###### Abstract  
The assessment of collision risk at intersecting air routes is a crucial method for determining safe separation during aircraft route flights. This paper employs the Monte Carlo method to analyze the operational characteristics of aircraft on intersecting air routes in two stages, integrating the influence of the human-operated adjustment process on the distance between the two aircrafts, so as to propose a collision risk assessment model more aligned with the actual operational process. The initial stage considers the positional errors to establish a distance distribution function between trailing and leading aircraft when the latter reaches the intersection. The subsequent stage calculates the minimum distance between the aircraft by combining the kinematic equations with the controller's thinking and reactions, communications, the pilot's reactivity, and time required for aircraft maneuvering. Ultimately, based on the intersection course configuration, Monte Carlo simulations were employed to assess the impact of variables, such as magnetic course and aircraft speed distribution, on collision risk. The study's findings highlight the significance of aircraft  
==WARNING: Truncated because of repetitions==
& 37 & 7 & 23 & 53 & 83 \\ \(\theta(^{\circ})\) & 186 & 156 & 126 & 96 & 66 & 36 & 6 & 24 & 54 \\ D2 (km) & 25.888 & 27.881 & 32.285 & 40.389 & 58.28 & 169.048 & 99.073 & 54.225 & 42.537 \\ \hline \hline \end{tabular} \end{table}
Table 7: D2 corresponding to different magnetic course \(\theta_{2}\) at TLS. Figure 12: Graph of collision risk variation with trailing aircraft distance to intersection and crossing angles \(\theta\). +++  
they will be mapped to different hash values and have a greater impact on the safety of the operation. On the other hand, it can be found that the smaller the standard deviation, the greater the rate of change of collision risk with distance, and given the TLS, it can be found that its corresponding distance from the trailing aircraft to the intersection will be smaller, which means that the more stable the cruising speed on the airway is, the higher the safety will be, and the safe separation that needs to be maintained will be reduced. While the histogram of cruising speeds seems to align more closely with a normal distribution, it is important to consider that speeds less than 0 are not feasible. Therefore, the speed should vary within a specific range. When compared with actual operational data, these characteristics suggest that a uniform distribution might be a more accurate representation of cruising speed variations. When the speed obeys the uniform distribution, that is, \(V_{1}\sim U(a,b),V_{2}\sim U(c,d)\), so that \(V=1/V_{1}\), \(f_{V}(v)=f_{V_{1}}(\frac{1}{v})\cdot\left|\frac{dv}{dv_{1}}\right|=\frac{1}{( b-a)^{\omega^{2}}}\). After a simple transformation, \(Z=V_{2}/V_{1}=V_{2}*V\) can be obtained; by the convolution formula, the following can be obtained:  
\[f_{Z}(z)=\int\limits_{c}^{d}f_{V_{2}}(v_{2})\cdot f_{V}(\frac{z}{v_{2}})dv_{2 }=\frac{c^{2}+d^{2}+cd}{3(b-a)z^{2}},\frac{c}{b}<z<\frac{d}{a} \tag{10}\]  
Building on Equation (4), we can derive the probability density function of the distance of the trailing aircraft from the intersection when the leading aircraft is at the intersection. The key aspect of the uniform distribution is the determination of its endpoint values. Our approach involves two steps: We first examine the impact of speed variability on collision risk. This is done by simulating the movement safety of the aircraft at the intersection, assuming both aircraft follow the same uniform distribution. The results of this simulation are illustrated in Figure 14a. Next, we keep the speed distribution of the leading aircraft constant and simulate the collision risk by varying the speed distributions of the trailing aircraft. This step is aimed at understanding the effects of speed differences between the two aircraft. The findings from this analysis are depicted in Figure 14b.