## 4 Apron U-Shaped Area Traffic Operation Model
### Traffic Modeling under Partition-Shared Operation Procedure  
We can also divide the aircraft waiting time into three categories by \(\sigma_{A}^{f},\sigma_{D}^{f}\), and the aircraft operating in the U-shaped area based on the description in Sections 3.3 and 4.2. The details are as follows. **Case 1:** Same as case 1 in the Exclusive Operation Procedure, the arrival or departure aircraft can directly enter the U-shaped area or push out, i.e., \(w_{f}^{p}=0,f\in\mathcal{F}_{A}\cup\mathcal{F}_{D}\). Based on the description in Section 3.3, in this scenario, there is no need to wait if there is no aircraft in operation in the U-shaped area. We also note that the partition-shared operation procedure is different from the exclusive operation procedure. There is no need to wait for aircraft \(f\) when the partitional area of the parking stand of \(f\) is different from that of the aircraft \(f^{\prime}\) in operation, as shown in Equation (6). \[w_{f}^{p}=0 \begin{cases}&\delta_{f^{\prime}}^{p^{\prime}}>X_{23},p\in\mathcal{ P}_{1}\cup\mathcal{P}_{2},p^{\prime}\in\mathcal{P}_{3},f,f^{\prime}\in \mathcal{F}_{A}\\ &\delta_{f^{\prime}}^{p^{\prime}}>X_{12},p\in\mathcal{P}_{1},p^{\prime}\in \mathcal{P}_{2}\cup\mathcal{P}_{3},f,f^{\prime}\in\mathcal{F}_{A}\\ &\delta_{f^{\prime}}^{p^{\prime}}<X_{12},p\in\mathcal{P}_{3},p^{\prime}\in \mathcal{P}_{1}\cup\mathcal{P}_{2}\cup\mathcal{P}_{3},f,f^{\prime}\in\mathcal{F }_{D}\end{cases}  \tag{6}\]  
where \(X_{12}\) is the distance from the middle area boundary to the U-shaped area entrance/exit, \(X_{23}\) is the distance from the inner area boundary to the U-shaped area entrance/exit. It is worth noting that there cannot be a queued aircraft before that aircraft, so \(\left|\sigma_{A}^{f}\right|=\left|\sigma_{D}^{f}\right|=0\) also needs to be satisfied in Case 1. **Case 2:** Except for Case 1, the arrival aircraft \(f\) (departure aircraft \(f\)) can enter the U-shaped area directly (can be pushed out directly), but needs to wait for the aircraft \(f^{\prime}\) operating in the U-shaped area to leave the U-shaped area (or park on arrival). We obtain  
\[w_{f}^{p}=\frac{L_{p^{\prime}}-\delta_{f^{\prime}}^{p^{\prime}}}{V_{f^{\prime }}}+\sigma_{A}^{f}\ \ \ f\in\mathcal{F},f^{\prime}\in\mathcal{F}_{A},\forall p,p^{\prime}\in \mathcal{P} \tag{7}\]  
\[w_{f}^{p}=\frac{\delta_{f^{\prime}}^{p^{\prime}}}{V_{f^{\prime}}}\ \ \ f\in \mathcal{F},f^{\prime}\in\mathcal{F}_{D},\forall p,p^{\prime}\in\mathcal{P} \tag{8}\]  
We can conclude that departure aircraft located in the middle area can use long push as well as push-and-pull strategies to reduce the delay time based on the description in Section 3.3. From this, we define the time of the long push of the departure aircraft in the middle area as \(\varpi_{DL}^{f}\); when \(p\in\mathcal{P}_{2}\), the delay time \(w_{f}^{p}\) in the long push case is shown in the following equation  
\[w_{f}^{p}=t_{early}^{f^{\times}}+\varpi_{D}^{f^{\times}}-t_{early}^{f}-\varpi_{ DL}^{f},p\in\mathcal{P}_{2},p^{\times}\in\mathcal{P}_{1},f,f^{\times}\in \mathcal{F}_{D} \tag{9}\]  
where \(f^{\times}\) is the aircraft queued next to and after \(f\) on FCFS principle. It is worth noting that there cannot be a queued aircraft before that aircraft, so \(\left|\sigma_{A}^{f}\right|=\left|\sigma_{D}^{f}\right|=0\) also needs to be satisfied in Case 2. **Case 3:** The arrival aircraft \(f\) (departure aircraft \(f\)) can enter the U-shaped area directly (can be pushed out directly), but needs to wait for the aircraft \(f^{\prime}\) operating in the U-shaped area, \(\sigma_{A}^{f}\) and \(\sigma_{D}^{f}\) to leave the U-shaped area (or park on arrival), and the operation separations for \(\sigma_{A}^{f}\) and \(\sigma_{D}^{f}\) are considered. We obtain  
\[w_{f}^{p}=w_{f}^{p^{\prime}}+w_{f}^{p}* \tag{10}\]  
where \(w_{f}^{p}\) consists of two components. \(w_{f}^{p}\) is the waiting time for an aircraft in operation and is calculated by Equations as shown in Case 2.