zone positions in \(A\). The measure of all motions of \(A\), such that \(A\cap B\), is [29]  
\[m_{j}(A\cap B\
eq 0)=\int_{A\cap B\
eq 0}f(x,y)dx\wedge dy\wedge d\phi. \tag{15}\]  
Finally, the sought probability is given by  
\[p_{C}=\frac{\int_{P\in A\cap B}f(x,y)dx\wedge dy\wedge d\phi}{\int_{A\cap B \
eq 0}f(x,y)dx\wedge dy\wedge d\phi}, \tag{16}\]  
and can be computed for a particular form of \(A\), \(B\), and \(f(x,y)\). The numerator in (16) is computed as [29]  
\[m_{j}(P\in A\cap B) =\int_{P\in B}dx\wedge dy\wedge d\phi=\] \[=\int_{P\in B}dx\wedge dy\int_{0}^{2\pi}\hskip-14.226378 ptd\phi=2 \pi E[S_{B}], \tag{17}\]  
where \(E[S_{B}]\) is the mean area of LoS visibility zone provided in (5). The measure of motions of \(A\) is such that \(A\cap B\
eq 0\) is [29]  
\[m_{A}=\int_{K}dA=\int_{K}dx\wedge dy\wedge d\phi. \tag{11}\]0 \[m_{A}=\int_{K}dA=\int_{K}dx\wedge dy\wedge d\phi. \tag{11}\]1  
where \(E[L_{B}]\) is the perimeter of LoS visibility zone, \(S_{A}\) and \(L_{A}\) are the area and the perimeter of \(A\), given by  
\[ \begin{array}{c}S_{A}=[M_{V}*(l+b_{l})+b_{l}]*[M_{H}(l+b_{w})+b_{l}],\\ L_{A}=2[M_{V}*(l+b_{l})+b_{l}]+2[M_{H}(l+b_{w})+b_{l}].\end{array}  \tag{19}\]  
Substituting (17), (18) into (12) we obtain  
\[p_{C}=\frac{2\pi E[S_{B}]}{2\pi(S_{A}+E[S_{B}])+L_{A}E[L_{B}]}. \tag{20}\]  
Recall that mmWave BSs are deployed randomly along the streets. When mmWave BS is deployed on the crossroad it creates two LoS visibility zones as illustrated in Figure 2(b). Let \(u\) be the probability that mmWave BS is at the crossroad.