《1. Introduction》

1. Introduction

Air transport is pivotal to modern traffic, as it facilitates our lives. Characterized by its high level of safety and fast speed, air transport is prioritized by travelers, resulting in the phenomenal growth of air traffic demands [1,2]. Air transport is associated with air transport systems, which encompass manifold components interacting with one another, usually in a nonlinear fashion [3,4]. The complex dynamics of air transport systems, together with a variety of uncertainties such as inclement weather, airspace restriction, mechanical failures, and so forth, result in intractable air traffic delays [5] which can be extremely difficult for air traffic controllers to deal with. This situation could worsen in the coming decade, as air traffic demands are envisaged to increase.

Air traffic delay is one of the most challenging tasks being addressed by modern air traffic management (ATM). It not only harms entities such as passengers, airlines, and airports, but also results in economic losses [6,7]. Furthermore, air traffic delay increases the pollution load on the natural environment [8,9]. It should be pointed out that air traffic delay is bound to happen, due to a wide range of factors [5,10]. Many efforts have been made in the past several decades to probe the reasons for delays and their internal causalities [11,12]. After building a comprehensive understanding of the relationships among the causal factors and various components of air traffic delays, scientists have spared no efforts in seeking remedies to mitigate them [13–15]. Representative initiatives include Air Traffic Flow Management (ATFM) [16,17], Ground Delay Programs (GDPs) [18,19], and Collaborative Decision-Making (CDM) [20–22]. All these procedures have proven to be valuable in reducing air traffic delays.

While tremendous collaborative efforts are still needed in the research on air traffic delay mitigation, a very fundamental yet challenging task pertaining to modern ATM is delay prediction [23,24]. Prediction of flight delays is significant to those working in aviation, especially during their decision-making process [25,26]. Thus far, researchers have developed many delay prediction methods. In particular, the data sciences represented by machine learning techniques are gaining notable momentum in delay prediction [27–30]. A comprehensive literature survey on delay prediction can be found in Ref. [26]. Despite the abundance of methods for delay prediction, it is cumbersome to develop models or methods that can predict precisely, given the vast amount of flight operation data available and the high complexity of air transport systems [26].

In reality, airlines construct flight schedules with the intent of maximizing passenger movements. As a result, an aircraft normally operates a number of flight ‘‘legs” or ‘‘hops” (where a leg or hop refers to a flight between two airports that is part of the aircraft’s itinerary). Consequently, the delay of an upstream flight can spread to downstream flights causing reactionary delays, and the reactionary delays of one aircraft will continue to cause a cascade of reactionary delays for other aircraft. Air traffic delays therefore display the ‘‘ripple effect,” which is detrimental to aviation workers [31]. According to the ripple effect, also known as delay propagation [32] primary flight delays grow and propagate within complex air transport systems. Studies on delay propagation have attracted an enormous amount of attention from researchers not only in the field of aviation, but also in the fields of computer science, management science, system science, and more [33,34]. Compared with studies on delay prediction, studies on delay propagation may be more appealing, as they can assist in locating the origins of delays, calculating reactionary delays, and understanding how delays evolve, making it possible for efficient measures to be taken to counterbalance the ripple effect [31].

Dozens of studies have investigated flight delay propagation. In the literature, researchers have explored delay propagation phenomena in regions such as the United States [32,35,36], Europe [37], and China [38]. In order to gauge the amount of propagated delays, researchers have devised several metrics, including the delay multiplier (DM) index [39]. In order to trace how primary delays propagate, scientists have mainly utilized agent-based methods to model the propagation process [32,35,37]. Because of delays, multiple flights may simultaneously request services. To resolve this conflict, scientists have predominantly applied the queuing theory [36].

Delay propagation is a collective phenomenon, as air traffic involves a variety of interacting components [5,40,41]. It is natural and straightforward to introduce complex network theories and tools to research on air traffic [40,42,43]. Although complex network modeling for air transport has a short history [44,45], its systemic view is injecting new blood into ATM. While the majority of existing studies on delay propagation focus on the estimation of reactionary delays, several studies on network models for delay propagation analysis have already shown great potential [35,36,46]. However, to the best of our knowledge, Refs. [35,36,46] failed to make use of the spatial–temporal properties of the constructed networks. Consequently, those works cannot provide a comprehensive understanding of delay propagation dynamics.

In this study, we present a dynamic network perspective of the propagation dynamics of air traffic delays. More specifically, for a given set of aircraft itineraries, each of which has a one-day cycle, we model the daily traffic scenario with a spatial–temporal network, given that each aircraft has a departure delay for the first flight leg. We suggest a simple yet effective delay propagation mechanism to transfer the delays as the aircraft implement their rotations. In case multiple flights request services simultaneously at the same airport due to delays, we apply our developed delay assignment strategies to prioritize the flights. At a given time point, we construct a spatial airport network with the delayed departure flights as the edges. For a one-day duration, we then construct a spatial–temporal network. Next, we analyze the degree properties of the constructed spatial–temporal network to quantify the delay propagation dynamics in terms of magnitude, speed, and severity. To verify the effectiveness of the proposed network perspective, we carry out case studies on domestic flights in the SAR and the United States that were operated in the last part of 2016. The experimental results demonstrate that the proposed network-based method can provide spatial–temporal details of the delay propagation process. The experiments also reveal that the delay propagation dynamics of flights in the SAR differ substantially from those in the United States in terms of magnitude, severity, and speed.

《2. Related backgrounds》

2. Related backgrounds

《2.1. Spatial networks》

2.1. Spatial networks

Network modeling has proven to be a potent tool for capturing the systemic behaviors of complex systems [44,45]. Generally speaking, a network is a set of nodes and edges. The nodes of a network denote the components of the focal network or networked system, while the edges represent the interactions or relationships between components. Mathematically, a network is denoted by G = { V, E }with n = |V| being the cardinality of the node set V, and m = |E|the cardinality of the edge set E. A network is generally represented by its adjacency matrix A, with the entry quantifying the relationship between nodes i and j.

In this work, we study the delay propagation dynamics of air traffic. In order to capture systemic delays, we construct airport networks with airports as the nodes. For an airport network G, if a flight flies from airport i to airport j, then an edge e_ij E connecting nodes and is created.

The constructed airport networks can be weighted and/or directed, depending on the specific calculation purposes. As the airports include geographical information, the constructed networks are spatial networks [47].

《2.2. Temporal networks》

2.2. Temporal networks

Complex systems in reality are usually time-evolving; that is, their structures change over time. In order to trace their evolution, scientists have developed an effective tool: temporal networks [48]. Mathematically, for a given time period [t₀, t_end ], a temporal network G can be denoted by a network sequence, that is,  , with   being the snapshot at time point t_i .

In this study, we construct airport networks. The edge constructions of the networks depend on the air traffic demands, which are time-evolving. Therefore, we can generate a sequence of airport networks to form temporal airport networks. As mentioned above, airports include geographical information. Consequently, the constructed airport networks are spatial–temporal networks.

《2.3. Air traffic delay》

2.3. Air traffic delay

The notion of delay is common in the transportation domain. In the air traffic domain, delay is normally defined as the difference between the scheduled and actual flight operation times. According to the definitions provided by the Federal Aviation Administration (FAA), a flight is considered to be delayed if it is 15 min or more past its scheduled time [49]. For research purposes, we consider a flight to be delayed in this work if its delay is positive.

The flying process of a flight generally encompasses two phases: the ground phase, which includes the departure and approach; and the airborne phase (from wheels off to wheels on), which covers the flight stages of climbing, cruising, and descending. Flight delay can occur in every phase. In the ground phase, the following five types of delay are mainly encountered:

• Departure delay: There is a difference between the actual and scheduled gate-out times.

• Taxi-out delay: There is a difference between the actual and scheduled taxi-out times.

• Taxi-in delay: There is a difference between the actual and scheduled taxi-in times.

• Arrival delay: There is a difference between the actual and scheduled gate-in times.

• Turn time delay: There is a difference between the actual and scheduled turn times.

In the airborne phase, the airborne delay is determined to be the difference between the actual and scheduled airborne times.

《3. Research problem and contribution》

3. Research problem and contribution

《3.1. Problem description》

3.1. Problem description

This work is dedicated to investigating the propagation dynamics of air traffic delays by using dynamic network modeling and analysis. Fig. 1 presents a graphical illustration of the research problem and the core idea of the network-based approach.

Fig. 1(a) depicts an air traffic scenario in which a set of aircraft are implementing their flight itineraries. The condition is that the first flight leg for each aircraft has a departure delay. For example, as shown in the bottom part of Fig. 1(a), the flight departing from Makassar to Jakarta is delayed for 10 min. Due to the flights’ rotations, the primary delays can propagate and elicit delays in other flights. Our research purpose is to understand the delay propagation dynamics. More specifically, we aim to answer these questions: ① How much is the propagated delay? ② How long will the delay propagation last? ③ How fast can the delay propagate?

In order to probe answers to these questions, we suggest a dynamic network perspective. We first convert the air traffic scenario into a spatial–temporal airport delay network, with the nodes being the spatial airports and the edges being the delayed departure flights (Fig. 1(b)). We then analyze the degree properties of the constructed spatial–temporal networks in order to quantify the delay propagation dynamics in terms of magnitude, severity, and speed.

《Fig. 1》

Fig. 1. A conceptual diagram of the studied problem and the core idea of the proposed dynamic network approach for understanding the propagation dynamics of air traffic delays. (a) A traffic scenario in which four aircraft (distinguished by color) are implementing their flight itineraries, and each aircraft has a departure delay for the first flight leg. (b) A summary of the idea of the proposed network approach for understanding the delay propagation dynamics.

《3.2. Research contribution》

3.2. Research contribution

This work suggests a complex network perspective towards understanding the delay propagation dynamics of air traffic. This work contributes to ATM in the following aspects.

(1) It provides a fine-grained view of delay propagation dynamics. Complementary to existing studies on delay propagation, which only provide a coarse-grained view, this work provides a fine-grained view of spatial–temporal resolution by making use of dynamic network modeling and analysis. The network-based approach can trace the evolution process of delay propagation at given airports at given times and time durations.

(2) It assists with strategic ATM. This study investigates delay propagation dynamics in terms of magnitude, severity, and speed. The proposed network approach can be implemented in real time. Therefore, the outcome of this study could facilitate airlines in enabling quality pre-assessment of flight schedules in terms of reactionary delays, such that they could adjust the schedules accordingly in order to mitigate air traffic delays. Furthermore, since the network approach can trace the delay propagation process with a spatial–temporal resolution, the outcome of this study will assist air navigation service providers and airports to provide better service.

(3) It contributes to CDM. CDM is recognized as a promising paradigm for modern air traffic control. This work carries out case studies on domestic flights in the SAR and the United States, and reveals that only hub airports encounter significant delay propagation. As the proposed approach can estimate temporal propagated delays in real time for given airports, aviation players can choose the proper time to apply CDM to hub airports in order to mitigate the impact of delay propagation. Upon the implementations of several CDM initiatives, the proposed approach can be re-applied to the updated traffic scenario in order to assist decision-makers with further ATM.

《4. Research methodology》

4. Research methodology

《4.1. Method overview》

4.1. Method overview

In order to solve the research problem, we first present an overview of the proposed approach and delineate its procedures in Fig. 2.

It can be seen from Fig. 2 that the core idea of the proposed approach is to construct a spatial–temporal airport network. To achieve this goal, we develop a delay propagation mechanism and a delay assignment mechanism. In what follows, we will elaborate each of the two key components in detail.

《Fig. 2》

Fig. 2. Flowchart of the proposed network approach for the propagation dynamics of air traffic delays.

《4.2. Primary departure delay》

4.2. Primary departure delay

In this work, we refer to the primary departure delay as the delay of the first flight leg in an aircraft’s itinerary. The primary departure delay is utilized as the stimulus to elicit the reactionary departure and arrival delays. There are two ways to work out the primary departure delays: obtaining them from data or sampling from a given distribution.

Here, we directly derive the primary departure delays from the real-world flight operation data. The reasons for doing so are twofold. First, it is straightforward and easy to obtain the primary departure delays from real data. Second, the sampling-based method requires prior knowledge such as the delay distribution.

《4.3. Delay propagation mechanism》

4.3. Delay propagation mechanism

The delay propagation mechanism is utilized to capture the reactionary delays. Putting it another way, for all flights , the propagation mechanism is used to estimate the actual departure time and actual arrival time , where N_f is the total number of flights. The delay propagation mechanism works on the basis of the following assumptions.

(1) No flight can depart more than 5 min earlier. In real-world scenarios, airlines can bring forward their flight plans, resulting in earlier departures (these can be several hours ahead of schedule, as reported in real cases). Apart from flight plan adjustment, some flights can depart slightly earlier than scheduled once all necessary preparation procedures—such as refueling, payload loading, and passenger embarkation—have been done. In our model, we hypothesize that the earliest departure cannot be more than 5 min before the scheduled departure.

(2) Departures from and arrivals at the same airport are served by different runways. Flight departures and arrivals require airport runway services. In this work, we assume that each airport accommodates the departures and arrivals separately by using different runways. In real life, some airports may only have single runways. However, this assumption simplifies the subsequent modeling.

(3) The minimum time separation to alleviate the wake turbulence is set to be 2 min. To alleviate the wake turbulence, we fix the minimum separation time for departures/arrivals to be 2 min. As a result, each runway can serve a maximum of 30 flights per hour. The setting of the minimum separation time is based on the wake turbulence category [50] promulgated by the International Civil Aviation Organization (ICAO), based on the fact that the majority of the aircraft in this study are of medium size.

(4) Delay can occur to a flight in the air. In reality, arrival delay can occur to a flight due to various factors such as convective weather, airspace restriction, airport congestion, and so forth. Unlike existing studies on delay analysis, which assume that delay does not occur during the en-route phase, we introduce the airborne delay of flights to our model. More specifically, we assume that the airborne delay of a flight is a nonlinear function of its departure delay and its scheduled flying time. This assumption also provides the probabilities for flights to absorb delays in the air.

Based on the above assumptions, we then estimate and for all in the following way.

(1) If is the first flight leg of an aircraft’s itinerary, then we estimate as follows:

where  and  represent the scheduled departure time and departure delay of flight .

(2) If is an intermediate flight leg of an aircraft’s itinerary, then we estimate  in the following ways.

where  represents the minimum turnaround time for flight  , which is the previous leg of .

Note that different aircraft have different minimum turnaround times. Normally, aircraft with larger sizes require longer minimum turnaround times. For a given set of flight plans, we categorize the aircraft into three classes: medium size, large size, and heavy size. We then set the minimum turnaround times T_TA respectively to 60, 90, and 120 min.

(3) Based on , we then calculate actual arrival time of flight as follows:

in which  is a random variable signifying the deviation between scheduled flying time  and estimated flying time .

In this study, we use multivariable nonlinear regression to estimate . Specifically, we assume that is calculated as follows:

with the parameters  being learned from the historical flight operation data.

《4.4. Delay assignment mechanism》

4.4. Delay assignment mechanism

Due to air traffic delays, multiple flights may request departure services simultaneously at the same airport. Likewise, multiple flights may arrive at the final approach fix at the same time. Consequently, it is necessary to prioritize all departure and arrival flights, and assign delays to the flights that request departure/arrival services. With regard to this, we suggest the following mixed departure and arrival delay assignment mechanisms to further update and .

Algorithm 1 defines the way to update when multiple flights request to depart simultaneously from the same airport, as flight does. In Step 6 of Algorithm 1, the variable represents the minimum time separation to avoid the wake turbulence of departures/ arrivals.

In order to update when multiple flights request to arrive simultaneously at the same airport, as flight does, we further develop the arrival delay assignment mechanism. The proposed arrival delay assignment mechanism works in exactly the same manner as what is shown in Algorithm 1. Both of the mechanisms are implemented simultaneously.

《4.5. Spatial–temporal network construction》

4.5. Spatial–temporal network construction

The purpose of the above exhausted delay propagation and assignment mechanisms is to estimate the actual departure/arrival times for all the flights. Based on the estimated actual departure times, for a given time horizon [t₀, t_end] , we construct a spatial temporal network G =  at a time resolution of one minute, that is, t₁ - t₀ = 1 min. Each snapshot captures the delayed flights that depart at time t_i . Specifically, for network , we build the edges between their origin–destination (OD) pairs with the weights of the edges being the departure delays (measured in minutes) of the corresponding flights that depart at time t_i . For example, if a flight departs at time t_i from airport , is 10 min later than scheduled, and is heading for airport b, then we construct an edge between nodes and b of the network with the edge weight being 10. Note that could be empty, since all the flights could be in the air and/or on the ground at time t_i .

As per FAA instructions, a flight is considered to be delayed if, assuming that the departure is considered, ≥ 15. When constructing , we construct an edge between the OD pair of flight  as long as  ≥ 1. The benefit from doing so is that the constructed spatial–temporal network can analyze the delay propagation at different delay levels by extracting the corresponding subnetwork from with respect to a given delay threshold, such as 15 min, 30 min, and so forth.

《4.6. Delay propagation dynamics》

4.6. Delay propagation dynamics

In the literature, the most widely used metric for gauging reactionary delay is the DM index, which can be formulated as DM = ( D + I )/I, where I and D respectively denote the primary delays and the reactionary delays. In this work, we take the predicted departure delay as the D for flight , since the prediction of is the outcome of the collaborative behavior of the delay propagations of all the flights.

The DM metric is straightforward and effective for providing an overall view of the magnitude of the reactionary delays. However, its main advantage is also its main disadvantage. The DM metric cannot reflect the spatial–temporal dynamics of the delay propagation process, and this is the very motivation for proposing the spatial–temporal network-based perspective.

For the constructed spatial–temporal network G, we use to denote the adjacency matrix of its snapshot . Let be a time duration. We further define a matrix as follows:

where   and is an integer within the range of . The matrix represents the cumulative network of the snapshot within time period .  As pointed out in Subsection 4.5, the snapshot could be empty. By defining we can avoid analyzing empty networks directly, since they contribute little to the research problem.

We then analyze the spatial–temporal degree properties of and quantify the delay propagation dynamics in the following ways.

(1) Magnitude. We quantify the delay propagation magnitude in terms of the number of flights that suffer from reactionary delays, hereafter denoted by DP-mag1, and the amount of delays, hereafter denoted by DP-mag2. Let be the entry of . Then DP-mag1 and DP-mag2 with respect to are calculated as follows:

In the above equations, elements respectively represent the unweighted and weighted degrees of node i of the network characterized by matrix . The magnitude metric therefore captures the number of delayed flights as well as the amount of delays each single airport will encounter during the given time period .

(2) Severity. Based on the definitions of magnitude, we further define the delay propagation severity for the time window [ t₀, t_end ]as follows:

where the terms  are respectively calculated as follows:

One can observe from the above equations that the delay severity practically measures the time duration between the time point from which the magnitude starts to surpass its mean and the time point from which the magnitude starts to decrease from the mean. Therefore, the delay severity captures the duration of time the propagation magnitude can last.

(3) Speed. Based on the definitions of magnitude and severity, for a given time window [ t₀, t_end ], we quantify the propagation speed as follows:

As can be seen from Eq. (12), specifies the average delay propagation magnitude across all the airports in terms of the number of delayed flights. Note that the definition of the delay propagation speed with respect to is omitted, as it has a similar form as the definition formulated above.

《5. Experimental study》

5. Experimental study

《5.1. Testing instances》

5.1. Testing instances

Section 4 elaborated in detail the proposed network approach for understanding the propagation dynamics of air traffic delays. This section will demonstrate the applications of the proposed approach to the real-world flight plan data.

5.1.1. Flights in the SAR

In this study, we use the flights in the SAR as the first case study. The SAR is a subregion of Asia that consists of 11 countries with a total land area of ~4.5 × 10⁶ km² . After South Asia and East Asia, SAR is now the third most populous geographical region in the world, with a total population of over 641 million. Therefore, the SAR plays an important role both in the world’s economic development and in air transport (there are more than 700 airports in the SAR).

We abstract the flight schedules for the SAR from the automatic dependent surveillance-broadcast (ADS-B) data provided by the Civil Aviation Authority of Singapore (CAAS). The ADS-B data provided by CAAS covers domestic and global flights and spans a period of six months (June, July, September, October, November, and December) in the calendar year 2016. From the six-month ADS-B data, we filter out the schedules for the flights in the SAR.

Fig. 3 displays the spatial airport network derived from the sixmonth ADS-B data. The weight of an edge is equal to the number of flights that have flown between its two connected airports. The network shown in Fig. 3 has 139 nodes and 376 edges. For the SAR airport network, we further calculate the degrees of the nodes. We then rank the nodes based on their degrees.

《Fig. 3》

Fig. 3. Overview of the SAR airport network constructed using six-month ADS-B data collected for the calendar year 2016. An edge is created if there is a flight between its two endpoints.

The top part of Table 1 summarizes the basic information for the top 22 busiest airports in the SAR. It can be seen from Table 1 that node WMKK has the largest unweighted degree of 51, followed by node WSSS with a degree of 42. The Malaysian airport WMKK has the greatest number of connections with other airports in the SAR. Although the Singapore airport WSSS is the second largest tie, the weighted degree k_w values as recorded in Table 1 show that the Indonesian airport WIII accommodates more flights than the Singapore airport does. In general, Malaysia and Thailand have the busiest airports in the SAR.

《Table 1》

Table 1 Properties of the top 22 airports in the SAR and the top 22 airports in the United States.

k: unweighted degree; k_w: weighted degree.

5.1.2. Flights in the United States

The second case study was carried out on US flights. In the literature, validations on US flight data have been widely implemented. The key reason is that the collection strategy of US flight data is well established and the data is available to the public. It is easy to obtain ten-year US flight operation data from a handful of websites.

For the sake of better comparison, we obtained US domestic flight schedules from the Bureau of Transportation Statistics (BTS) website for the same time period as the SAR data. Based on the six-month US domestic flight schedules, we constructed a corresponding airport network (Fig. 4).

The network shown in Fig. 4 has 302 nodes and 2160 edges. The US airport network is more complicated than the SAR airport network in terms of the number of nodes and edges. The bottom part of Table 1 summarizes the basic information for the top 22 busiest airports (sorted by their weighted degrees) in the United States.

It can be seen from Table 1 that the values of unweighted degree k for the US airports are much larger than those for SAR airports. Each of the 22 top airports in the United States has more connections with other airports than each top airport in the SAR does. Each airport in the SAR is connected to 2.7 airports on average, while US airports are connected to an average of 7.2 airports. The greater numbers of airports and airport connections lead to larger throughputs, as reflected by the weighted degree k_w.

《Fig. 4》

Fig. 4. Overview of the US airport network constructed using six-month flight schedule data collected for the calendar year 2016. An edge is created if there is a flight between its two endpoints.

《5.2. Flight itinerary construction》

5.2. Flight itinerary construction

This work aims to investigate the propagation dynamics of reactionary delays. To do so, we need to work out the flight itineraries so as to trace the delay propagation between consecutive flight legs. The flight operation data obtained above contains tail numbers for all the flights. A tail number is a unique identifier that specifies a certain aircraft. Flights with the same tail number correspond to the same aircraft. Based on the tail numbers of the flights, we then construct the flight itineraries by sorting all the flights chronologically based on their scheduled departure time.

《5.3. Delay statistics overview》

5.3. Delay statistics overview

The SAR and the US flight schedules provide both the scheduled and actual times of the flights, based on which we know the real delays for the flights. In what follows, we first present the fundamental statistical studies on the delays.

It can be clearly seen from the upper part of Fig. 5 that the number of operated airports in the SAR in each of the studied six months is around 130, which is almost one third of the number operated in the United States. Furthermore, the number of monthly operated flights in the SAR is around 50 000, which is nearly one tenth of the number in the United States. It can also be seen from the upper part of Fig. 5 that the flights in the SAR have smaller arrival and departure punctualities than those in the United States. Nevertheless, the average flight delays in the SAR are much lower than those in the United States.

The middle and bottom parts respectively visualize the probabilistic distribution of the flight delays in the SAR and the United States, together with the curve fittings for the distributions of the positive delays. During the curve fittings, we have utilized three types of distribution: the Weibull, LogNorm, and exponential distributions, with their probabilistic distribution functions respectively being formulated as follows: , , and   , where  are all constants.

Although the average delays vary in different months, we can see that the delay distributions over different months in both the SAR and the United States share many resemblances. With regard to the curve fittings, the Weibull distribution obtains the best fittings in terms of the statistical metrics of R² and root mean squared error (RMSE). The curve-fitting results indicate that the positive delays obey the Weibull distributions.

《Fig. 5》

Fig. 5. Delay statistics for the SAR and US flight operation data recorded in the calendar year 2016. Arr: arrival; Dep: departure; PDF: probabilistic distribution of the fight delays; RMSE: root mean squared error.

《5.4. Problematic days selection》

5.4. Problematic days selection

The original flight schedule data is recorded by month. With regard to the fact that flight delays on different days in a month can possess unique properties, since air traffic is fraught with technical, operational, and meteorological issues, we analyze the delay propagation dynamics at a resolution of one day. More specifically, we abstract the daily flight schedules from a given monthly data as the basic unit for subsequent analysis.

In this study, we choose four days for every month—two days with the highest average departure delays and two days with the lowest average departure delays—as the problematic days to be studied. The selected days, together with their corresponding basic information, are listed in Table 2.

《Table 2》

Table 2 Selected problematic days for investigating the delay propagation dynamics of flights in the SAR and the United States.

: number of flights with ≥ 1; : number of flights with ≥15.

《5.5. Starting time point selection》

5.5. Starting time point selection

When a problematic day is determined, we then extract the 24 h flight schedules from the original flight data. To do so, we must determine the starting time point, from which we eventually build the flight itinerary. Different selections of starting time point could lead to different flight itineraries, which may affect the final results.

In order to determine the starting time point, we first visualize the six-month traffic demands over the time horizon from 00:00 to 24:00 at intervals of one hour. The statistics are shown in Fig. 6.

It can be seen from Fig. 6(a) that 20:00 p.m. can be regarded as the peak-off hour, as there are relatively few flights, while Fig. 6(b) suggests that the peak-off time for the US air traffic is 09:00 a.m. In this study, we choose the peak-off hours observed in Fig. 6 as the starting time points to respectively build the daily flight itineraries from the SAR and US flight schedules.

《Fig. 6》

Fig. 6. Distribution of the six-month traffic demands over the time horizon from 00:00 to 24:00 (UTC time) at intervals of one hour. (a) SAR; (b) United States.

《5.6. Reactionary delay analysis》

5.6. Reactionary delay analysis

For each selected problematic day, together with the primary flight delays, we implement the delay propagation and assignment mechanisms to work out the reactionary delays for the remaining flight legs. In order to investigate the difference between the reactionary delays of flights in the SAR and the United States, we calculate the values of the DM metric. Furthermore, based on the estimated reactionary delays, we calculate the classification accuracy for the delayed flights.

Fig. 7(a) visualizes the distributions of the values of the DM metric when applied to the reactionary delays for the studied 24 days. It can be seen from Fig. 7 that the majority of the DM values are in the range 1–2, which indicates that a primary delay of 1 min would cause a reactionary delay of around 1 min.

As for the delay accuracy, we observe from the middle part of Fig. 7 that the delay accuracies for flights with ≥ 1 and flights with ≥ 15 are respectively around 78% and 80% for the flights in the SAR, while the corresponding accuracies for flights in the United States are respectively around 72% and 85%. In the literature, the classification accuracies for delayed flights with ≥ 1 as reported in Refs. [29,30,35] are respectively 70%, 74.2%, and DD 82.7%. The results recorded in Fig. 7 indicate that the proposed delay propagation and assignment mechanisms make sense, and the subsequent analysis on the delay propagation dynamics is therefore reliable.

《Fig. 7》

Fig. 7. (a) Distribution of the values of the DM metric and (b) the classification accuracies for the delayed flights for the studied 24 problematic days.

《5.7. Delay propagation dynamics》

5.7. Delay propagation dynamics

The DM metric is a widely applied index to quantify the delay propagation effect. However, as can be seen from Fig. 7, the DM metric hardly provides useful insights into the magnitude, severity, and speed of the delay propagation dynamics, which is the very motivation for our suggested dynamic network approach.

Based on the reactionary delays, we then construct the spatial– temporal networks. In order to capture the delay propagation dynamics, we construct the spatial–temporal networks at a granularity of per hour. Specifically, in the network modeling process, we set δ = 60 min; as a consequence, we have Φ = [ 1, 24 ].

5.7.1. Delay propagation magnitude

In this paper, we only focus on the departure delays, since the results shown in Fig. 5 indicate that the arrival and departure delays have an approximately linear relation. Fig. 8 exhibits the delay propagation magnitude with respect to DP-mag1. On 18 July, among the top eight airports in the SAR, the three airports WMKK, WIII, and WSSS are found to be most congested, as there are about eight delayed flights per hour at each of these airports, while there are around three delayed flights per hour for each of the remaining top eight airports. On 30 November, since the average delay is smaller than that on 18 July, the top eight airports are relatively less congested. The top three airports are still found to be the most congested.

Regarding the US airports, it can be clearly seen from Fig. 8 that the delay propagation magnitude is higher than that in the SAR. In the United States, of the 40 busiest airports, about 30% are congested on each of the studied days, that is, 21 July and 25 November, while the top two airports, namely ATL and DEN, are the most congested, with around 30 delayed flights in each airport.

《Fig. 8》

Fig. 8. Delay propagation magnitude with respect to DP-mag1. (a, b) The values of DP-mag1 for the top 22 airports operated in the SAR (a) on 18 July and (b) 30 November. (c, d) The values of DP-mag1 for the top 40 airports operated in the United States on (c) 21 July and (d) 25 November.

The delay propagation magnitude in terms of the number of delayed flights shows a positive relation with the degrees of the airports. Next, we investigate the delay propagation magnitude in terms of the amount of delays occurring at each airport. The corresponding results are demonstrated in Fig. 9.

It can be seen from Fig. 9 that the WMKK airport suffers from significant delays on 18 July (74 min on average) and 30 November (64 min on average). Given that the airport operates at a maximum throughput, that is, with 30 flights taking off in 1 h, each flight on 18 July has a propagated delay of 2.5 min on average. On 30 November, the average delay is decreased to 2 min, since the traffic on that day is less congested than that on 18 July. The WIII and WSSS airports have significant delays on both of the studied days. Interestingly, for some airports, such as VTCC and WMKP, even though there are not too many delayed flights on the ground, there are still massive delays, as flights are heavily delayed due to reactionary delays.

Figs. 9(c) and (d) reveal that the delay propagation magnitude in terms of delays for the US flights is more significant than that of the SAR. On 21 July, the most congested airport, ATL, suffers from huge delays. As can be seen from Fig. 9, the hourly delays for the ATL airport are around 300 min. Putting it another way, each flight on the ground has a propagated delay of 10 min on average. On 25 November, the average propagated delay per flight is 2 min. Of the investigated 22 top airports in the SAR, around 25% have significant delays. However, around 40% of the 40 top US airports suffer from departure delays.

《Fig. 9》

Fig. 9. Delay propagation magnitude with respect to DP-mag2. (a, b) The values of DP-mag2 for the top 22 airports operated in the SAR on (a) 18 July and (b) 30 November. (c, d) The values of DP-mag2 for the top 40 airports operated in the United States on (c) 21 July and (d) 25 November.

5.7.2. Delay propagation severity

An analysis of the spatial–temporal airport networks provides a comprehensive understanding of the delay propagation dynamics in terms of magnitude. Here, we continue to analyze the delay propagation dynamics in terms of severity.

In this work, we define the delay propagation severity as the time duration for which the propagation magnitude can last. As the magnitude is quantified in two ways, the severity is also measured in two ways. It can be seen from the upper part of Fig. 8 that the peak time (the time when the number of delayed flights starts to exceed the mean value) starts at 03:00 a.m. and ends at 13:00 p.m. for the air traffic in the SAR. The middle and bottom parts of Fig. 8 indicate that the peak time starts at 13:00 p.m. and ends at 03:00 a.m. for the air traffic in the United States. Similar phenomena can be observed in Fig. 9. We therefore conclude that the delay propagation for the flights in the United States lasts longer than that of the flights in the SAR, since the propagation severities are 14 and 10 h, respectively.

It can be seen from the above results that the magnitude metric helps to estimate the amount of delays for each flight and airport, given the scheduled flight plans, while the severity metric indicates the duration of time the delay propagation process can last. With regard to this, airlines can estimate in advance whether scheduled flight plans will suffer from significant delays by utilizing the proposed network model and applying the magnitude metric. By doing so, airlines can adjust flight plans to counterbalance the delay propagation effect. Furthermore, air traffic controllers can utilize the proposed metrics to assist with timely monitoring of the air traffic situation. For example, the above results show that the airport ATL suffers from significant delays in terms of magnitude and severity. In this case, air traffic controllers may consider to take strategic ATM measures, such as ATFM, to balance the traffic demand, airspace, and airport capacities. In addition, airlines, air traffic controllers, airports, and other stakeholders can collaborate with each other to come up with effective CDM initiatives for better ATM.

5.7.3. Delay propagation speed

Apart from the delay propagation dynamics in terms of magnitude and severity, aviation workers may want to know how fast the delay can propagate. As per the definition of DP-spe presented in Subsection 4.6, we now calculate the delay propagation speed for the flights in the SAR and the United States.

Fig. 10 shows the delay propagation speed for the flights operated on the four days investigated above. In Fig. 10, the red solid curves record the values of with , that is, the averaged degrees of the spatial–temporal network characterized by matrix . As mentioned in Subsection 4.5, the construction of the network can analyze delay at different levels. Here, we also calculate the average degrees of the network characterized by , excluding edges with T_DD < 15, which are represented by the blue solid curves.

Note that reflects the hourly average degree of the network characterized by matrix . In order to better compare the delay propagation speed, we further calculate the mean of , which is represented by the red and blue dashed lines in Fig. 10. Table 3 records the mean values of for the selected 24 days for the flights in the SAR and United States. It can be seen from the left part of Table 3 that the average delay propagation speeds, Spd1, for the flights in the SAR are around 14, which means that primary delays can elicit delays in 14 flights per hour. When the situation in the United States is compared with that in the SAR, the delay propagation speeds for flights in the United States are about eight times faster than those in the SAR.

《Table 3》

Table 3 Delay propagation speeds for the flights in the SAR and United States.

Spd1: average propagation speed for the case of T_DD ≥ 1; Spd15: average propagation speed for the case of T_DD ≥ 15.

《Fig. 10》

Fig. 10. Delay propagation speed with respect to DP-spe. (a, b). The values of d/d for the flights operated in the SAR on (a) 18 July and (b) 30 November. (c, d) The values of d/d for the flights operated in the United States on (c) 21 July and (d) 25 November.

《6. Conclusion》

6. Conclusion

Air traffic inevitably suffers from internal/external perturbations, giving rise to air traffic delays that harm both the aviation industry and the natural environment. Studies on how delays happen and promising initiatives to mitigate delays are pivotal to aviation workers in modern ATM. Air traffic delays are associated with the high complexity of air transport systems. In order to maximize passenger movements, airlines normally plan an aircraft schedule with multiple flight legs during the aircraft’s rotation. As a consequence, a delay for one flight is likely to elicit reactionary delays for other flights, triggering the ripple effect, also known as delay propagation.

Delay propagation is detrimental to air traffic. An understanding of how delays propagate throughout air transport systems is crucial in order to achieve optimal structural design of air transport systems and to delay mitigation. This work presented a complex network perspective on air traffic delay propagation dynamics. Complementary to existing studies, the proposed network perspective can help decision-makers to acquire a comprehensive understanding of delay propagation dynamics in terms of magnitude, severity, and speed.

To validate the effectiveness of the proposed methodology, we carried out extensive case studies on flights in the SAR and the United States. We discovered that delay propagation dynamics for the flights in the SAR vary considerably from those in the United States. The proposed network-based method provides temporal details for the delay propagation dynamics for each airport and therefore contributes to strategic ATM and CDM.

《Acknowledgements》

Acknowledgements

The authors would like to thank Prof. Michael Ball from the University of Maryland and Mr. Narendra Pratap Singh, a senior air traffic controller, for their technical support and suggestions, which helped a lot to improve the quality of this work. This work was supported by SUG Research Grant M4082126.050 by the School of Mechanical and Aerospace Engineering (MAE), Nanyang Technological University (NTU), Singapore, and NTU-CAAS Research Grant M4062429.052 by the ATM Research Institute, School of MAE, NTU, Singapore.

《Compliance with ethics guidelines》

Compliance with ethics guidelines

Qing Cai, Sameer Alam, and Vu N. Duong declare that they have no conflict of interest or financial conflicts to disclose.