## 4 Results and Discussion
The Gaussian model, with its probabilistic nature, allows for a more sophisticated understanding and calculation of potential conflicts, dynamically adjusting to the movements and position updates of the UAVs. However, care must be taken to ensure the assumptions of normality are met and sufficient computational resources are available for real-time calculations. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Consensus Mechanism** & **Number of UAS** & **Number of Shards** & **Validation Time (ms)** & **Error Rate \%** \\ \hline \multirow{4}{*}{Proof of Work (PoW)} & \multirow{2}{*}{100} & 4 & 21 & 7.7 \\ \cline{3-5}  & & 16 & 18 & 7.2 \\ \cline{2-5}  & & 4 & 420 & 9.3 \\ \cline{2-5}  & & 16 & 365 & 8.8 \\ \hline \multirow{4}{*}{Proof of Stake (PoS)} & \multirow{2}{*}{100} & 4 & 11 & 13.1 \\ \cline{2-5}  & & 16 & 8 & 12.6 \\ \cline{1-1} \cline{2-5}  & & 4 & 121 & 17.6 \\ \cline{1-1} \cline{2-5}  & & 16 & 101 & 16.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The airspace allocation performance using The PoW and PoS. For considering conflict probability as the performance metric, firstly, it is needed to determine encounter situations between UASs. Figure 9 shows in 2D the conflict between two UASs flying at a specific altitude. As shown in Figure 9, two aircraft (UAS\({}_{i}\) and UAS\({}_{j}\)) are coming to a probable collision area, where \(\alpha\) denotes the difference between their heading angles, and their distance to the collision point 0 is represented as \(d_{i}\) and \(d_{j}\), respectively. The Bayesian geometrical approach [59] was used for computing conflict probability between UASs, which is represented by Equation (4):  
\[P_{conflict}=\frac{1}{2}\left(f\bigg{(}\frac{s-\mu}{\sigma\sqrt{2}}\bigg{)}-f \bigg{(}\frac{-s-\mu}{\sigma\sqrt{2}}\bigg{)}\right) \tag{4}\]  
where \(s\) is the minimum allowed aircraft horizontal separation in an airspace. The minimum distance is normally distributed with the mean \(\mu\) and variance \(\sigma^{2}\), and \(f\) can be obtained from Equation (5) [59]. \[f(x)=\frac{2}{\sqrt{\pi}}\eta^{x}\,exp(-t^{2})\;dt \tag{5}\]  
The mean \(\mu\) and variance \(\sigma^{2}\) are computed through Equation (6) [59]. \[\mu=\lambda\big{(}d_{j}-\rho d_{i}\big{)},\;\;\sigma^{2}=\lambda^{2}\overline {a}(t)^{2}\;(1+\rho^{2}) \tag{6}\]  
and Equation (7) [59]. \[\lambda=\frac{\sin(\alpha)}{\sqrt{\rho^{2}-2\rho\,\cos\,(\alpha)+1}},\;\;\rho =\frac{v_{j}}{v_{i}} \tag{7}\]  
where UAS\({}_{i}\) and UAS\({}_{j}\) are flying to the conflict point 0 with \(v_{i}\) and \(v_{j}\) speeds; their distance to the conflict point 0 is represented by \(d_{i}\) and \(d_{j}\), respectively. While PoS exhibited superior performance in validation time, PoW surpassed PoS in terms of error rate, which resulted in a lower probability of conflict between UASs using the PoW methodology, as presented in Figure 10.  
Figure 10: Probability of conflicts according to the PoS (left) and the PoW (right) methodologies [44]. Figure 9: Two-dimensional conflict situation between two UASs [59]. Figure 10 depicts the probability of conflicts among UASs at a specific altitude. Each cube size was considered to be \(100\) m \(*100\) m \(*100\) m in the cubic airspace. One hundred UAS-S4s were arranged at \(x=4-8\) km (latitude), \(y=8-12\) km (longitude), \(z=3.75\) km (altitude) considering 16 shards. This figure was extracted from the 10 th shard, in which 7 UAS-S4s had conflicts in 16 km\({}^{2}\) composed of 1600 airspaces sector (40 sectors along the \(x\)-axis and 40 sectors along the \(y\)-axis). Comparing the plot on the left (corresponding to PoS) with the one on the right (related to PoW) confirms that PoW outperformed PoS in terms of conflict probabilities and safety. The average conflict probability for the PoS was 34.1%, and for the PoW, it was 25.4%. The blockchain is a safe consensus protocol that can store trajectory data in an untrusted environment, even when nodes corresponding to new UAVs are added to the P2P topology.