Modeling trajectories as stochastic processes, we generate a conflict distribution among aircraft $i$ and $j$ defined by their relative schedule $t_j - t_i$. A conflict ratio is estimated by fixing the relative schedule of the two aircraft and computing the ratio of conflicting trajectories to the total number of sampled feasible trajectories. Conflicts are defined when trajectories come into close spatial proximity along their route. The conflict distribution is estimated by computing a conflict ratio at every whole second, see the below Figure.

For conflicts that arise between two aircraft that travel through the same merge node(departure vs departure or arrival vs arrival conflicts) there exists a well-defined sequence such that either aircraft $i$ comes prior to aircraft $j$, or vice versa. When aircraft $i$ is followed by $j$, define the minimum-time separation constraint $\delta_{ij} (\delta_{ij}^*)$ that ensures departure (arrival) pairwise separation constraints. This value can be estimated from the upper bound of the conflict distributions, see left image of Figure. The value $\delta_{ji} (\delta_{ji}^*)$ can be estimated from the lower bound of the conflict distribution. The minimum-time separation constraints are defined as strictly non-negative. Therefore, if departing aircraft $i$ is followed by departing aircraft $j$ we should separate the aircraft at the merge node by the value $\delta_{ij}$, else we separate the aircraft by the value $\delta_{ji}$.

For conflicts that arise between two aircraft that do not travel through the same merge node(departure vs arrival conflicts) there does not exist a well defined sequence, see the right panel of the above Figure. In this figure we assume departing aircraft $i$ always arrives at the merge node P1 at time $t_i = 0$ and arriving aircraft $j$ is released from the merge node P2 at the time defined by the value on the horizontal axis. Define the lower bound separation constraint $\Delta_{ij}^{LB}$ as the left boundary of the conflict distribution. Define the upper bound separation constraint $\Delta_{ij}^{UB}$ as the right boundary of the conflict distribution. Given that the values of $\Delta_{ij}^{LB}$ and $\Delta_{ij}^{UB}$ can both be negative, i.e. whether using the lower or upper bound separation constraint we release the arrival from P2 prior to the time that the departure is scheduled at node P1. Therefore, we can not simply select which separation constraint to use defined by the sequencing of aircraft at the merge node as we did before. This implies that to separate the aircraft we should release the arriving aircraft to the left of the value $\Delta_{ij}^{LB}$ or to the right of the value $\Delta_{ij}^{UB}$.

Using the sampled trajectories in the CLT center alley we estimated the following conflict distributions: