## 5 Conclusions  
In this paper, we combined the analysis of the time required for controllers and pilots to cooperate to complete the process of human-operated adjustment to calculate the minimum distance between two aircraft. We then assessed the collision risk of intersecting routes using the Monte Carlo method, ensuring that our simulation scenarios closely resembled actual aircraft operations. Unlike most previous research that focused predominantly on positioning errors, our study emphasizes the crucial role of human factors in operational processes. Based on our simulation results, we provide a graph depicting the relationship between collision risk and the distance of the trailing aircraft from the intersection, specifically when the leading aircraft is 10 km from the intersection. Also, the safe separation (D2) that the trailing aircraft should maintain from the intersection in different simulation scenarios for a given TLS was calculated:  
1. By simulating variations in the angles of each magnetic course, it was found that: as the angle \(\theta\) of the two aircraft heading towards the intersection keeps increasing, the collision risk decreases and then increases, and is minimized at about 135\({}^{\circ}\). When the leading aircraft is heading away from the intersection, the smaller the projected angle \(\varphi\) of the trailing aircraft on its route is, the higher the collision risk. Taken together, when the two take larger values, the corresponding collision risk is relatively smaller, i.e., the trailing aircraft needs to maintain a smaller safe separation from the intersection to meet the existing TLS;
2. Minimizing both position and speed errors in aircraft is crucial, as this not only reduces the need for safe separation but also leads to a more controllable operational process;
3. The simulation results for safe separation are influenced by the characteristics of the speed distribution. This study used real operational data to simulate both normal and uniform distributions, finding that the rate of change in collision risk was greater with the uniform distribution, largely due to its higher proportion of tail data. The inverse transformation method can be used to generate random numbers for simulating collision risk under different speed distributions on intersecting routes, which can contribute to enhancing air traffic management and flight safety;  
Figure 15: Graph of collision risk variation with trailing aircraft distance to intersection for different overall human-operated adjustment times. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Overall Time Spent on Human-Operated Adjustment (s)** & **10** & **12** & **14** & **16** & **18** & **20** & **22.242** \\ \hline D2 (km) & 30.998 & 32.796 & 34.691 & 36.193 & 38.133 & 39.865 & 41.646 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: D2 corresponding to different overall time spent on human-operated adjustment \(\tau\) at TLS. * In response to different human-operated adjustment times, as the time shortens, the required safe separation correspondingly decreases. This situation places higher demands on the qualities of both air traffic controllers and pilots. They need to not only possess solid professional knowledge but also exceptional emergency handling capabilities. To ensure flight safety by quickly and accurately assessing situations and taking appropriate measures in emergencies, the following measures can be implemented:
* Enhance Professional Training: conduct regular and comprehensive training for air traffic controllers and pilots, such as updating them on the latest aviation regulations, the use of advanced technologies, and the development of emergency handling skills.