## 4 Apron U-Shaped Area Traffic Operation Model
### Traffic Modeling under Global-Shared Operation Procedure  
We can 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 3.4 and 4.2. The details are as follows:  
**Case 1:** Same as case 1 in the Exclusive Operation Procedure and Partition-Shared Operation Procedure, the arrival or departure aircraft can enter or push out from the U-shaped area directly, i.e., \(w_{f}^{p}=0,f\in\mathcal{F}_{A}\cup\mathcal{F}_{D}\). Based on the description in Section 3.4, 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 global-shared operation procedure is different from the exclusive operation procedure. There is no need to wait for aircraft \(f\) as long as the lowest separation of the aircraft is satisfied, the detail is shown in Equation (12). \[w_{f}^{p}=0\left\{ \begin{aligned} \delta_{f^{\prime\prime}}^{p^{ \prime}}>d_{\min},f,f^{\prime}\in\mathcal{F}_{A}\\ \left|L_{p}-\delta_{f^{\prime}}^{p^{\prime}}\right|>d_{\min},f,f^{ \prime}\in\mathcal{F}_{D}\end{aligned} \right. \tag{12}\]  
where \(d_{\min}\) is the minimum safe separation between aircraft in a U-shaped zone and \(f^{\prime}\) is the aircraft in the U-shaped area that is operating whose corresponding parking stand is \(p^{\prime}\). **Case 2:** the arrival aircraft \(f\) (departure aircraft \(f\)) can enter the U-shaped area directly (can be pushed out directly), but it needs to wait until there is a safe separation between \(f\) and the aircraft \(f_{1}^{\prime},\cdots f_{n}^{\prime}\) in operation in the U-shaped area. When the arrival aircraft reaches the entrance of the U-shaped area (when the departure aircraft is ready to be pushed out), there may be more than one aircraft in operation in the U-shaped area. The waiting time of the arrival aircraft at the entrance (the waiting time of the departure aircraft at the parking stand) shall be determined according to the number of aircraft in operation and their status (arrival or departure). We obtain  
\[w_{f}^{p}=\left\{ \begin{aligned} \max(\max(\frac{\int_{f_{1}^{ \prime}}^{\sigma_{f^{\prime}}^{\prime}}}{\int_{f_{2}^{\prime}}^{\sigma_{f^{ \prime}}^{\prime}}},\frac{\delta_{f^{\prime\prime}}^{\prime\prime}}{\int_{f_{ 2}^{\prime}}^{\sigma_{f^{\prime}}^{\prime\prime}}},0)-t_{early^{\prime}}^{f}f \in\mathcal{F}_{A},f_{1}^{\prime}\cdots f_{n}^{\prime}\in\mathcal{F}_{D}\\ \max(\max(\frac{\frac{d_{\min}-\delta_{f^{\prime}}^{\prime}}{\int_ {f_{1}^{\prime}}^{\prime}}}{\int_{f_{1}^{\prime}}^{\sigma_{f^{\prime}}^{\prime \prime}}},\frac{d_{\min}-\delta_{f^{\prime\prime}}^{\prime\prime}}{\int_{f_{ 2}^{\prime}}^{\sigma_{f^{\prime}\prime}}},0)-t_{early^{\prime}}^{f}f,f_{1}^{ \prime}\cdots f_{n}^{\prime}\in\mathcal{F}_{A}\\ \max(\max(\frac{\frac{L_{f^{\prime}}-\delta_{f^{\prime}}^{\prime }}{\int_{f_{1}^{\prime}}^{\sigma_{f^{\prime}}^{\prime}}},\frac{L_{f^{\prime}}- \delta_{f^{\prime\prime}}^{\sigma_{f^{\prime}\prime}}}{\int_{f_{2}^{\prime}}^ {\sigma_{f^{\prime}\prime}}},\cdots,\frac{L_{f^{\prime}}-\delta_{f^{\prime \prime}}^{\prime\prime}}{\int_{f_{n}^{\prime}}^{\sigma_{f^{\prime}\prime}}},0)- t_{early^{\prime}}^{f}f\in\mathcal{F}_{D},f_{1}^{\prime}\cdots f_{n}^{\prime}\in \mathcal{F}_{A}\\ \max(\max(\frac{L_{f^{\prime}}-\delta_{f^{\prime}}^{\prime}+d_{ \min}}{\int_{f_{1}^{\prime}}^{\sigma_{f^{\prime}}^{\prime}}},\frac{L_{f^{\prime }}-\delta_{f^{\prime\prime}}^{\sigma_{f^{\prime}}^{\prime}}+d_{\min}}{\int_{f_{ 2}^{\prime}}^{\sigma_{f^{\prime}\prime}}},0)-t_{early^{\prime}}^{f}f,f_{1}^{ \prime}\cdots f_{n}^{\prime}\in\mathcal{F}_{D}\end{aligned} \right. \tag{13}\]  
where \(t_{early}^{f}\) is the earliest time to enter the U-shaped area (the earliest off-block time) for \(\forall f\in\mathcal{F}_{A}\) (\(\forall f\in\mathcal{F}_{D}\)). 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.