## 4 Simulation and Results
### Further Investigation  
In this section, we conduct additional experiments to test the scalability of the proposed algorithms. In [23], scalability is defined as the ability of an algorithm, system, or model to deal with increasing numbers of missions at a reasonable cost, or its ability to be effectively enlarged to commensurate that growth.  
In practical applications, we may encounter different settings of UAVs, which depend on different values of densities, variable \(R_{\textit{gf}}\) and sense range. For instance, with the increase of applications such as surveillance in which various UAVs owning different capabilities and properties would cooperate and compete for a certain target, it is expected that the airspace traffic density will grow considerably in certain reduced areas during short time periods. Since the CA system is applied independently to each cluster, the increasing UAV number in the whole airspace may improve the encounter possibility, thereby raising the number of clusters, rather than the number of UAVs involved in a specific cluster.  
To validate the scalability, we use the computation time as a parameter to record the performance of the proposed CA system in handling different-density scenarios. Based on the simulation results in Table 3, the average computation time (over 100 runs of random flight tests) for low density (20 UAVs), medium density (60 UAVs), and high density (100 UAVs) traffic do not grow exponentially as the airspace density increases, attributing the high scalability to the independent clusters, which can be frequently processed.  
In addition, several extensive simulations with varying values of tuning parameters \(R_{\textit{gf}}\) and sense range (SR) are constructed to observe the performance in terms of efficiency. The CA maneuvers should be optimal so that UAVs can arrive at their respective destination in minimum possible time. Define \(t_{i}^{e}\) as the expectant flight time (not to make any deviation of the planned trajectory) and \(t_{i}^{r}\) as the realistic flight time (fully follow the CA maneuvers) of \(UAVi\) (\(i=1,2,..,n\)), and the efficiency is given as [24]:  
\[Efficiency=\frac{1}{n}\sum_{i=1}^{n}\frac{t_{i}^{e}}{t_{i}^{r}} \tag{15}\]  
Figure 10: Pitch angle profile of both UAVs.  
\begin{table}
\begin{tabular}{c c} \hline
**UAV Number** & **Computation Time (s)** \\ \hline
20 & 0.92 \\
60 & 5.28 \\
100 & 17.62 \\ \hline \end{tabular}
\end{table}
Table 3: Time taken in different densities.  
Figure 11 depicts the simulation results of efficiency with variable parameters. We could conclude the following:  
* The efficiency is decreasing as the \(R_{\mathit{cf}}\) value increases, because extensive conflict volume leads to more threats that would be detected by the CD algorithm, thus more waypoints have to be amended to avoid potential collisions resulting in the higher realistic flight time.
* Compared to SR = 9 km, the efficiency is lower when SR is equal to three kilometers with the same \(R_{\mathit{cf}}\), because the restricting sensor range may ignore the neighboring threats that could be synthetically resolved, and lately detect the intruder UAV, causing the number of amended waypoints to be higher.
* With the same parameters of \(R_{\mathit{cf}}\) and sense range, high efficiency is obtained if the complete CR algorithm including the causal analysis is utilized rather than only the DDOA. With the positive impact of causal analysis, the average increment of efficiency is respectively 2.83% and 3.47% for SR = 9 km and SR = 3 km. Consequently, the causal analysis is effective for the selection of optimal resolution trajectory.