# Characterization of Dynamic Blockage Probability in Industrial Millimeter Wave 5G Deployments
## 6 Numerical Results
### BS to UE Communications Scenario  
The utilization of UE-to-BS communications can efficiently solve the problem of high direct UE-UE blockage probability. Here, the additional critical parameters that define the blockage probability are machine height, \(H_{B}\), UE height, \(h_{U}\), and BS height \(h_{A}\). We start exploring their effects in Figure 7, where the blockage probability \(p_{B}(l_{x},l_{y})\) is shown as a function of exponentially distributed machine height \(H_{B}\) and constant UE height \(h_{U}\) for different BS heights. Note that in Figure 7a, UE height is 1 m, while in Figure 7b the mean machine height is also 1 m. The rest of the parameters are: site occupation probability \(\
u=0.5\), machine width and length 3 m, and transparency probability \(\kappa=0.5\), UE coordinates \((10,10)\).  
By analyzing the presented results, we observe that increasing the BS height allows the blockage probability to be drastically improved. Specifically, even having a BS height of 10 m allows the blockage probability to be decreased from approximately 0.8 corresponding to the UE-to-UE case to just 0.45, see Figure 7a. Another important takeaway is that randomness in machine heights may even make the situation worse as compared to direct UE-UE communications. In both, Figure 7a,b we observe that even for BS height of 4 m the LoS blockage probability is slightly higher than that of indirect UE-UE communications, 0.84 vs. 0.8. Finally, by comparing the results for different UE height and mean machine height, we see that the former produces more impact on the blockage probability. This implies that in practical deployments, all efforts need to be made to install communications modules at higher possible locations at machines.  
Figure 6: UE to UE communications scenario. (**a**) The effect of machines dimensions. (**b**) The effect of machine’s transparency.  
In Figure 7 we observed that the randomness of the machines' height produces substantial impact on blockage probability. To quantify this impact, Figure 8 compares the blockage probabilities for constant and random machines height with coinciding means as a function of the cite occupancy probability, \(\
u\). The rest of the parameters are the same as in Figure 7. Here, we observe that the difference may indeed be drastic, reaching 0.6 (for \(E[H_{B}]=H_{B}=2\) m and \(\
u=0.2\)). However, this difference diminishes as the height of blockers increases. Nevertheless, whenever possible, an attempt needs to be made to ensure that the height of the deployed machines is constant.  
Figure 7: BS to UE communications with random machine heights. (**a**) The effect of machine height. (**b**) The effect of UE height.  
Finally, in this section, we present the average blockage probability for a square compartment, calculated according to (11). Recall that the average blockage probability is a typical measure utilized for dimensioning of wireless access networks, where a geometrical location of the session is expected to be random instead of fixed [30]. Using this metric, one may further utilize the proposed model in capacity planning and dimensioning of such industrial systems. Figure 9 shows the average blockage probability for UE randomly and uniformly distributed in a square factory compartment with side \(R\) as a function of \(R\) and for multiple BS heights. The rest of the parameters are: site occupancy probability \(\
u=0.5\), mean width and length of the machine of \(1/\lambda_{x}=1/\lambda_{y}=3\) m, transparency probability \(\kappa=0.5\), \(h_{A}=4\) m, UE height \(h_{L}=1\) m, and constant machine height of \(H_{B}=2\) m. The presented results show the expected behavior--the average blockage probability increases with the factory side length \(R\). For the chosen parameters, the increase is rather mild for all considered BS heights and the probability never reaches 0.8 even for BS height \(h_{A}=4\) m and \(R=50\) m. This implies that BS-to-UE communications, even in the rather dense considered deployment, is actually feasible.