## 4 Performance Evaluation
### Multicast Group Formation  
We propose to form multicast groups according to the maximum HPBW angle \(\alpha\) for a given coverage radius \(R\). To evaluate \(\alpha\), we first define the appropriate antenna gain as  
\[G_{A}=\frac{S(R)[N_{0}+M_{T}]}{P_{A}G_{U}R^{-\xi_{T}}e^{-KR}p_{B}(R)}.\]  
Then, once we obtain \(G_{A}\), we can easily define the HPBW value for specified antenna parameters the way it is presented in Table 2. Here we denote \(\alpha(G_{A})\) as HPBW angle corresponding to the next nearest predefined value of \(G_{A}\). Now, we can establish the number of multicast groups \(N\) as  
\[N=\big{[}\Theta/\alpha(G_{A})\big{]},\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Array** & **Gain, dBi** & **HPBW, \({}^{o}\)** \\ \hline
64 \(\times\) 4 & 17.58 & 1.59 \\
32 \(\times\) 4 & 14.58 & 3.18 \\
16 \(\times\) 4 & 11.57 & 6.37 \\
8 \(\times\) 4 & 8.57 & 12.75 \\
4 \(\times\) 4 & 5.57 & 25.50 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Antenna parameters utilized in calculations. where \(\Theta\) is the arc segment of the BS antenna. To employ [PERSON]'s model, we convert the HPBW angle to the length of the corresponding arc segment circumference  
\[\xi=\frac{\alpha(G_{A})\pi R}{180},\]  
as shown in Figure 2.  
Following [27], the gaps between two neighbor shadows follow exponential distribution with parameter \(\frac{\lambda K^{2}}{2R}\), allowing us to find the probability \(q_{k}\) that a gap \(\omega\) is confined between \(k\) and \(k+1\) segments as  
\[q_{k}=e^{-\frac{\lambda K^{2}q_{k}^{2}}{2R}}-e^{-\frac{\lambda K^{2}(k+1)^{2} }{2R}},\]  
and can eventually not be covered by beams. This allows us to estimate the mean number of void beams \(\Delta\) as  
\[\Delta=\frac{\pi\lambda K^{2}}{3\big{(}e^{\lambda B(R-Q)+\lambda}\big{)}}\sum _{k=1}^{N}kq_{k}. \tag{2}\]  
Finally, the mean number of required beams is calculated as \(N-\Delta\). The demand for resource of a single beam depends on whether all the covered UEs are in LoS conditions, or there at least one UE is blocked and requires the whole group to lower its MCS [28]. Thus, to estimate the mean resource required by an active beam, we first calculate the probability of having \(u\) UEs covered by the beam that follows Poisson distribution in accordance with the PPP as  
\[q_{u}=\frac{e^{-\lambda_{u}}\lambda_{n}^{u}}{u!},\]  
where \(\lambda_{n}=\lambda\pi(R^{2}-Q^{2})(\alpha/360)\). Eventually, the mean demand for an active beam is calculated as  
\[b_{M}=\sum_{u=1}^{\infty}q_{u}\Bigg{[}b_{M,L}\prod_{i=1}^{u}\left(1-p_{B,i} \right)+\left(1-\prod_{i=1}^{u}\left(1-p_{B,i}\right)\right)b_{M,B}\Bigg{]},\]  
where \(b_{M,L}\) and \(b_{M,B}\) are the demands for resource blocks in LoS non-blocked and LoS blocked conditions respectively, while \(p_{B,i}\) is the LoS blockage probability of the UE that is \(i\)-s closest to the NR BS. Figure 2: Illustration of the proposed multicast group formation procedure. Following [25], LoS blockage probability depends on both intensity of blockers entering the blockage zone, i.e.,  
\[\mu_{B,i}=\int\limits_{0}^{\infty}f_{i}(x)\frac{(x[h_{B}-h_{U}]+r_{B}[h_{T}-h_{U }])}{(2r_{B}\lambda_{B}v)^{-1}(h_{T}-h_{U})}dx.\]  
where \(h_{A}\), \(h_{U}\), and \(h_{B}\) are the heights of NR BS, UE, and blockers respectively, \(r_{B}\) and \(v\) are the radius and speed of blockers; and the distance between a UE and NR BS that follows the pdf of the distance to the \(i\)-s nearest neighbor in PPP [24]  
\[f_{i}(x)=\frac{2(\pi\lambda)^{i}}{(i-1)!}x^{2i-1}e^{-\pi\lambda x^{2}},\ x>0,\ i=1,\ldots\]  
Then, LoS blockage probability of the UE can be found as  
\[\xi=\frac{\alpha(G_{A})\pi R}{180},\]0  
where \(2r_{B}/v\) is the mean time for a blocker to cross the LoS blockage zone at right angle to its long side. Another grouping approach based on multi-session multicasting is presented in [29]. A grouping algorithm allows to achieve better spectral efficiency and obtain an optimal resource allocation using convex optimization.