## 4 Apron U-Shaped Area Traffic Operation Model
### Notation
\begin{table}
\begin{tabular}{c c} \hline \hline
**Sets** & \\ \hline \(\mathcal{F}\): & Set of aircraft in study period \\ \(\mathcal{F}_{A,}\mathcal{F}_{D}\): & Set of arrival and departure aircraft in study period, respectively \\ \(\mathcal{P}\): & Set of parking stands in U-shaped area \\ \(\mathcal{P}_{1}\), \(\mathcal{P}_{2}\), \(\mathcal{P}_{3}\): & Set of parking stands in outer, middle, and inner area of U-shaped area, \\  & respectively, \(\mathcal{P}_{1}\subset\mathcal{P}\), \(\mathcal{P}_{2}\subset\mathcal{P}\), \(\mathcal{P}_{3}\subset\mathcal{P}\) \\ \(\varpi^{A}\),\(\varpi^{D}\): & Arrival and departure aircraft queueing set, respectively \\ \hline Parameters & \\ \hline \(V_{f}\): & Speed of the taxiing phase of \(f\), which can be considered to be related to the aircraft type \\ \(L_{p}^{f}\): & Distance for \(f\) to travel from parking stand \(p\) to the entrance/exit of U-shaped area \\ \(\varpi_{A}^{f}\varpi_{D}^{f}\): & Duration required to park into and push out of a parking position \\ \(X_{12,}X_{23}\): & Distance from middle (12) and inner area boundary (23) to entrance/exit of U-shaped area in partition-shared operation procedure, respectively \\ \(d_{\min}\): & Minimum safe separation between aircraft in the U-shaped area \\ \(\varpi_{DL}^{f}\): & Duration of the long push of the departure aircraft \(f\) in the middle area \\ \hline Variables & \\ \hline \(t_{n}(f)\), \(t_{p}(f)\), & Time when aircraft \(f\) reaches node \(n\), parking stand and entrance/exit of U-shaped area, \(f\in\mathcal{F}\) \\ \(t_{entry}(f)\): & Parking stands of partition area in partition-shared operation procedure, \\ \(p_{n}\in\mathcal{P}_{nt}\), \(n=1,2,3\) & \\ \(w_{f}^{p}\): & Waiting duration of \(f\) entering or exiting U-shaped area, and the \\  & corresponding stand of \(f\) is \(p\), \(p\in\mathcal{P}\), \(f\in\mathcal{F}_{A}\cup\mathcal{F}_{D}\) \\ \(\delta_{f}^{p}\): & Distance from entrance/exit to the aircraft \(f\) in operation in the U-shaped \\  & area and the corresponding stand of \(f\) is \(p\), \(p\in\mathcal{P}\), \(f\in\mathcal{F}_{A}\cup\mathcal{F}_{D}\) \\ \(t_{entry}^{f}\): & Earliest time to enter the U-shaped area or the earliest off-block time for arrival or departure aircraft, respectively, \(f\in\mathcal{F}_{A}\cup\mathcal{F}_{D}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Sets, parameters, and variables of Apron U-shaped Area Traffic Operation Model. \[t_{entry}(f)-t_{p}(f)=\frac{L_{p}^{f}}{V_{f}}+w_{f}^{p}+\varpi_{D}^{f}\ \ f\in\mathcal{F}_{D} \tag{2}\]  
where Equation (1) represents the time for an arrival aircraft to travel from the U-shaped area entrance to the parking stand, and Equation (2) represents the time for a departure aircraft to travel from the parking stand to the U-shaped area exit. Considering that aircraft operate on the same route and at the same speed in the U-shaped area, the aircraft through taxi time is the same for different U-shaped area operating procedures, but the difference in operating rules makes a difference in the additional taxi time of aircraft due to waiting under different procedures. Therefore, the following traffic model is carried out for each of the three types of operation procedures.