2.1 Aircraft arrival operations

In this section we describe the aircraft arrival operations at Tokyo International Airport. Tokyo International Airport is the fifth busiest airport in the world with respect to passenger traffic, processing a total of 83,189,933 passengers in 2017. An increase of 4.32% in the number of passengers was registered in 2017 relative to 2016⁽⁴⁾. The maximum number of departures and arrivals accommodated per year is 447,000, with a maximum of 80 operations per hour. Over 60% of the domestic flights in Japan are concentrated at this airport. The airport makes use of four runways on a daily basis, with the choice of runway configuration depending on the wind direction (see Fig. 1). For operations at times of a northerly wind, runway 34L is used for arrivals from the south-west direction, whereas runway 34R is used for arrivals from the north. Runway 34R is also used for departure traffic. For operations when a southerly wind prevails, runway 22 is used for arrivals from the south-west, and runway 23 for arrivals from the north.

Figure 1. Departure and arrival operations depending on wind direction.

In this study we consider flight plans and track data for a period of 71 days, which are randomly selected from the odd months of 2016 and 2017. In 2016, there were a total average of 608 arrivals per day, of which 530 flights were domestic and 78 flights international. In 2017, eight additional international flights were accommodated every day compared to 2016. Thus, in 2017, there were a total average of 614 arrivals per day, of which 530 flights were domestic and 84 flights were international. The total number of arrivals between 8:00AM and 11:00PM is slightly below the maximum allowed daily traffic threshold, and the most congested period is between 5:00PM and 10:00PM.

Figure 2. Example of flight tracks during an entire day in September 2017. The red and blue tracks represent the traffic flow from the south-west and the north, respectively. Concentric circles are drawn every 100NM for the airspace with a radius from 100 to 500NM centered at Tokyo International Airport.

Figure 3. Number of aircraft arriving per hour during the hours 5:00PM–10:00PM in 2016 and 2017. (a) Traffic flow from the north. (b) Traffic flow from the south.

2.2 Data-driven analysis of the aircraft arrival flows

One of the features of the arrival traffic flow at Tokyo International Airport is that the arrivals are clustered either in the northern or the southwestern part of the airport. The period 8:00AM–11:00PM has a total of 569 arrivals, of which 422 flights arrive from the south-west, and 147 flights arrive from the north. Figure 2 shows the flight track data for one day of operation in September 2017. The flight tracks of arrivals that use runway 22 are drawn in red, whereas arrivals at runway 23 are drawn in blue.

Air traffic flow in Japan is strategically controlled with a central focus on arrivals at Tokyo International Airport. Figure 3 shows the number of arrival flights crossing the concentric circles with radii ranging from 10 to 100NM every 10NM centered at Tokyo International Airport per hour from 5:00PM to 10:00PM. The total number of flight arrivals is controlled within a maximum of 40 (30 arrivals from the south-west and 10 arrivals from the north). Figure 4 shows the distribution of departure airports for both domestic and international flights depending on the number of arrivals in a day. It is obvious that a larger number of domestic flights arrive from the south-west. Additionally, international arrivals from airports in other Asian countries also approach from the south-west. As global air traffic passenger demand is expected to increase by 250% in the next 20 years according to Asian economic growth forecasts⁽¹⁸⁾, a further increase in arrivals from the south-west is estimated to occur.

Figure 4. Distributions of departure airports depending on the numbers of arrivals in a day. (a) Departure airports of domestic flights. (b) Departure airports of international flights.

Table 1 shows the types of aircraft, expressed as a percentage, that arrive and depart. The majority of the aircraft are used for short and medium-distance passenger transportation using A737-800 (B738), A320, and B767-300 (B763). The second category includes long-distance passenger aircraft such as B777-200 (B772) and B787-8 (B788). Finally, less than 1% of the flights are business jets including Gulfstream and Bombardier products.

Table 1 Aircraft types as a percentage of the total number of aircraft arrivals

3.1 Wake vortex categories

The minimum aircraft inter-arrival time is decided based on wake vortex categories, which define aircraft separation minima between the preceding and succeeding aircraft according to their aircraft type. Table 2 compares the aircraft types by categorising them according to the maximum take-off mass (MTOM) and length of wind span (WS) of the aircraft based on four different standards: ‘Current ICAO’ is the current international standard, ‘RECAT-Dubai’ is the new standard implemented at Dubai International Airport⁽³⁾, ‘RECAT-US 1.5’ is the new US standard⁽²⁾, and ‘RECAT-EU’ is the new EU standard^{(Reference Rooseleer and Treve1)}. In these standards^{(Reference Rooseleer and Treve1-3)}, the minimum distance separation is decided using combinations of preceding and succeeding aircraft types. Specifically, ‘RECAT-Dubai’ employs the combination of low separation minima in ‘RECAT-US1.5’ and ‘RECAT-EU,’ and is proposed as the new ICAO standard. Thus, the aircraft types arriving at a destination airport (see Table 1) determine the runway throughput following these separation standards.

Table 2 Wake vortex categories depending on maximum certified take-off mass (MTOM) and wing span (WS)

3.2 Distance separation minima under RECAT-Dubai

Table 3 summarises the distance-based wake turbulence separation minima based on the RECAT-Dubai standards⁽³⁾. The radar separation minimum (3NM in Japan) is applied for category sets not included in the table. Figure 5 shows the frequency of a specific type of preceding-succeeding aircraft set when the RECAT-Dubai standard is applied to current traffic flow arriving at Runways 22 and 34L between 8:00AM and 10:00PM. Arrival sequencing is determined by the radar data. The majority of aircraft (see Table 1) arriving at Tokyo International Airport are of types D, B, and C with a ratio of approximately 5:3:2. When arrival sequencing is optimised by means of, for instance position shifting, one or two additional aircraft arrivals per hour are accommodated, relative to the current first come first served (FCFS) sequencing. However, these position-shifting operations are not necessarily equitable from the airlines’ point of view. Furthermore, it is not clear how such optimisation operations can be achieved without increasing the workload of air traffic controllers and/or without any automation support for the air traffic controllers. Equally important, these optimisation operations may also propagate traffic congestion to succeeding and/or surrounding aircraft traffic. For these reasons, the optimisation of arrival sequencing was not addressed in the study presented in this paper.

Table 3 Distance-based wake turbulence separation minima under RECAT-Dubai separation standards

Figure 5. Current arrival sets of aircraft categories according to RECAT-Dubai standards. (a) Runway 22, southerly wind operation, 8–22h. (b) Runway 34L, northerly wind operation, 8–22h.

3.3 Aircraft time separation minima under different standards

In this section we determine the time separation associated with the current ICAO, RECAT-Dubai, RECAT-US 1.5, and RECAT-EU standards. The same arrival sequences are given to estimate the time separation. Figure 6 shows the minimum time separation, when these four standards are applied to the arrival traffic flow at runway 22 and runway 34L. We note that these runways are used only for arrivals during the congested time period 5:00PM–10:00PM at Tokyo International Airport. To determine the aircraft time-based separation for each of the four standards, the distance-based separation minima are divided by the aircraft ground speed at 5NM prior to the runway threshold. We obtained this aircraft speed from the flight track data recorded during the period 2016–2017. Table 4 compares the mean and STD values of the time-based separation when the four standards are taken into consideration. As shown in Fig. 6 and Table 4, RECAT-Dubai has the best potential to minimise aircraft inter-arrival time, whereas RECAT-EU shows slightly larger values. However, it is unrealistic to analyse arrival traffic flows only by considering the time-based separation minima. This is because the air traffic controllers use safety margins, together with the separation minima, to ensure that the separation requirements are always maintained during operations. As such, the next section presents an analysis of the separation minima, together with the safety margins.

Table 4 Comparing the mean and STD of the arrival time separation

Figure 6. Comparing the time-separation when four different wake vortex categories are applied to the arrivals at Tokyo International Airport. (a) Runway 22, southerly wind operation, 17–22h. (b) Runway 34L, northerly wind operation, 17–22h.

3.4 Safety margins depending on the levels of automation support

As a benchmark for safety margins, the Inter-Aircraft Time (IAT) was introduced^{(Reference Arbuckle19)}. This safety margin corresponds to one standard deviation (STD) value of the aircraft arrival time, i.e. 68% of the time the aircraft arrives within the mean of the assigned arrival time $\pm$ 1/2 IAT. The value of IAT depends on the level of automation of the tools that support air traffic controllers, as provided in Table 5: ‘No metering’ is the baseline capacity, i.e. the amount traffic the air traffic controllers are able to handle without using any automation tools; ‘Metering’ is the capability when air traffic controllers are supported by arrival management (AMAN) systems, which automatically suggest the ideal arrival time schedules; ‘GIM-S, TSAS’ is the capability supported by advanced levels of AMAN (refer to Ref. (Reference Thipphavong, Jung, Swenson, Martin, Lin and Nguyen7)); and ‘IM’ is its combination with aircraft self-separation (refer to Ref. (Reference Van Tulder8)). In general, as the level of automation support increases, the values of IAT decrease. Therefore, a higher level of automation support can accommodate an increased aircraft arrival rate.

Table 5 Inter-Aircraft Time (IAT) when applying different levels of automation^{(Reference Arbuckle19)}

3.5 Estimating the arrival rate at Runway 22 under various separation standards and safety margins resulting from automation

Figure 7 shows the arrival rate (number of arrival aircraft per hour) at Runway 22 $\lambda_{\text{RW22}}$ , under the four separation standards and considering the safety margins for different levels of automation. We determine this arrival rate as follows:

Figure 7. Estimation of arrival rates according to combinations of wake vortex categories and automation levels.

(1) \begin{align}{\lambda}_{\text{RW22}} = \frac{3600.0}{{\mathbb{E}{[}t_{\text{min}{]}}} + t_{\text{IAT}}},\end{align}

where $\mathbb{E}{[}t_{\text{min}}{]} $ is the mean of the minimum time separation (sec) at Runway 22 for southerly wind operation during the period 5:00–10:00PM, corresponding to each wake vortex category in Table 8, and $t_{\text{IAT}}$ is IAT in seconds corresponding to each automation level in Table 5.

Figure 7 shows that the introduction of the RECAT categories has the potential to support an increase in the aircraft arrival rate up to 133% (40 arrivals per hour) with the highest automation support ‘IM,’ when compared with the current arrival rate of 30 arrivals per hour from the south-west (see Fig. 3(b)). When considering the highest level of automation support ‘IM,’ the current ICAO standard supports an increase in the arrival rate of up to 116% (35 arrivals per hour). Without any automation support, the current ICAO standard supports an increase in the arrival rate up to 103% (31 arrivals per hour). However, RECAT-Dubai and RECAT-EU lead to an increase in up to 116% (35 arrivals per hour) relative to the current arrival rate (30 arrivals per hour). In the case of ‘metering’ automation support, RECAT-Dubai and RECAT-EU increase the arrival rate up to 120% (36 arrivals per hour), compared to the current ICAO standards that allow the arrival rate to increase to a maximum rate of 103% (31 arrivals per hour).

4.1 Modeling the flow of arrival air traffic using a $\textbf{\textit{G\,/\,G\,/\,c}}$ queuing model

We propose a queuing model for traffic arriving from the south-west at Tokyo International Airport between 5:00PM and 10:00PM, which represents the most congested arrival period. We specify our queuing model using the arrival flow, service time, and the number of servers as follows. The aircraft arrive in a given airspace according to an hourly rate, i.e. a specified number of arrivals per hour, referring to Figs 3(b) and 7. The time between two consecutive arrivals in an airspace area is termed the aircraft inter-arrival time. Upon arrival, the aircraft receives a service time, i.e. the time the aircraft spends flying in the given airspace. We consider a multi-server queuing model, in which the number of servers indicates the number of aircraft that are allowed to be present at any time in the given airspace, i.e. the capacity of the given airspace. We apply this model to nine disjoint airspace areas, defined as concentric circles centered at the airport and with radii increasing in increments of 10NM (see Fig. 8).

We formulate this aircraft arrival process by means of a $G/G/c$ queuing model as follows. We consider the circular airspace with a radius of 100NM centered at the airport. We partition this area using concentric circles (see Fig. 8) centered at the airport, with radii at increments of 10NM. This partitioning defines nine airspace areas $1, 2, \ldots, 9$ , where area 1 is the airspace area centered at the airport within the circular rings with radii of 10 and 20NM, area 2 is the airspace area centered at the airport within the circular rings with radii of 20 and 30NM radii, and so on. An aircraft arriving at area 1 crosses the circle with radius of 20NM, whereas an aircraft arriving at area 2 crosses the circle with radius of 30NM, and so on.

The $G/G/c$ queuing model allows us to estimate potential aircraft delay time under the assumption that both service time and inter-arrival time follows general distributions. This paper applies the $G/G/c$ model because this model fits the data characteristics of the both distributions of service time and inter-arrival time. An airborne delay would occur when the aircraft is requested to fly a longer route, i.e. vectoring/metering/speed adjustments.

Figure 8. Illustration of the service time and inter-arrival time in the airspace divided into nine areas.

Let $D_i$ denote the delay an aircraft experiences in airspace area $i, 1 \leq i \leq 9$ because of the lack of capacity in area $i-1$ . Then we approximate the total airborne delay in airspace area i as follows^{(Reference Whitt20-Reference Kingman23)}:

(2) \vspace*{-3pt}\begin{align} \mathbb{E}{[}D_i^{G/G/c}{]} \simeq \mathbb{E}{[}D_i^{M/M/c}{]} \frac{C_{A_i}^2 +C_{B_i}^2}{2} ,\vspace*{-3pt}\end{align}

where $C_{A_i} = \frac{\sqrt{\sigma{[}A_i{]}}}{\mathbb{E}{[}A_i{]}}$ is the coefficient of variation of the aircraft inter-arrival time in airspace area i and $C_{B_i} = \frac{\sqrt{\sigma{[}B_i{]}}}{\mathbb{E}{[}B_i{]}}$ is the coefficient of variation of the aircraft service time in airspace area i. Here $\mathbb{E}{[}A_i{]}$ and $\sigma{[}{A_i}{]}$ are the mean and variance of aircraft inter-arrival time, and $\mathbb{E}{[}B_i{]}$ and $\sigma{[}{B_i}{]}$ are the mean and variance of aircraft service time in the airspace area i.

In Equation (2), $\mathbb{E}{[}D_i^{M/M/c}{]}$ denotes the expected aircraft delay time in airspace area i when a $M/M/c$ queuing model is considered, which is well known^{(Reference Itoh and Mitici16,Reference Itoh and Mitici17,Reference Hillier24,Reference Adan and Resing25)} . This means that the arrival process is considered a Poisson process with a parameter $\lambda_i$ , with service times are assumed to follow an exponential distribution with a parameter $\mu_i$ and there is a fixed number of parallel servers $c_i$ . Identical servers is well-known and determined, following Refs. (Reference Hillier24) and (Reference Adan and Resing25).

Moreover, for stability, we have that

(3) \vspace*{-3pt}\begin{align} \rho_i=\frac{\lambda_i}{ c_i \cdot \mu_i}<1 .\vspace*{-3pt}\end{align}

If $\rho_i$ is not less than 1, then the queue becomes unstable and the estimated delay time for incoming aircraft tends to become extraordinarily long.

4.2 Parameter estimation for the $\textbf{\textit{G\,/\,G\,/\,c}}$ queuing model based on flight track data analysis

We estimate the parameter of the $G/G/c$ queuing model as follows. For each airspace area $i \in\break \{1,2,\ldots, 9\}$ , we estimate the values of $c_i$ to be the number of aircraft present at the same time in this area, as recorded in the historical flight track data. Figure 9 shows the number of aircraft present in airspace area $i\in \{1,2,\ldots, 9\}$ every 10min during the time period 5:00PM–10:00PM during the 2 years (2016–2017) of flight track data. An analysis of the historical flight data shows that the highest mean number of aircraft in airspace $i, i \in \{1, 2, \ldots, 9\}$ is at most two. Thus, we choose $c=2$ to represent the current capacity for airspace $i, i \in \{1, 2, \ldots, 9\}$ .

The value $c=2$ is also relevant from the perspective of safety requirements and minimum aircraft separation standards because the inter-arrival aircraft distance is approximately 5NM (i.e. two aircraft within a 10NM range in-trail), which is exactly the minimum aircraft separation currently required for radar separation in the en-route airspace.

The mean and variance of the inter-arrival time, $\mathbb{E}{[}A_i{]}$ and $\sigma{[}A_i{]}$ , are estimated from the data distribution of the time-separation between the preceding and succeeding aircraft when entering airspace area $i \in \{1,2,\ldots, 9\}$ . The mean and variance of the service time, $\mathbb{E}{[}B_i{]}$ and $\sigma{[}B_i{]}$ , are estimated from the data distribution of the flight time in each airspace area i. Table 6 lists the values of these parameters with $\rho_i$ when $c=2$ is given. As shown in Table 6, all values of $\rho_i$ satisfy the stability condition in Equation (3).

Table 6 Parameters of exmpirical distributions of the inter-arrival time and service time in seconds and squared seconds, for the G/G/c model

Figure 9. Data-driven estimates of the number of servers, c, for the $\textit{G}/\textit{G}/\textit{c}$ model.

5.1 Estimating the minimum required airspace capacity

As discussed in Section 4.1, Equation (3) shows the condition for queue stability. In this section we estimate the minimum required airspace capacity while ensuring that the queue remains stable, i.e. that $\rho < 1$ .

Figure 10 shows the estimation of the maximum arrival rate $\lambda_i, i \in \{1, 2, \ldots, 9\}$ , under a fixed service rate $\mathbb{E}{[}B_i{]}$ (see Table 6) and airspace capacity $c_i \in \{1, 2, 3\}$ , such that $\rho \rightarrow 1$ . As shown in Fig. 10, increasing the airspace capacity $c_i$ allows for higher arrival rates for all airspace $i \in \{1, \ldots, 9\}$ . For $c=1$ , 32 arrivals per hour is not allowed because the queue stability condition for airspace $i\in \{1,2,\ldots, 5\}$ no longer holds. When $c=2$ , the maximum arrival rate allowed in airspace $i=1$ is 32 arrivals, while 40 arrivals are not allowed in airspace $i=3$ . If $c=3$ , then for all airspace $i\in \{1,2,\ldots, 9\}$ it is possible to have 40 arrivals per hour. Thus, for airspace $i=1$ , we need to ensure a capacity of at least $c_1=2$ to be able to handle 32 arrivals per hour, while $c_1=3$ is required to be able to handle 40 arrivals per hour.

Figure 10. Comparison of maximum arrival rate satisfying $\rho<1$ .

Table 7 summarises the minimum $c_i$ values to stabilise the $G/G/c$ queuing model for airspace $i, i \in \{1, 2 , \ldots, 9\}$ . Table 7 shows that the airspace closer to the airport requires more capacity (larges values for $c_i$ ), especially in the airspace $i \in \{1, 3\}$ . This is because close to the destination airport, i.e. airspace $i=1$ , the aircraft slow down. Thus, the mean service rate is reduced. In the airspace $i=3$ and 4, especially $i=3$ , the inter-aircraft arrival time is controlled by using vectoring operations. As a result, in this airspace the aircraft change their heading direction and adjust the spacing between succeeding aircraft. Therefore, the service rate $\mu_i$ is reduced in the airspace $i=1, 3, 4$ . In $i=2$ , the aircraft inter-arrival time is already controlled, and most arriving aircraft follow in-trail because the airspeed is faster than in the $i=1$ airspace.

Table 7 Minimum $c_i$ values that enable the arrival rates of 32, 36, and 40 arrivals per hour, respectively, to be handled in airspace $\textit{i}, \textit{i} \in \{\text{1}, \text{2}, \ldots, \text{9}\}$

Table 8 Comparison of the mean and STD of the arrival time separation for the metering operation

5.2 Impact on aircraft arrival delay when increasing airspace capacity

In this section we estimate the arrival delay time in airspace $i\in \{1, 2, 3, 4\}$ , where the maximum arrival rate is limited in comparison with the airspace $i, i \in \{5, \ldots, 9\}$ (see Fig. 10). As shown in Fig. 11, we consider five combinations of arrival rates and airspace capacities, which satisfy the minimum $c_i$ in Table 7, to estimate the arrival time delay. First, for the current arrival rate of 30 arrivals per hour, the capacity $c=2$ results in an increased arrival time delay in comparison with the cases when $c=3$ . Thus, increasing the airspace capacity to $c_i=3$ , $i\in \{1, 2, 3, 4\}$ , is an efficient solution for mitigating arrival delay. However, even when considering $c_i=3$ , for high arrival rates of 40 aircraft per hour, the arrival delay time remains high, especially in the case of airspace $i=1$ .

Figure 11. Comparison of arrival delay time in airspace $\textit{i}\in \{\text{1,\, 2,\, 3,\, 4}\}.$

5.3 Aircraft arrival delay for various values of the variance of the service time and the variance of the inter-arrival time

Generally, reducing the variance of the service time and inter-arrival time are options to mitigate the delay in the arrival time in each airspace $i \in \{1, 2,\ldots, 9\}$ . In this section we determine the arrival delay for various values of the variance of the service time and the inter-arrival time, respectively. We consider different aircraft arrival rates and airspace capacities $c_i$ such that the requirements in Table 7 are satisfied.

Figure 12 shows the arrival delay for various values of the airspace capacity $c_1$ and aircraft arrival rates. Figure 12(a) indicates that, when $c_1=2$ , the arrival delay dramatically increases even though the variance decreases. Thus, $c_1=3$ is necessary at airspace $i=1$ to enable 32 arrivals per hour to be handled (see Fig. 12(b)). Increasing the variance of the service time is critical for the arrival delay, especially when considering higher arrival rates, e.g. 32 or 36 arrivals per hour. In general, the variance of the service time at airspace $i=1$ is larger than that in the other airspace $i, i \in \{2, 3, \ldots, 9\}$ (see also Table 6) because in this area the arriving aircraft slow down before landing, and they are also affected by the wind velocity and wind direction associated with the used runway configuration. Because of this, in airspace $i=1$ , significant traffic control is necessary to ensure that the arrival delay is limited to the minimum.

Figure 12. Aircraft arrival delay for various values of the variance of the inter-arrival time and service time – airspace $\textit{i}\text{\,=\,1}$ . (a) $\textit{c}_{\text{1}}\text{\,=\,2},$ 32 arrivals per hour. (b) $\textit{c}_{\text{1}}\text{\,=\,3},$ 32 arrivals per hour. (c) $\textit{c}_{\text{1}}\text{\,=\,3},$ 36 arrivals per hour.

Figures 13-15 show the arrival delay by varying the variance of the service time and inter-arrival time, respectively, at airspace $i\in \{2, 3, 4, 5\}$ . We consider the case when $c_i \in \{2,3\}$ and an arrival rate of 32 and 36 aircraft arriving per hour. The lowest service rate (the service time with the highest mean) is in airspace $i=3$ among the airspace $i\in \{2, 3, 4, 5\}$ (see Table 6), and the second lowest service rate is in airspace $i=4$ (see Table 6). This can be explained by the fact that currently, in these two areas, the air traffic controllers space the arriving aircraft. Thus, in the airspace $i=3,4$ , when $c_i=2$ , the arrival delay time increases even for small values of the service and inter-arrival variance, as shown in Figs 13 and 14. Increasing the airspace capacity $c_i=3$ is the most realistic solution to mitigate the arrival delay time, rather than minimising the variance of the service time and the inter-arrival time (see Fig. 15).

Figure 13. Arrival delay under various values for the variance of the inter-arrival time and service time – $\textit{c}_{\textit{i}}\text{\,=\,2}$ , 32 arrivals per hour, at airspace $\textit{i}\in \{\text{2,\, 3,\, 4,\, 5}\}$ . (a) $\textit{i}\text{\,=\,2}$ . (b) $\textit{i}\text{\,=\,3}$ . (c) $\textit{i}\text{\,=\,4}$ . (d) $\textit{i}\text{\,=\,5}$ .

Figure 14. Arrival delay for various values of the variance of the inter-arrival time and the service time, $\textit{c}_{\textit{i}}\text{\,=\,2}$ , 36 arrivals per hour, at airspace $\textit{i}\in \{\text{2,\, 3,\, 4,\, 5}\}$ . (a) $\textit{i}\text{\,=\,2}$ . (b) $\textit{i}\text{\,=\,3}$ . (c) $\textit{i}\text{\,=\,4}$ . (d) $\textit{i}\text{\,=\,5}$ .

Figure 15. Arrival delay under various values for the variance of the inter-arrival time and the service time, $\textit{c}_{\textit{i}}\text{\,=\,3}$ , 36 arrivals per hour, at airspace $\textit{i}\in \{\text{2,\, 3,\, 4,\, 5}\}$ . (a) $\textit{i}\text{\,=\,2}$ . (b) $\textit{i}\text{\,=\,3}$ . (c) $\textit{i}\text{\,=\,4}$ . (d) $\textit{i}\text{\,=\,5}$ .

Figures 16-18 show the arrival delay for various values of the variance of the service time and inter-arrival time at airspace $i\in \{6, 7, 8, 9\}$ . Although the minimum airspace capacity in these areas is estimated to be $c_i=1$ , $i\in \{6, 7, 8, 9\}$ , as shown in Table 7, in the airspace $i\in \{6, 7, 8, 9\}$ , $c_i=2$ is required to minimise the arrival delay time for both of the arrival rates of 32 and 36 arrivals per hour.

Figure 16. Arrival delay for various values of the variance of the inter-arrival time and the service time, $\textit{c}_{\textit{i}}\text{\,=\,1}$ , 32 arrivals per hour, at airspace $\textit{i}\in \{\text{6,\, 7,\, 8,\, 9}\}$ . (a) $\textit{i}\text{\,=\,6}$ . (b) $\textit{i}\text{\,=\,7}$ . (c) $\textit{i}\text{\,=\,8}$ . (d) $\textit{i}\text{\,=\,9}$ .

Figure 17. Arrival delay for various values of the variance of the inter-arrival time and the service time, $\textit{c}_{\textit{i}}\text{\,=\,2}$ , 32 arrivals per hour, at airspace $\textit{i}\in \{\text{6,\, 7,\, 8,\, 9}\}$ . (a) $\textit{i}\text{\,=\,6}$ . (b) $\textit{i}\text{\,=\,7}$ . (c) $\textit{i}\text{\,=\,8}$ . (d) $\textit{i}\text{\,=\,9}$ .

Figure 18. Arrival delay under various values of the variance of the inter-arrival time and the service time, $\textit{c}_{\textit{i}}\text{\,=\,2}$ , 36 arrivals per hour, at airspace $\textit{i}\in \{\text{6,\, 7,\, 8,\, 9}\}$ . (a) $\textit{i}\text{\,=\,6}$ . (b) $\textit{i}\text{\,=\,7}$ . (c) $\textit{i}\text{\,=\,8}$ . (d) $\textit{i}\text{\,=\,9}$ .

These results clarify that increasing the airspace capacity is a higher-priority strategy than the effort of minimising the variance of the service time, except for the case of airspace $i=5$ . Figure 14(d) shows that, for the current inter-arrival time with a variance of 3,488 sec $^2$ (see Table 6) the arrival delay time is large. This shows that for airspace $i=5$ , to limit the arrival delay we have two possible strategies when considering an arrival rate of 36 arrivals per hour: (1) increase the airspace capacity $c_5=3$ (see Fig. 15(d)) or, (2) maintain $c_5=2$ and limit the inter-arrival time variance (see Fig. 14(d)).

5.4 Service time envelopes estimating the arrival delay

Optimising the service time in each airspace i is the other potential solution for mitigating the arrival delay time while increasing the arrival rate. Figure 19 shows the estimated envelope of the maximum service time satisfying the condition $\rho < 0.85$ . Figure 19 shows that an increase in the arrival rate reduces the service time at the same capacity c. At a larger capacity c, the service time is increased.

Figure 19. Estimation of service time $\mathbb{E}{[\textit{B}]}.$

The estimated service time shown in Fig. 19 is given to airspace $i\in \{1, 2,\ldots, 9\}$ for estimating the arrival time delay corresponding to airspace capacity $c _i$ and the arrival rate as shown in Fig. 20. One of the features clarifies that the arrival time delay increases while i increases if the same amount of service time is allowed. This means that the delay in the arrival time increases if the same amount of flight time associated with the airspace closer to the arrival airport would be associated with the airspace farther away. In other words, it is more efficient to extend the flight time closer to the arrival airport than in the airspace farther away. One of the reasons is that the variance of the inter-arrival time becomes larger in the case of the airspace farther away, as shown in Table 6. To increase the service time is more effective when the variance of the inter-arrival time is smaller. This means that extending the flight time in airspace farther away is not the solution to reduce the arrival delay time.

Figure 20. Estimating the impact of the service time at airspace $\textit{i}\in \{\text{1,\,2,\,3},{\ldots}\,,\text{9}\}$ . (a) $\textit{i}\text{\,=\,1}$ . (b) $\textit{i}\text{\,=\,2}$ . (c) $\textit{i}\text{\,=\,3}$ . (d) $\textit{i}\text{\,=\,4}$ . (e) $\textit{i}\text{\,=\,5}$ . (f) $\textit{i}\text{\,=\,6}$ . (g) $\textit{i}\text{\,=\,7}$ . (h) $\textit{i}\text{\,=\,8}$ . (i) $\textit{i}\text{\,=\,9}$ .