## 4 Apron U-Shaped Area Traffic Operation Model
### Traffic Modeling under Exclusive Operation Procedure  
In the following part, we introduce the U-shaped area arrival aircraft queueing set \(\sigma^{A}\) and the departure aircraft queueing set \(\sigma^{D}\), respectively, because we need a quantitative representation of the aircraft waiting time. \(\sigma_{A}^{f}\) is the set of arrival aircraft that are lined up at the entrance to the U-shaped area when the aircraft \(f\) arrives in the U-shaped area (when ready to push back out of the block, \(f\in\mathcal{F}_{D}\)). \(\sigma_{D}^{f}\) is the set of departure aircraft in the U-shaped area that are preparing to push back out of block when the aircraft \(f\) reaches the U-shaped area (when preparing to pull out of block, \(f\in\mathcal{F}_{D}\)). \(\sigma_{A}^{f}\), \(\sigma_{D}^{f}\), and the aircraft \(f^{\prime}\) in operation in the U-shaped area are of great interest because we can use them to classify the aircraft waiting time into three categories. The details are as follows. **Case 1:** The arrival aircraft can directly enter the U-shaped area (the departure aircraft can directly push out), i.e., \(w_{f}^{p}=0,f\in\mathcal{F}_{A}\cup\mathcal{F}_{D}\). Based on the description in Section 3.2, when an arrival aircraft \(f\) reaches the U-shaped area entrance (when the departure aircraft \(f\) is ready to push back), there is no need to wait if there is no aircraft in operation in the U-shaped area. 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. Figure 7: Network and Key Nodes Diagram of Airport Surface. Red triangles indicate the different types of airport surface nodes. **Case 2:** 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  
\[w_{f}^{p}=\frac{L_{p^{\prime}}-\delta_{f^{\prime}}^{p^{\prime}}}{V_{f^{\prime}}} +\alpha_{A}^{f}\ \ \ f\in\mathcal{F},f^{\prime}\in\mathcal{F}_{A},\forall p,p^{ \prime}\in\mathcal{P} \tag{3}\]  
\[w_{f}^{p}=\frac{\delta_{f^{\prime}}^{p^{\prime}}}{V_{f^{\prime}}}\ \ \ f\in \mathcal{F},f^{\prime}\in\mathcal{F}_{D},\forall p,p^{\prime}\in\mathcal{P} \tag{4}\]  
where \(f^{\prime}\) is the aircraft in the U-shaped area that is operating, and the corresponding parking stand of \(f^{\prime}\) is \(p^{\prime}\). The following explanations of \(f^{\prime}\) and \(p^{\prime}\) are the same. Based on the description in Section 3.2, when the arrival aircraft \(f\) reaches the U-shaped area entrance (when the departure aircraft \(f\) is ready to push out), it needs to wait for the U-shaped area to be cleared before it can operate in the U-shaped area. 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). We obtain  
\[w_{f}^{p}=\frac{L_{p^{\prime}}^{f^{\prime}}-\delta_{f^{\prime}}^{p^{\prime}}} {V_{f^{\prime}}}+\sum_{\sigma_{A}^{f}}\frac{L_{p}^{f}}{V_{f}}+\sum_{\sigma_{ A}^{f^{\prime}}}\omega_{A}^{f}+\sum_{\sigma_{D}^{f^{\prime}}}\frac{L_{p}^{f}}{V_{f}} +\sum_{\sigma_{D}^{f^{\prime}}}\omega_{D}^{f}\ \ \ f\in\mathcal{F},f^{\prime}\in \mathcal{F}_{A},\forall p,p^{\prime}\in\mathcal{P} \tag{5}\]  
where \(f^{\prime}\) is the aircraft in the U-shaped area that is operating, and the corresponding parking stand of \(f^{\prime}\) is \(p^{\prime}\).