# Analytical Model for Enhancing the Adoptability of Continuous Descent Approach at Airports
## 4 Model Development
### Adopting Queueing Theory to Our Model
#### 4.2.2 Capacity of the Stacking Space for Aircraft Arrivals  
To maximize airport capacity, especially during periods of high demand, the controller longitudinally aligns and separates approaching aircrafts (i.e., positions arriving aircrafts in the queue) for landing on the same runway in a predetermined airspace, according to a predefined requirement for minimum separation between aircrafts that typically operates under IFR [36]. Optimal spacing refers to the efficient implementation of separation requirements by the controller, such that spacing delivers seamless and efficient air traffic control services while maintaining safety. As the controller often emphasizes sequencing (ordering of aircrafts approaching based on their sizes), this should not be the case with CDA operations. During CDA operations, the optimal spacing between aircrafts is more important than optimal sequencing [45]. Thus, we principally assume that the separation distance (mapped as the horizontal distance in Figure 6) between two, same-weight-class, consecutively  
Figure 6: Parameters of our model for aircrafts approaching airports with CDA. arriving aircrafts conducting CDA is _greater_ than when these two consecutive arriving aircrafts are conducting SDA, thus  
\[d_{CDA}>d_{SDA} \tag{16}\]  
where:  
\(d_{CDA}=\) minimum separation distance between aircrafts conducting CDA; and  
\(d_{SDA}=\) minimum separation distance between aircrafts conducting SDA. Assuming that the space available to stack aircraft arrivals (i.e., the maximum number of aircrafts in the queuing system) at an airport is \(S_{p}\), and the minimum allowable horizontal separation distance between a pair of same-weight-class aircraft arrivals is \(d\), then the number of aircrafts stacked for approach, \(k\), must fit safely within the allowable stack space, as follows:  
\[k\leq\frac{S_{p}}{d} \tag{17}\]  
The largest integer value of \(k\) that satisfies Equation (17) is a key output of our modeling. When CDA is used, that largest integer value of \(k\) is the maximum number of aircrafts that can fit within the stacking space under the CDA assumption. The air traffic controller may be able to compare this largest integer with the number of CDA aircrafts presently in the tracking space to ascertain whether there is available stacking space to permit the next arriving aircraft to use CDA or not. Assuming that the aircraft approach speed, measured in knots, on average, is \(V_{app}\) and that the distance the aircraft covers during descent from the TOD point to touchdown, measured in nautical miles, is \(d_{des}\), then the _average_ descent time, \(t_{des}\), could be estimated as  
\[t_{des}=\frac{d_{des}}{V_{app}} \tag{18}\]  
We assume that an airport's nominal capacity, \(AAR\), is sufficiently large to handle all aircrafts that can fit safely within the stacking space. Therefore, when implementing CDA, this assumption is represented as follows:  
\[AAR\times t_{des}\geq k \tag{19}\]  
Essentially, stacking space is a contained airspace with predefined boundaries based on traffic and/or obstacles limitations with the purpose to stack aircraft arrivals up to a certain capacity. As the separation distance between aircrafts increases, the stacking space capacity, in terms of the number of aircrafts that could be stacked, \(k\), will decrease. Moreover, as the airport arrival rate increases, typically during periods of high demand when many airport staff are working and the airport operates at near capacity, stacking space capacity may decrease as well. This is due to the high level of traffic causing stress and cognitive pressures on the air traffic controllers who, thus, may decide to increase the minimum separation distance between aircrafts as a safety buffer to reduce stress and the possibility of a safety error. Furthermore, we assume that almost all aircraft arrivals at the airport are expected to successfully land on a runway, regardless of their descent profile type.