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Ground-Air Traffic Congestion Propagation Model Based on Hierarchical Control Interdependent Network

Ground-Air Traffic Congestion Propagation Model Based on Hierarchical Control Interdependent Network Hindawi Journal of Advanced Transportation Volume 2023, Article ID 4602148, 22 pages https://doi.org/10.1155/2023/4602148 Research Article Ground-Air Traffic Congestion Propagation Model Based on Hierarchical Control Interdependent Network Furong Jiang , Zhaoning Zhang, and Xiaoxu Dai Civil Aviation University of China, Tianjin 300300, China Correspondence should be addressed to Furong Jiang; frjiang@cauc.edu.cn Received 6 March 2023; Revised 11 April 2023; Accepted 27 April 2023; Published 5 May 2023 Academic Editor: Tao Liu Copyright © 2023 Furong Jiang et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A multilayer network approach to model and analyze air trafc networks is proposed. Tese networks are viewed as complex systems with interactions between airports, airspaces, procedures, and air trafc fows (ATFs). A topology-based airport-airspace network and a fight trajectory network are developed to represent critical physical and operational characteristics. A multilayer trafc fow network and an interrelated trafc congestion propagation network are also formulated to represent the ATF connection and congestion propagation dynamics, respectively. Furthermore, a set of analytical metrics, including those of airport surface (AS), terminal controlled airspace (TCA), and area-controlled airspace (ACA), is introduced and applied to a case study in central and south-eastern China. Te empirical results show the existence of a fundamental diagram of the airport, terminal, and intersections of air routes. Moreover, the dynamics and underlying mechanisms of congestion propagation through the AS- TCA-ACA network are revealed and interpreted using the classical susceptible-infectious-removed model in a hierarchical network. Finally, a high propagation probability among adjacent terminals and a high recovery probability are identifed at the network system level. Tis study provides analytical tools for comprehending the complex interactions among air trafc systems and identifes future developments and automation of layered coupled air trafc management systems. improve air trafc system resilience and alleviate air trafc 1. Introduction congestion to a considerable extent. Trafc fow is a ubiquitous dynamic process in human so- Diferent from urban trafc congestion, which is gen- ciety. Air trafc, which difers from most trafc variations, is erated by the confict between the vehicles driven by each strictly monitored by air trafc management (ATM). Air independent decision-maker (i.e., driver) and the limited trafc congestion is a scientifc problem restricting the road trafc resources, air trafc fow (ATF) congestion is development of civil aviation and fight safety. a combination of the air trafc control system (ATCS) and Te ATM system’s operational supervision initiatives the airline operation system in a limited space environment. are the determining factors for solving air trafc con- Terefore, as shown in Figure 1, ATF congestion generally occurs in a spatiotemporal environment with limited trafc gestion problems. To cope with increasing air trafc de- mand and congestion, the global ATM data management operation resources (such as the airport surface (AS) (in- system is being upgraded and transformed from sub- cluding aprons, taxiways, and runways), terminal controlled regional management to unifed management at the na- airspace (TCA), and area-controlled airspace (ACA) with tional level. By 2021, China has implemented a national route convergence). fow management system that plays the role of a brain In recent decades, several studies on ATF modeling have center in the operation and management of civil aviation been conducted. As an important location for the outfow air trafc. Te promotion and implementation of ATM and infow of air trafc, the AS operation has various system planning, such as the Single European Sky ATM constraints, such as release time, available apron, and Research (SESAR) and NextGen, are anticipated to runway assignment, which cause trafc congestion. Studies 2 Journal of Advanced Transportation high-altitude air routes must remain safely spaced and or- derly while fying. Bilimoria and Lee proposed a spatial clustering method based on the relative distance of aircraft and identifed the air trafc status by establishing the congestion index of “Gaggle Density” [16]. Lee et al. eval- ACA uated and identifed the degree of congestion caused by aircraft entering the control sector with diferent headings using the minimum course change parameter of aircraft collision avoidance [17]. Various recent studies highlight the importance of considering network modeling to study the propagation of trafc congestion and fight delays [18–21]. Lin et al. propose TCA a new fight delay model to capture the impact of en-route congestion on fight delays over an air trafc network [22]. Wang et al. provide an algorithm, which using sliding correlation windows to extract the airport pairs with stable AS delay lags, to estimate statistically signifcant time lags be- tween airport delays from noisy, aggregate operational data [23]. Wu et al. developed an Airport-Sector Network Delays Figure 1: Hierarchical airspace structure. model which takes both airports and airspace capacities into account [24]. on the trafc congestion on ASs have mainly focused on the Other studies have focused on airspace sector capacity assessments [25–27] and aviation network modeling based queuing problem connecting the apron, taxiway, and run- way [1–3]; fight delays [4–7]; and taxiing management on complex networks [28–34]. Most studies that reveal the topology and dynamic behavior of air trafc network [8–10]. Yang et al. discussed the congestion of departing trafc fow but ignored the infuence of arriving trafc fow congestion are based on scale-free small-world network modeling in which airports and fights are represented by [11]. Two distinct phases are involved, namely, free-fow and congested phases, which are represented by a density- nodes and edges, respectively. However, this modeling speed-fow relationship [12]. When the AS congestion in- method only considers airport nodes; it ignores the root tensifes and continues for some time, trafc congestion cause of congestion in actual air trafc network fows. Te spreads to the airspace upstream of the trafc fow, causing entire process of civil aviation fight operation is under the congestion in the terminal area. Few studies have system- monitoring and command of the ATCS, which controls the atically examined the characteristics of trafc fow conges- AS, TCA, and ACA. A review of literature indicates that existing studies focus tion on ASs, especially concerning simultaneous taxiing after landing and before takeof. on the modeling and evaluation of air trafc systems without fully comprehending the inherent airspace fow dynamics Congestion in the TCA is another key manifestation of air trafc dynamics. Te trafc fow congestion propagation and rules that govern their coevolution. Moreover, reports within the TCA has motivated the formulation of several on the congestion propagation of air trafc at the universal ATF control models in recent years to predict the overall network system level are limited. impact on air trafc delays and support fow management in Te air trafc network fow system refers to the for- the TCA of hub airports. Inspired by the fundamental di- mation of an air trafc network among airports through air agram of trafc fow [13, 14], Zhang et al. proposed a cell routes. Te network and movement of aircraft in this net- transmission model-based terminal airspace fow model work generates ATF; the two comprises the air trafc net- with an assumed “fow density-velocity” relationship [15]. work fow system. When fow input is absent, that is, no aircraft fies in the airspace, the air trafc network is a static More detailed trafc fow phase transitions (free-fow, smooth, semistable, and congested) were identifed by Yang trafc network with only spatial characteristics. In contrast, when aircraft is fying in the airport and air route, the non- et al. [12]. In contrast to AS trafc congestion, which can be analyzed using a two-dimensional model, TCA congestion is negative fow endows the air trafc network with certain a time-dependent change process in a three-dimensional dynamic characteristics. With the support of the network space. Hence, a strong correlation between the TCA and AS transmission of fow, the operation of the air trafc network congestions is observed. Accordingly, most studies do not fow system also has certain dynamic characteristics. distinguish the mechanism of congestion between these two. Moreover, the quantitative characterization of the com- However, from the perspective of trafc fow, ground fow plexity of air trafc systems from the perspective of trafc fow and airspace interactions is lacking. To conduct such and ATF have diferent motion states, fow environments, and restrictive conditions. Terefore, the AS trafc fow and characterization at both the macroscopic and microscopic levels, a novel multilayer network approach is proposed. Te arrival/departure trafc fows in the TCA can be analyzed separately in terms of their motion states. multilayer network consists of two physical networks, namely, a topology-based airport-airspace network (TAAN) Another critical area of air trafc congestion is the ACA. In this space, aircraft operating in medium-altitude and and a fight trajectory network (FTN) and two hierarchical Journal of Advanced Transportation 3 networks, namely, a multilayer trafc fow network (MTFN) network modeling method can simplify the actual route and an interrelated trafc congestion propagation network network and abstract the complex route network distribu- tion into a simple network with node and edge connections. (ITCPN). To support such a multilayer network framework and reveal the internal ATF node congestion propagation For the airspace around and above an airport, the trafc dynamics, diferent classes of analytical metrics are de- fow movement space is divided into AS, TCA, and ACA veloped as follows: trafc fow metrics of (i) AS, (ii) TCA, according to the trafc fow rule and control range of the (iii) ACA, (iv) trafc congestion metrics, and (v) hierarchical ATCS. Te adjacent nodes are also connected with edges, as congestion propagation metrics. Te main contribution of shown in Figure 4. this study is the formulation of the multilayer coupled congestion propagation network model. It also contributes 2.2. FTN. Consider a Tianjin-Shanghai fight fow as an the following fndings obtained through the verifcation of example. When an aircraft departs from Tianjin Airport and the theoretical and empirical data of a representative region climbs to a certain altitude, the departure procedure ends, in China. Te content and fow of this study are shown in and the TCA ATC (Air Trafc Controller) transfers the fight Figure 2. to the ACC ATC. After climbing out of the ZBTJ (Tianjin) (1) Empirical fow diagram (FD) of AS, TCA, and ACA: TCA, the aircraft passes through ZBAA (Beijing) ACC, ZSJN At the level of the air trafc network based on MTFN, (Jinan) ACC, and ZSSS (Shanghai) ACC and fnally ap- AS, TCA, and ACA are used as network nodes for the proaches and lands through ZSSS (Shanghai) TCA. Te frst time, and a bidirectional fundamental FD based route passing the ACCs encounters three route interchanges, on empirical data is created. Tree distinct fow states which can be represented as ACA nodes, as shown in were identifed for the nodes of the three levels, Figure 5. Accordingly, the airspace traversed by the entire namely, free, smooth, and congested. fight process is represented in the form of nodes to simplify the movement path of ATF. (2) Time slot-based congestion propagation network: At the regional level, the layered coupled congestion propagation network model is verifed by analyzing 2.3. MTFN. To defne an MTFN, TAAN and FTN as well as the trafc fow data of the AS, TCA, and ACA on three types of nodes (AS, TCA, and ACA) are combined. a typical busy day. Te introduction should be Considering the coupling network after the fusion of the succinct, with no subheadings. Limited fgures may TAAN and FTN, an MTFN was established according to the be included only if they are truly introductory and trafc fow motion connectivity. Subnetwork A represents contain no new results. the AS layer, and the AS nodes are not connected. Sub- network B represents the TCA layer; if the terminals are 2. Multilayer Network for Congestion close to or contain each other, the adjacent TCA nodes are connected by edges. Subnetwork C represents the ACA Propagation Modeling layer. According to the sector distribution of low/high- In region-based operations, the complex ATF dynamics altitude airspaces, adjacent ACA nodes are connected by between AS, TCA, and ACA and the operational status edges. As shown in Figure 6, the nodes among the layers are changes in AS, TCA, and ACA arise from the interactions linked according to the connection relationship of between the regional airway distribution, air trafc, and trafc fow. ATCS. To analyze the trafc fow dynamics of AS-TCA comprehensively, a novel analytical framework based on 2.4.ITCPN. Te nodes in Figure 6 correspond to ASs, TCAs, a multilayer network is established. Te framework consists and ACAs, signifying the primary zones where air trafc of two physical networks, namely, TAAN and FTN, and two converges and becomes concentrated. When the volume of hierarchical networks, namely, MTFN and ITCPN. Tese air trafc within these zones reaches high levels, congestion networks are elaborated in the following sections. emerges. Connected air trafc convergence areas tend to facilitate congestion spread across multiple areas in a cas- cade efect. Tis congestion propagation follows a pattern 2.1. TAAN. Conventionally, airspace networks are modeled as graphs with waypoints as nodes and route segments similar to the transmission of an epidemic among in- dividuals residing in proximity. Consequently, the infectious (links) as edges. However, as presented in this section, a topology-based airspace network is constructed, utilizing disease model is an apt approach to illuminating the the dense interchanges of air routes in the airspace and transmission of air trafc congestion. surrounding airspace as airspace nodes to form the ACA In studies of trafc congestion propagation mechanisms, network. As shown in Figure 3, the nodes in the ACA infectious disease models such as susceptible-infectious- network represent major air route interchanges or air route susceptible (SIS), susceptible-infectious-removed (SIR), and interchanges formed by multiple interchanges. Te con- susceptible-exposed-infectious-removed (SEIR) are com- monly employed [5, 7, 35]. It is crucial to note that disease nectivity in the ACA network refects the adjacency of air routes between interchanges and dense air routes. High- propagation is irreversible, i.e., epidemiology spread is a unidirectional difusion that mirrors congestion propa- altitude air trafc congestion typically occurs during route interchanges. Terefore, the topology-based airspace gation. For this reason, traditional infectious disease models 4 Journal of Advanced Transportation Empirical Data Network Modeling Analytical Metrics Empirical Results AS Traffic Flow Metrics Airspace Airway-based Air traffic flow Configuration Airspace Network TCA Traffic Flow dynamic at network system level Metrics Fuse Flight Trajectory ACA Traffic Flow Flight Network Metrics Schedule Congestion Propagation Data- Driven Traffic Congestion Traffic Congestion Traffic Congestion Metrics propagation Propagation Radar mechanism at Network Multi-layer Trajectory SIR Model network system level Congestion Propagation Metrics Figure 2: Framework of proposed network-based air trafc congestion propagation analysis. ZBTJ ZSSS AS AS ZBTJ ZSSS TCA TCA ACA 1 ACA 3 Figure 3: Topology-based ACA network. ACA 2 Figure 5: FTN of Tianjin-Shanghai fight fow. MTFN to describe trafc congestion propagation both be- ACA tween nodes of each layer and nodes of diferent layers. Furthermore, trafc node states can be classifed into ACA free, congested, and removed states which correspond to susceptible, infected, and removed states in SIR model, TCA respectively. In Section 3, defnitions of free, congested, and removed states will be investigated based on node type, TCA connecting distribution patterns of nodes within, and AS among distinct layers. Te connection relationship between the nodes AS and layers in the subnetwork is constructed based on the TAAN and FTN. Te defnitions of parameters are shown in Table 1. To represent the probability that node A in subnet A Figure 4: Interlayer connection structure. is randomly selected to have j edges connected to B, P (j) is defned. Similarly, P (i, j, k) represents the probability that are deemed appropriate for explaining air trafc congestion a node is randomly selected in subnet B with i edges con- propagation caused by single source of congestion. nected to A, j edges to B, and k edges to C. Te probability of Nonetheless, congestion propagation between nodes is randomly selecting a node in subnet C that has j edges solely associated with node degree and congestion trans- connected to B and k edges connected to C is represented by mission rate within traditional infectious disease models, P (j, k). with no diferentiation between node types or direct con- According to the foregoing parameter defnitions, the gestion transmission rates of various node types. Te string- total number of nodes in the three subnetworks and the coupled three-layer network model [28] overcomes this number of nodes in the crowded, free, and removed states shortcoming in the SIR model and can be implemented in can be expressed as follows: MTFN Journal of Advanced Transportation 5 AS 13 AS 14 AS 12 AS 15 TCA 10 TCA 11 TCA 12 Layer A- AS Layer B- AS 11 ACA 8 TCA Layer C- TCA 9 ACA AS 10 TCA 8 ACA 9 AS 9 ACA 10 ACA 7 ACA 11 TCA 6 TCA 7 AS 8 ACA 2 AS 7 ACA 1 ACA 6 ACA 3 TCA 1 AS 1 ACA 5 TCA 2 AS 2 ACA 4 TCA 3 TCA 4 AS 3 TCA 5 AS 5 AS 4 AS 6 Figure 6: MTFN example. Table 1: Defnition of ITCPN parameters. Parameter Meaning N Number of nodes in layer A with degree (0, j) A A A I (or S or R ) Number of infected (or susceptible or removed) nodes in layer A with degree (0, j) j j j N Number of nodes in layer B with degree (i, j, k) i,j,k B B B I (or S or R ) Number of infected (or susceptible or removed) nodes in layer B with degree (i, j, k) i,j,k i,j,k i,j,k N Number of nodes in layer C with degree (j, k) j,k C C C I (or S or R ) Number of infected (or susceptible or removed) nodes in layer C with degree (j, k) j,k j,k j,k n Maximum degree value of nodes in layer A connecting to layer B n (or n or n ) Maximum degree value of nodes in layer B connecting to layer A (or B or C) 21 22 23 n (or n ) Maximum degree value of nodes in layer C connecting to layer B (or C) 32 33 ⟨k⟩ Average degree value of nodes in layer A connecting to layer B ⟨k⟩ (or ⟨k⟩ or ⟨k⟩ ) Average degree value of nodes in layer B connecting to layer A (or B or C) 21 22 23 ⟨k⟩ (or ⟨k⟩ ) Average degree value of nodes in layer C connecting to layer B (or C) 32 33 6 Journal of Advanced Transportation A(orBorC) A(orBorC) A(orBor C) A(orBorC) Table 2: Equations of ITCPN parameters. ⎧ ⎪ N � I + S + R , ⎪ n 12 Parameter Value A A A A A A I 􏼐or S or R 􏼑 � 􏽘 I 􏼐or S or R 􏼑, ⎪ A A j j j ⎪ P (j) N /N ⎪ A j ⎪ j�0 B B P (i, j, k) N /N n n n B i,j,k 21 22 23 (1) B B B B B B I 􏼐or S or R 􏼑 � 􏽘 􏽘 􏽘 I 􏼐or S or R 􏼑, C C i,j,k i,j,k i,j,k P (j, k) N /N C j,k i�0 j�0 ⎪ k�0 n n ⎪ 22 23 P (i, ·, ·) 􏽐 􏽐 P (i, j, k) ⎪ B j�0 B n n k�0 32 33 C C C C C C n n 21 23 I 􏼐or S or R 􏼑 � 􏽘 􏽘 I 􏼐or S or R 􏼑. P (·, j, ·) 􏽐 􏽐 P (i, j, k) ⎪ j,k j,k j,k B B i�0 k�0 j�0 k�0 n n 21 22 P (·, ·, k) 􏽐 􏽐 P (i, j, k) B i�0 j�0 B Te joint degree distribution, marginal degree distri- P (j, ·) 􏽐 P (j, k) C C k�0 bution, average degree values (φ �1), and second moment of P (·, k) 􏽐 P (j, k) degree (φ � 2) are calculated; the results are summarized in C j�0 C Table 2. φ 12 φ ⟨k ⟩ 􏽐 j P (·, j) 12 j�0 A Te probability that the AS node of layer A is trans- φ 21 φ ⟨k ⟩ 􏽐 i P (i, ·, ·) 21 i�0 B formed into a congested state by the infuence of the TCA φ 22 φ node of layer B (connected layer) is λ . Te probabilities that 21 ⟨k ⟩ 􏽐 j P (·, j, ·) 22 j�0 B the TCA node of layer B is transformed into a congested φ 23 φ ⟨k ⟩ 􏽐 k P (·, ·, k) 23 B k�0 state by the infuence of the AS node of layer A, TCA node of φ φ layer B, and ACA node of layer C (connected layers) are λ , ⟨k ⟩ 􏽐 j P (j, ·) 12 32 j�0 C λ , and λ , respectively. Te probabilities that the ACA 22 32 φ 33 φ ⟨k ⟩ 􏽐 k P (·, k) 33 C k�0 node in layer C is afected by the TCA node in layer B and ACA node in layer C in the congested state are λ and λ , 23 33 C C C C N , i � I /N are similarly defned. According to the respectively. In subnetworks A, B, and C, the transition ι,κ ι,κ ι,κ ι,κ previous assumption, we build a networked mean-feld probabilities of the congested state nodes to the removed spreading model, which is composed of [(n + 1) state are μ , μ , and μ , respectively. 1 2 3 12 +(n + 1)(n + 1)(n + 1) + (n + 1)(n + 1)]-dimensio Based on the foregoing assumptions, the SIR density 21 22 23 32 33 A A A A A A nal ordinary diferential equations as follows: model of ITCPN is established. Let s � S /N , i � I /N , ε ε ε ε ε ε which represent the corresponding densities. Te other B B B B B C B C densities s � S /N , i � I /N , s � S / ϕ,c,η ϕ,c,η ϕ,c,η ϕ,c,η ϕ,c,η ϕ,c,η ι,κ ι,κ ⎪ δs (τ) ⎪ ε � −λ εs (τ)Θ (τ), ⎪ 21 21 δτ δi (τ) ⎪ ε A A � λ εs (τ)Θ (τ) − μ i (τ), 21 ε 21 1 ε δτ δs (τ) ϕ,c,η Β Β Β � −λ ϕs (τ)Θ (τ) − λ cs (τ)Θ (τ) − λ ηs (τ)Θ (τ), ⎪ 12 ϕ,c,η 12 22 ϕ,c,η 22 32 ϕ,c,η 32 δτ (2) ⎪ δi (τ) ⎪ ϕ,c,η Β Β Β Β ⎪ � λ ϕs (τ)Θ (τ) + λ cs (τ)Θ (τ) + λ ηs (τ)Θ (τ) − μ i (τ), 12 ϕ,c,η 12 22 ϕ,c,η 22 32 ϕ,c,η 32 2 ϕ,c,η δτ ⎪ δs (τ) ι,κ C C � −λ ιs (τ)Θ (τ) − λ κs (τ)Θ (τ), ⎪ 23 23 33 33 ι,κ ι,κ ⎪ δτ δi (τ) ι,κ C C C � λ ιs (τ)Θ (τ) + λ κs (τ)Θ (τ) − μ i (τ), 23 ι,κ 23 33 ι,κ 33 3 ι,κ δτ where ε � 0, 1, ..., n ; ϕ � 0, 1, ..., n ; c � 0, 1, .., n ; η � 0, 1, In the traditional SIR model with average degree value 12 21 22 ..., n ; ι � 0, 1, ..., n ; κ � 0, 1, ..., n . equals ⟨k⟩, Θ (τ) is the probability that any adjacent contact 23 32 33 To simplify the model further, the coupled network is can be afected to transform into a crowded state. It can be assumed to be degree-independent, that is, the degree of any expressed as the average joint probability of connecting from node is independent of the degrees of its neighboring nodes. a node of degree α to a node of degree β, P(β | α)i (τ), β β Journal of Advanced Transportation 7 which is abbreviated as Θ(τ) � 􏽐 P(β | α)i (τ) � 1/ Te classical SIR model was used to analyze the prop- β β ⟨k⟩􏽐 kP(k)i (τ). agation of the crowded state of the node in the ITCPN. Te k k basic regeneration number, R In the string-coupled three-layer network model, the , is an important parameter formulas are expressed as follows: for revealing the propagation characteristics. In this study, R represents the number of nodes converted into a crowded n 0 ⎧ ⎪ A state; the conversion is due to the infuence of one congested ⎪ Θ (τ) � 􏽘 jP (j)i (τ), 12 A ⎪ j ⟨k⟩ ⎪ 12 node propagation in all the free-state network nodes. Te j�0 ⎪ fundamental reproduction number, R , can be calculated −1 ⎪ n n n using the next generation matrix, Γ � FV [35], where F is 21 22 23 ⎪ Θ (τ) � 􏽘 􏽘 􏽘 iP (i, j, k)i (τ), the rate of new occurring congestion, and V is the rate of 21 B ⎪ i,j,k ⟨k⟩ ⎪ 21 i�0 j�0 k�0 transferring congested nodes from the free nodes group. ⎪ Matrix Γ can be rewritten using similar transformations as n n n 21 22 23 follows: Θ (τ) � jP (i, j, k)i (τ), ⎪ 􏽘 􏽘 􏽘 ⎪ 22 B i,j,k ⟨k⟩ i�0 j�0 k�0 (3) n n n 21 22 23 ⎪ Θ (τ) � 􏽘 􏽘 􏽘 kP (i, j, k)i (τ), ⎪ 23 B i,j,k ⟨k⟩ ⎪ 23 i�0 j�0 k�0 ⎪ n n 32 33 ⎪ Θ (τ) � 􏽘 􏽘 jP (j, k)i (τ), 32 C ⎪ j,k ⟨k⟩ j�0 ⎪ k�0 ⎪ n n 32 33 ⎪ 1 Θ (τ) � 􏽘 􏽘 kP (j, k)i (τ). ⎩ 33 C j,k ⟨k⟩ j�0 k�0 λ ⟨k ⟩ λ λ 21 21 21 21 ⎡ ⎢ 0 ⟨k⟩ ⟨k⟩ 0 0 ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 22 23 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ μ ⟨k⟩ μ μ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 2 21 2 2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ λ ⟨k ⟩ ⎥ ⎢ ⎥ ⎢ 12 12 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 0 0 0 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ μ ⟨k⟩ ⎥ ⎢ ⎥ ⎢ 2 12 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ λ λ ⟨k ⟩ λ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 22 22 22 22 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 ⟨k⟩ ⟨k⟩ 0 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 21 23 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ μ μ ⟨k⟩ μ ⎥ ⎢ ⎥ ⎢ 2 2 22 2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ Γ � ⎢ ⎥. (4) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ λ ⟨k ⟩ λ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 32 32 32 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 0 0 0 ⟨k⟩ ⎥ ⎢ ⎥ ⎢ 33 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ μ ⟨k⟩ μ ⎥ ⎢ ⎥ ⎢ 3 32 3 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ λ λ λ ⟨k ⟩ ⎥ ⎢ ⎥ ⎢ 23 23 23 23 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 ⟨k⟩ ⟨k⟩ 0 0 ⎥ ⎢ ⎥ ⎢ 21 22 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ μ μ μ ⟨k⟩ ⎥ ⎢ ⎥ ⎢ 2 2 2 23 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ λ λ ⟨k ⟩ ⎦ 33 33 33 0 0 0 0 ⟨k⟩ μ μ ⟨k⟩ 3 3 33 Te fundamental reproduction number of the model is operating network, and region-level interrelated trafc � ρ (Γ), where ρ (Γ) is the spectral radius of matrix Γ. congestion propagation characteristics are thoroughly in- vestigated using diferent network representations of such 2.5. Summary. Te primary aim of this study is to explore complex systems. Te MTFN is a three-layer network representing the the complex dynamics of ATF and the operating states of airports and airspaces through which trafc fows; empirical connection between the AS, TCA, and ACA. As described in Section 2.3, to model the MTFN, generating the cross-layer and comprehensive approaches are used. To achieve this objective, the ATF trajectory, AS-TCA-ACA air trafc linkage between the TAAN and FTN is crucial. Tis 8 Journal of Advanced Transportation identifes the aircraft fying along each airspace sector and We set the length of the time window presented in derives the trafc fow metrics presented in Section 3. An Sections 3 to 30 min. Te nonlinear relationships between ITCPN is developed based on the integrated TAAN-FTN PretTTD, PoldTTD, normalized PretTS, and normalized and the propagation law of the node congestion state. Te PoldTS are shown in Figure 7(a). For a detailed analysis of propagation parameters of the node congestion state in the the AS taxiing trafc fow dynamics, the three phases are three-layer AS-TCA-ACA network are represented based on divided by dissecting the fundamental diagram (Figures 7(b) the characteristics of the classical SIR model in the layered and 7(d)) and temporal-spatial diagrams (Figure 8). coupled network. Tis provides a mathematical model for (1) Free phase corresponds to relatively low PretTTD simulating the parameters and verifying empirical data. and PoldTTD, adequate taxiway availability, and low interaction among aircraft. Generally, the entry of 3. Analytical Metrics aircraft into the parking stand is not restricted if no fight follows after its landing. Consequently, the 3.1. AS Trafc Flow Metrics. An FD that showed the phase PoldTS in the free-state is typically higher than the transition of trafc fow on a road section by describing the PretTS in the same state. Tis is because departing one-to-one relationship among the fundamental parameters fights have more limiting factors, such as fight time, of trafc fow was proposed by Daganzo [13]. Considering runway, and airspace. that the dynamic monitoring of AS trafc fow mainly in- cludes the taxiing trafc fow of aircraft about to take-of and (2) Smooth phase corresponds to relatively high has landed, the following are defned: PretTTD and PoldTTD with a distinct reduction in both PretTS and PoldTS compared with the free (1) Take-of Trafc Flow, Q . Number of aircraft taking To phase owing to the high fow density. At this stage, of from the airport within a specifed time. the trafc fow in the AS signifcantly increases, the (2) Landing Trafc Flow, Q . Number of aircraft Ld average pre-takeof taxiing speed of the aircraft landing at airports within a specifed time. considerably decreases, and the waiting time for (3) Pre-takeof Taxiing Trafc Density (PretTTD), ρ . takeof increases, although the operation is within To-t Total number of aircraft taxiing before take-of the normal range. within a specifed time. (3) Congested phase corresponds to the drop in both (4) Postlanding Taxiing Trafc Density (PoldTTD), ρ . the PretTTD and PretTS as PoldTTD and PoldTS Ld-t Total number of aircraft taxiing after landing within continue to increase as a consequence of the a specifed time. higher priority of aircraft landing than of take-of. At this stage, the postlanding trafc fow moves to (5) Average Pre-takeof Taxiing Speed (PretTS), V . Te To-t the AS at high density and velocity. Meanwhile, average taxiing speed of all aircraft ready to take-of the pretake-of trafc fow is heavy because nu- at the airport in the taxiing state within a specifed merous aircraft are in the waiting state owing to time; this can be regarded as the average velocity of resource limitations. Moreover, the density and the entire trafc fow in the pre-takeof taxiing state. velocity are both low. (6) Average Postlanding Taxiing Speed (PoldTS), V . Ld-t Te average taxiing speed of all aircraft in the taxiing As for the ITCPN model presented in Section 2, both the state after landing at the airport within a specifed free and smooth phases are defned as a free state (sus- time; this can be regarded as the average velocity of ceptible state), and the congested phase is defned as the entire trafc fow in the postlanding taxiing state. a congested state (infected state). When the AS changes from a relatively distinct congested phase to a smooth phase, Airports are a major source of initial delays in fight. Te both the PretTTD and PretTS rise back to a relatively high operation for taxiing on the ground and fying in air can be level, whereas PoldTTD and PoldTS drop to an average level. regarded as a complete closed-loop trafc fow. Te trafc Tis can be considered as the end of the congested state and fow within the AS mainly includes fights taxiing on the defned as the removed state. ground after landing and fights preparing to take-of after leaving the parking stand. Te congestion of AS mainly occurs in the movement 3.2. TCA Trafc Flow Metrics. Similar to the approach of range of aircraft on the ground, such as aprons, taxis, and defning parameters in the AS node, the spatial range is runways. Te fight congestion caused by the limited resources defned from an altitude of 1000 m to the upper limit of the of the airport is represented by fight departure delay. Con- airspace of the terminal area (generally 6000 m). Moreover, sidering that the taxiing route and taxiing time in diferent the latitude and longitude boundaries of the horizontal airports cannot be directly compared because of the diferent range of the terminal area are considered as nodes. To study confgurations of runways, taxiways, and parking stands, the the motion state of trafc fow in TCA nodes, the following taxiing speed must be normalized. If the total taxiing process is are defned. 1 and the taxiing time is x (in minutes), the average taxiing speed after normalization is 1/x. Te length of the time slot is (1) Departure Trafc Density (DTD), ρ . Number of Dpt 30 min. Figure 7 shows ρ , ρ , and the nonlinear re- aircraft departing from the terminal area within To-t Ld-t lationship between the normalized V and V . a specifed time. To-t Ld-t PretTTD (aircraf/30mins) Journal of Advanced Transportation 9 Frequency 1.0 Congested 0.9 0.8 20 0.7 0.6 Smooth 0.5 0.4 0.3 0.2 Free 0.1 22 0 5 10 15 20 25 14 PretTTD (aircraf/30 min) Normalized PretTS Normalized PoldTS (a) (b) 0.35 0.35 1.4 1.4 0.30 0.30 1.2 1.2 0.25 0.25 1.0 1.0 0.20 0.20 0.8 0.8 0.15 0.15 0.6 0.6 0.10 0.10 0.4 0.4 0.05 0.05 0.2 0.2 0.00 0.00 0.0 0.0 0 5 10 15 20 25 0 5 10 15 20 25 PretTTD (aircraf/30 min) PoldTTD (aircraf/30 min) Normalized PretTS Normalized PretTS Normalized PoldTS Normalized PoldTS (c) (d) Figure 7: FD in pudong AS: (a) FD; (b) PretTTD-PoldTTD-frequency relationship; (c) PretTTD-PretTS-PoldTS relationship; (d) PoldTTD-PretTS-PoldTS relationship. 0.30 0.20 0.25 0.16 Congested 0.20 0.12 Congested 0.15 0.08 0.10 5 0.04 0.05 Smooth Free Smooth 0 0.00 0 0.00 06:00 08:00 10:00 12:00 14:00 16:00 18:00 20:00 22:00 00:00 06:00 08:00 10:00 12:00 14:00 16:00 18:00 20:00 22:00 00:00 PretTTD PretTTD PoldTTD Normalized PretTS Normalized PretTS Normalized PoldTS (a) (b) Figure 8: Continued. PoldTTD (aircraf/30mins) PretTTD/PoldTTD (aircraf/30 min) Normalized PretTS Normalized PretTS/PoldTS Normalized PoldTS PretTTD (aircraf/30 min) Normalized PretTS PoldTTD (aircraf/30 min) Normalized PoldTS Normalized PretTS 10 Journal of Advanced Transportation 0.28 0.24 Congested 0.20 0.16 0.12 0.08 0.04 Smooth Smooth Free 0 0.00 06:00 08:00 10:00 12:00 14:00 16:00 18:00 20:00 22:00 00:00 PoldTTD Normalized PoldTS (c) Figure 8: Temporal density-speed diagram of taxiing trafc fow in AS of pudong on 12/15/2019: (a) PretTTD-PoldTTD-PretTS-PoldTS- time relationship; (b) PretTTD-PretTS-time relationship; (c) PoldTTD-PoldTS-time relationship. (2) Approach Trafc Density (ATD), ρ . Number of that in Section 3.1, the three phases were divided by scru- Arv aircraft approaching the terminal area within tinizing the fundamental diagram (Figures 9(b)– 9(d)) and a specifed time. temporal-spatial diagrams (Figure 10). (3) Average Departure Speed (ADS), V . Te average Dpt (1) Free Phase corresponds to relatively low DTD and departure speed of all departure fights in the ter- ATD, large available space, and negligible interaction minal area within a specifed time can be regarded as among aircraft. Te departure speed of an aircraft is the average departure speed of the entire departure generally extremely high if no congestion occurs in trafc fow. the subsequent route after takeof; hence, the ADS in the free-state is generally higher than the AAS in the (4) Average Approach Speed (AAS), V . Te average Arv approach speed of all departure fights in the ter- same state. Tis is because the approach procedure has more restrictive conditions than the departure minal area within a specifed time can be regarded as the average velocity of the entire approach procedure in the terminal area. trafc fow. (2) Smooth Phase corresponds to a relatively high DTD and ATD with a distinct reduction in ADS compared Te terminal area is the main section in which increased with the free phase owing to the high fow density. At fight congestion occurs. Owing to the reduction in airspace this stage, the trafc fow of the terminal airspace capacity caused by the weather, convergence of waypoints, signifcantly increases, and the AAS of the aircraft trafc congestion in the air section, and other reasons, the considerably decreases. Moreover, the ADS fuctu- trafc fow in this area may turn into a crowded state. ates within a certain range, although it is within the Previous studies have shown that terminal trafc fow has normal operation range. the basic properties of a fuid, and the trafc capacity be- tween approach and departure has the property of a convex (3) Congested Phase corresponds to a drop in the AAS function [36]. as the ATD continues to increase. At this stage, the Owing to the diferent fight paths of the approach and departing trafc fow moves at a relatively high departure procedures in the terminal area, the approach and density and speed in the terminal area. Furthermore, departure processes cannot be directly compared. Terefore, the approach trafc fow has numerous aircraft in the trafc fow speed in the terminal area is assumed to be waiting states at low speed owing to resource limitations. normalized regardless of the diference in the basic time of approach and departure caused by fight procedure difer- For the ITCPN model presented in Section 2, the free ences. In the terminal area, if the total fight process is 1 and and smooth phases are defned as a free state (susceptible the approach (departure) time is x (in minutes), the average state), and the congested phase as a congested state (infected normalized approach (departure) speed is 1/x. Te length of state). Te TCA may change from a relatively distinct the time slot is 30 min. Figure 9 shows ρ , ρ , and the Dpt Arv congested phase to a smooth phase in which the ATD drops nonlinear relationship between the normalized V Dpt to a normal level and the AAS rises back to a relatively and V . Arv average level, whereas the DTD and ADS return to a smooth Figure 9(a) shows the nonlinear relationship between the phase. Tis can be considered as the end of the congested DTD, ATD, normalized ADS, and normalized AAS. For state; this is defned as the removed state. a detailed analysis of TCA trafc fow dynamics, similar to PoldTTD (aircraf/30 min) Normalized PoldTS DTD (aircraf/30 mins) Journal of Advanced Transportation 11 Frequency Congested Smooth Free 10 25 20 0 0 5 1015202530 30 0 DTD (aircraf/30 min) Normalized ADS Normalized AAS (a) (b) 0.20 0.20 0.20 0.20 Free 0.15 0.15 0.15 0.15 0.10 0.10 0.10 0.10 0.05 0.05 0.05 0.05 0.00 0.00 0.00 0.00 0 5 1015202530 0 5 10 15 20 25 DTD (aircraf/30 min) ATD (aircraf/30 min) Normalized ADS Normalized ADS Normalized AAS Normalized AAS (c) (d) Figure 9: FD in Shanghai TCA. (a) Fundamental diagram; (b) DTD-ATD-frequency relationship; (c) DTD-ADS-AAS relationship; (d) ATD-ADS-AAS relationship. 40 40 0.14 0.12 30 0.12 0.10 20 0.10 Congested 0.08 Free Smooth Congested Smooth 10 0.08 0.06 0 0.06 06:00 08:00 10:00 12:00 14:00 16:00 18:00 20:00 22:00 00:00 06:00 08:00 10:00 12:00 14:00 16:00 18:00 20:00 22:00 00:00 DTD DTD ATD Normalized ADS Normalized ADS Normalized AAS (a) (b) Figure 10: Continued. 0.20 0.15 0.10 0.05 ATD (aircraf/30 mins) Normalized ADS DTD/ATD (aircraf/30 min) Normalized AAS Normalized ADS/AAS DTD (aircraf/30 min) Normalized ADS ATD (aircraf/30 min) Normalized ADS Normalized AAS 12 Journal of Advanced Transportation 40 0.070 30 0.065 20 0.060 Free Congested Smooth Smooth 10 0.055 0 0.050 06:00 08:00 10:00 12:00 14:00 16:00 18:00 20:00 22:00 00:00 ATD Normalized AAS (c) Figure 10: Temporal density-speed diagram of trafc fow in Shanghai TCA on 15/12/2019: (a) DTD-ATD-ADS-AAS-time relationship; (b) DT-ADS-time relationship; (c) ATD-AAS-time relationship. 3.3. ACA Trafc Flow Metrics. Te studies that have been 32.6 conducted on congestion propagation in the ACA are limited. In general, if congestion occurs at the TCA and lasts 32.4 for a long time, congestion also occurs at the upstream ACA. Tis can be manifested by the decrease in the horizontal or 32.2 vertical interval of the aircraft to a minimum, adjustment in the altitude or speed of the aircraft, aircraft circling, and 32.0 waiting at the designated waypoint, etc. In the high-altitude airspace, ATF congestion mainly occurs at the air route α 31.8 interchange. Te area where multiple air routes cross is prone to a shortage of airspace resources, leading to con- 31.6 gestion. To simplify the representation of the ACA nodes, as shown in Figure 11, the intersection where the main route 31.4 crosses is selected as the ACA node. To study the motion state of the trafc fow in ACA nodes, the following are defned: 31.2 (1) Cruising Trafc Density (CTD), ρ R . Number of C us 31.0 aircraft cruising across the ACA node area within 116.6 116.8 117.0 117.2 117.4 117.6 117.8 118.0 a specifed of time. Longitude (°E) (2) Average Cruising Speed (ACS), V R . Te average C us Figure 11: Trajectory data of a typical intersection of air routes in cruising speed of all cruising fights across the ACA China. node area within a specifed time can be regarded as the average cruising speed of the entire cruising Te motion state of trafc fow at the ACA node is trafc fow. studied using the analysis method of FD in the AS and TCA. (3) Average Angular Velocity (AAV), ω R . Te average C us Because the cruising trafc fows fying into and out of ACA angular velocity of all cruising fights across the ACA nodes have varying angles, fight distances, and fight paths, node area within a specifed time can be regarded as the status of all cruising fights passing through ACA nodes the average angular velocity of the entire cruising cannot be directly compared. Terefore, the geometric trafc fow. center where the ACA node trafc fow converges is selected In high-altitude airspaces, the air route intersection is the as the center of the circle. Te length that can contain the main area of increasing fight congestion. Owing to the main convergence area is considered as the radius. Te reduction in airspace capacity caused by public navigation circular area and its center (orange), which is set as the ACA stations and the limited range of available airspace, the trafc node, are shown in Figure 11. Te cruise trafc fow velocity fow in this area may enter a crowded state. of ACA nodes is assumed to be normalized ignoring the time ATD (aircraf/30 min) Latitude (°N) Normalized AAS Journal of Advanced Transportation 13 diference between entering and leaving the ACA node 4. Numerical Simulation Analysis area owing to fight path variations. In the ACA node A congestion propagation simulation analysis is imple- area, if the total fight process is 1, and the time from mented based on the ITCPN model presented in Section 2 entering the node airspace range to leaving this range is x and the analysis of the phase transition of the AS-TCA-ACA min, the average normalized cruise speed is 1/x. If the node discussed in Section 3. Te following equation rep- angle of the total fight process is α and the time from resents the average crowded state density of subnetworks A, entering the node airspace range to leaving this range is x B, and C at time t: (in minutes), the average normalized angular velocity is α/x. Te length of the time slot is 15 min. Te nonlinear n n n n 12 21 22 23 A A B relationship between ρ , V , and ω are shown in i (t) � 􏽘 P (j) · i (t), i (t) � 􏽘 􏽘 􏽘 P (i, j, k) Crus Crus Crus A j B Figure 12. j�0 i�0 j�0 k�0 (5) Figure 12(a) shows the relationship between the CTD, n n 32 33 B C C ACS, and AAV. Te fgure indicates that the AAV is · i (t), i (t) � 􏽘 􏽘 P (j, k) · i (t). i,j,k j,k j�0 fundamentally proportional to the ACS, whereas the CTD k�0 is negatively correlated with the relationship between the two. Sections 3.1 and 3.2 elaborate the analysis of ACA trafc fow dynamics. Te three phases are divided by 4.1. Fundamental Parameter Characteristics of ITCPN Model. scrutinizing the fundamental diagram (Figures 12(b) and Set N � N � 80. Tis indicates that the number of airports A B 12(d)) and temporal-spatial diagram (Figure 13). is equal to the number of terminal areas, ignoring the case in which adjacent airports share terminal areas. Te status of (1) Free Phase corresponds to a relatively low CTD, the airport node has no relationship with those of other large available cruise space, and low interaction airport nodes; hence, the node degree in layer A is zero. among aircraft. When the cruise fight route is not Some nodes in the terminal area are distant from adjacent congested, the aircraft speed is generally overly high. nodes according to the distribution law of trafc fow in the Moreover, the ACS and AAV in the free-state are terminal area in an actual airspace. Teir mutual infuence considerably high. can be ignored and represented as isolated nodes in this (2) Smooth Phase corresponds to a relatively high CTD layer. In a part of the terminal area, mainly the terminal area with considerable reductions in ACS and AAV corresponding to the airport group, nodes are clustered in compared with the free phase owing to high fow a small range; they are represented as circular nodes con- density. At this stage, the trafc fow at the air route nected to each other in pairs. Accordingly, the following can interchange signifcantly increases, and the ACS and be assumed: n � <k> �1, n � <k> � 1, n � 3, and 12 12 21 21 22 AAV of the aircraft decrease to a certain extent. As <k> � 2. shown in Figure 13, the CTD, ACS, and AAV all A route node network of a layer is formed by connecting fuctuate within a certain range over time but within adjacent nodes. If N � 30, a random route node network in the normal operating range. which n � 6 and <k> � 4.47 is formed, as shown in 33 33 (3) Congested Phase in this case difers from that in Figure 14. the AS and TCA. Here, the CTD does not rise but One-to-one correspondence does not exist between falls when the ACA is crowded. Tis is because if the nodes of the route and terminal area. A terminal area the intersection of air routes is congested, the node is connected to at least one route node, and a route aircraft is directed to circle around or wait; node can be connected to multiple terminal area nodes. In consequently, both the ACS and AAV drop to the simulated random network, n �1, n � 4, <k> �1, 23 32 23 a low value. In the congested phase, owing to the and <k> � 2.67. large airspace occupied by circling or waiting aircraft, the number of aircraft cruising across the 4.2.NodeStateDensity(i,s,r)-Time(T). Let’s assume that the intersection of air routes decreases. Furthermore, initial delay occurs in the terminal area (i.e., i (0) � 0.01) and the CTD decreases compared with that in the that the propagation probability of crowded nodes (λ) is 0.5. smooth phase. Te changes in the crowded, free, and removed state den- Regarding the ITCPN model presented in Section 2, the sities of the nodes at each layer over time are shown in free and smooth phases are defned as free states (suscep- Figure 15. tible states), and the congested phase is defned as a con- Figure 15 indicates that if the propagation probability of gested state (infected state). Te ACA can change from a crowded state among nodes is the same, the ACA node of a relatively distinct congested phase to a smooth phase in layer C is closely related to each node owing to its high average which the ACS and AAV rise back to a relatively average degree of nodes. Te congested state density of the ACA node level, whereas the CTD returns to a smooth phase. Tis may increases with time at the highest rate, followed by the TCA be considered as the end of the congested state, which is node; the AS node had the lowest propagation rate. Under this defned as the removed state. condition, the basic regeneration number, R , is 3.21. 0 CTD (aircraf/30 min) 14 Journal of Advanced Transportation 010 20 30 40 CTD (aircraf/30 min) (a) (b) 1200 25 510 15 20 25 010 20 30 40 AAV (°/min) CTD (aircraf/30 min) (c) (d) Figure 12: FD in typical intersection of air routes in China: (a) FD; (b) CTD-ACS relationship; (c) AAV-ACS relationship; (d) CTD-AAV relationship. 35 17 35 880 30 16 30 25 15 25 820 20 14 20 15 13 15 760 Congested Congested 10 12 10 Free Free Smooth Smooth Smooth Smooth 5 11 5 700 0 10 0 06:00 08:00 10:00 12:00 14:00 16:00 18:00 20:00 22:00 00:00 06:00 08:00 10:00 12:00 14:00 16:00 18:00 20:00 22:00 00:00 CTD CTD AAV ACS (a) (b) Figure 13: Temporal density-speed diagram of trafc fow in typical intersection of air routes in China on 12/15/2019: (a) CTD-ACS-AAV- time relationship and (b) DTD-ADS-time relationship. AAV (°/min) CTD (aircraf/30 min) ACS (km/h) ACS (km/h) AAV (°/min) CTD (aircraf/30 min) AAV (°/min) ACS (km/h) ACS (km/h) Journal of Advanced Transportation 15 Figure 14: Simulated network confguration of layer C. 1.0 0.8 0.6 0.4 0.2 0.0 0 2 4 6 8 10 12 14 16 18 20 iA sA rA iB sB rB iC sC rC Figure 15: Node state density i, s, r-time T with assumption; N � N � 80, N � 30, n � <k> �1, n � <k> � 1, n � 3, <k> � 2, A B C 12 12 21 21 22 22 n � 6, <k> � 4.47, n �1, n � 4, <k> �1, <k> � 2.67. 33 33 23 32 23 32 In a real aviation network, the congestion propagation As shown above, the cross-infection rates (λ , λ and 12 21 rate among ACA nodes is considerably less than that of TCA λ , λ ) have the same infuence on R . Tis shows that 23 32 0 nodes. Tis is because the trafc fow in the route can be the degree of infuence of congestion propagation be- alleviated by speed regulation and the adjustment of the tween the AS and TCA is the same in both directions. Tis horizontal and longitudinal spacings among aircraft. means that the probability of congestion in the TCA resulting from the congestion in the AS and the proba- bility of fight congestion in the AS resulting from the 4.3. Infection Rate (λ) and Crowded State Density (i). All congestion in the TCA have the same infuence on the infection rates in the network are set as 0.5, 0.4, and 0.3, and large-area trafc fow congestion throughout the system. the recovery rate is assumed to be 1. With a decrease in In this network model, an increase in λ can considerably infection rate, as shown in Figure 16, the peak value of the increase R because of the large average degree of layer congestion rate decreases, and the time point of the peak C. Tis can be explained by the complex ACA distri- value subsequently shifts. bution; the trafc fow congestion propagation rate in the ACA has a more signifcant impact on the large-scale congestion of the entire system. However, trafc con- 4.4. Infection Rate (λ) and Fundamental Reproduction gestion in actual operations can be alleviated though Number(R ). Te efect of infection rate on the fundamental various means, such as rerouting, regulating speed, and reproduction number (R ) considering diferent network converting the altitude level. Tis can lead to a low nu- structures is investigated. As presented in Figure 17, when merical value of the system (λ ) such that the congestion one of the infection rates changes, the other rates are fxed propagation in the network decreases. at 0.1. Density 16 Journal of Advanced Transportation 0.30 0.25 0.20 0.15 0.10 0.05 0.00 0 2 4 6 8 10 12 14 16 18 20 iA λ=0.5 iA λ=0.4 iA λ=0.3 iB λ=0.5 iB λ=0.4 iB λ=0.3 iC λ=0.5 iC λ=0.4 iC λ=0.3 Figure 16: Congested density-infection rate relationship. 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 0.2 0.4 0.6 0.8 1.0 λ λ 22 23 λ λ 12 32 λ λ 21 33 Figure 17: Congested density-infection rate relationship. corresponding names are listed in Table 3. To explore the 5. Empirical Results: Case Study in Central and congestion propagation process of air trafc in a large South-Eastern China space based on the multilayer network framework and analytical metrics, the fight trajectory data on a typical 5.1. Data Description. Te central and south-eastern parts of busy day in December 2019 were collected. Te time China have 12 airports. Because the air networks are densely resolution of the trajectory data is 15 s. Te sample tra- distributed, the trafc operational scope of the 12 airports, jectory data for the TCA and ACA are shown in Figure 18. including 6 of the top 15 airports in China, is selected in An MTFN integrated with TAAN and FTN is established terms of fight volume. Tese airports include the Shanghai according to the data analysis method presented in Sec- Hongqiao, Shanghai Pudong, Hangzhou Xiaoshan, Nanjing tion 3. Te node distribution and MTFN are shown in Lukou, Wuhan Tianhe, and Zhengzhou Xinzheng airports Figures 19 and 20, respectively. and fve medium-sized airports. Te nodes number and Density Journal of Advanced Transportation 17 Table 3: Node number and corresponding name. AS no. Airport identifer TCA no. Approach control 1 PVG 1 Shanghai 2 SHA 2 Hangzhou 3 WUX 3 Ningbo 4 NTG 4 Nanjing 5 HGH 5 Hefei 6 NGB 6 Wuhan 7 NKG 7 Zhengzhou 8 CZX 9 XUZ 10 HFE 11 WUH 12 CGO 35 35 34 34 33 33 32 32 31 31 30 30 29 29 112 114 116 118 120 122 112 114 116 118 120 122 Longitude (°E) Longitude (°E) (a) (b) 112 114 116 118 120 122 Longitude (°E) (c) Figure 18: Trajectory data in central and south-eastern China for one typical busy day: (a) trajectory data of (a) ASs, (b) TCAs, and (c) ACAs. 5.2. Model Verifcation. Using the data analytical metrics and gray grids represent the free, congested, and removed described in Section 3, aircraft taxiing, takeof, and landing states, respectively. as well as the fight trajectory data for a day are analyzed. A comparison of the crowded node density between the Moreover, the set time slot is every 30 min. Te states of each empirical and simulation data is shown in Figure 22. In the node in the ITCPN are established according to real data simulation, the TCA2 node (i.e., the node whose degree changes over time, as shown in Figure 21. Te light, dark, values in layer B connecting A, B, and C are 1, 3, and 1, Latitude (°N) Latitude (°N) Latitude (°N) 18 Journal of Advanced Transportation 35 35 34 34 33 33 32 8 4 32 5 31 31 30 30 29 29 112 114 116 118 120 122 112 114 116 118 120 122 Longitude (°E) Longitude (°E) (a) (b) 7 4 112 114 116 118 120 122 Longitude (°E) (c) Figure 19: MTFN inner layer node distribution: layers (a) A-AS. (b) B-TCA, and (c) C-ACA. 10 9 12 6 11 5 Layer A- AS Layer B- TCA Layer C- ACA Figure 20: MTFN-integrated TAAN and FTN. Latitude (°N) Latitude (°N) Latitude (°N) Journal of Advanced Transportation 19 Timeslot 6:00 6:30 7:00 7:30 8:00 8:30 9:00 9:30 10:00 10:30 11:00 11:30 12:00 12:30 13:00 13:30 14:00 14:30 15:00 15:30 16:00 16:30 17:00 17:30 18:00 18:30 19:00 19:30 20:00 20:30 21:00 21:30 22:00 22:30 23:00 23:30 Node State AS1 AS2 AS3 AS4 AS5 AS6 AS7 AS8 AS9 AS10 AS11 AS12 TCA1 TCA2 TCA3 TCA4 TCA5 TCA6 TCA7 ACA1 ACA2 ACA3 ACA4 ACA5 ACA6 ACA7 ACA8 ACA9 ACA10 ACA11 Figure 21: Node state of ITCPN based on empirical data. 0.4 0.3 0.2 0.1 0.0 15:00 16:00 17:00 18:00 19:00 20:00 21:00 22:00 23:00 ρA iA ρB iB ρC iC Figure 22: State density of node i based on simulation results (square) and empirical data (triangle). respectively) is initially in a crowded state. We assume that infuence of the congestion propagation between two ad- other nodes are in free-state at initial and there is only one jacent terminals is the highest. In contrast, λ , λ , and λ 12 21 23 trafc congestion cause. Te propagation probability values are relatively low, indicating that the possibility of trafc fow of the crowded state among the nodes are set as λ � 0.2, congestion at the AS and intersection trafc congestion in air λ � 0.1, λ � 0.8, λ � 0.2, λ � 0.4, and λ � 0.4; the re- routes caused by terminal area congestion is low. In the ACA 21 22 23 32 33 covery rates are set as μ � 0.3, μ � 0.3, and μ � 0.7. Te layer network, the nodes are connected by air routes, in- 1 2 3 maximum value of λ in the simulation indicates that the cluding long air routes that can bufer and absorb the Density 20 Journal of Advanced Transportation congested trafc fow at the intersection. Accordingly, the Tis study presents a comparison of the crowded node recovery rate of ACA nodes, μ , considerably exceeds those density based on empirical and simulation data in Section 5. of μ and μ . Results show that the congestion propagation between two 1 2 adjacent terminals has the highest infuence, while the 6. Conclusions possibility of congestion in the AS and intersection trafc congestion in air routes caused by terminal area congestion Te study of the state of complex ATF in the range of is low. Tis underscores the pivotal role played by the in- ground and air operations is essential to understand the terplay of trafc management within airport terminal areas nature of the interaction between the ATF, airport, and in instigating and exacerbating air trafc congestion. By airspace and to reveal the technical potential of advancing prioritizing the management of trafc fow within airport ATM. Among the various analytical approaches, network terminal areas, it is possible to mitigate the adverse con- analysis is an efective and intuitive method for examining sequences of large-scale trafc congestion. the behavior of complex systems with varying granular- In future research, testing the generality of the fndings ities. Te objective is to depict the characteristics of the and verifying the key characteristics of more complex avi- operational status of ATF and explain the internal ation networks and airport layouts are necessary. Tis model mechanism of trafc fow congestion state propagation in is useful for aviation network layout design, congestion diferent ground and air operation spaces. Accordingly, prediction, and stability evaluation. a comprehensive multilayer network is constructed to integrate and present factors, such as airports, terminal Nomenclature areas, air route networks, aircraft trajectories, trafc fow space states, and congestion state propagation laws. To ACA: Area-controlled airspace characterize the congestion propagation among diferent ACC: Area control center ATF regions further, an ITCFN is established. Te fol- AS: Airport surface lowing are implemented: ATC: Air trafc controller (1) Te node state parameters and crowded state ATCS: Air trafc control system ATF: Air trafc fow propagation parameters in the ITCPN network are ATM: Air trafc management modeled and analyzed. FTN: Flight trajectory network (2) Te AS and terminal area of Shanghai Pudong ITCPN: Interrelated trafc congestion propagation Airport and the typical air route crossing nodes network in central China are considered as examples. MTFN: Multilayer trafc fow network Te proposed network node state and analysis SIR: Susceptible-infectious-removed infectious disease index, including trafc fow dynamics and three models fow stages (free, congested, and removed), are TAAN: Topology-based airport-airspace network analyzed. TCA: Terminal controlled airspace. (3) Empirical analysis is performed on the fight data of 12 airports and 7 terminal areas and route networks Data Availability in central and south-eastern China on a typical busy day to verify the accuracy of the ITCPN network Te data used to support the fndings of this study are model parameters. available from the corresponding author upon request. Tis study uses general methods to study the fight data in some regions of China empirically. Te efectiveness of Conflicts of Interest using string-coupled three-layer network model to describe Te authors declare that they have no conficts of interest. congestion at the convergence of air trafc fows both on ground and in the airspace is verifed. Acknowledgments In the numerical simulation of Section 4, the study further reveals that cross-infection rates have the same in- Tis research was supported by the National Natural Science fuence on R , suggesting that the degree of infuence of Foundation of China (Grant no. 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Ground-Air Traffic Congestion Propagation Model Based on Hierarchical Control Interdependent Network

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10.1155/2023/4602148
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Hindawi Journal of Advanced Transportation Volume 2023, Article ID 4602148, 22 pages https://doi.org/10.1155/2023/4602148 Research Article Ground-Air Traffic Congestion Propagation Model Based on Hierarchical Control Interdependent Network Furong Jiang , Zhaoning Zhang, and Xiaoxu Dai Civil Aviation University of China, Tianjin 300300, China Correspondence should be addressed to Furong Jiang; frjiang@cauc.edu.cn Received 6 March 2023; Revised 11 April 2023; Accepted 27 April 2023; Published 5 May 2023 Academic Editor: Tao Liu Copyright © 2023 Furong Jiang et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A multilayer network approach to model and analyze air trafc networks is proposed. Tese networks are viewed as complex systems with interactions between airports, airspaces, procedures, and air trafc fows (ATFs). A topology-based airport-airspace network and a fight trajectory network are developed to represent critical physical and operational characteristics. A multilayer trafc fow network and an interrelated trafc congestion propagation network are also formulated to represent the ATF connection and congestion propagation dynamics, respectively. Furthermore, a set of analytical metrics, including those of airport surface (AS), terminal controlled airspace (TCA), and area-controlled airspace (ACA), is introduced and applied to a case study in central and south-eastern China. Te empirical results show the existence of a fundamental diagram of the airport, terminal, and intersections of air routes. Moreover, the dynamics and underlying mechanisms of congestion propagation through the AS- TCA-ACA network are revealed and interpreted using the classical susceptible-infectious-removed model in a hierarchical network. Finally, a high propagation probability among adjacent terminals and a high recovery probability are identifed at the network system level. Tis study provides analytical tools for comprehending the complex interactions among air trafc systems and identifes future developments and automation of layered coupled air trafc management systems. improve air trafc system resilience and alleviate air trafc 1. Introduction congestion to a considerable extent. Trafc fow is a ubiquitous dynamic process in human so- Diferent from urban trafc congestion, which is gen- ciety. Air trafc, which difers from most trafc variations, is erated by the confict between the vehicles driven by each strictly monitored by air trafc management (ATM). Air independent decision-maker (i.e., driver) and the limited trafc congestion is a scientifc problem restricting the road trafc resources, air trafc fow (ATF) congestion is development of civil aviation and fight safety. a combination of the air trafc control system (ATCS) and Te ATM system’s operational supervision initiatives the airline operation system in a limited space environment. are the determining factors for solving air trafc con- Terefore, as shown in Figure 1, ATF congestion generally occurs in a spatiotemporal environment with limited trafc gestion problems. To cope with increasing air trafc de- mand and congestion, the global ATM data management operation resources (such as the airport surface (AS) (in- system is being upgraded and transformed from sub- cluding aprons, taxiways, and runways), terminal controlled regional management to unifed management at the na- airspace (TCA), and area-controlled airspace (ACA) with tional level. By 2021, China has implemented a national route convergence). fow management system that plays the role of a brain In recent decades, several studies on ATF modeling have center in the operation and management of civil aviation been conducted. As an important location for the outfow air trafc. Te promotion and implementation of ATM and infow of air trafc, the AS operation has various system planning, such as the Single European Sky ATM constraints, such as release time, available apron, and Research (SESAR) and NextGen, are anticipated to runway assignment, which cause trafc congestion. Studies 2 Journal of Advanced Transportation high-altitude air routes must remain safely spaced and or- derly while fying. Bilimoria and Lee proposed a spatial clustering method based on the relative distance of aircraft and identifed the air trafc status by establishing the congestion index of “Gaggle Density” [16]. Lee et al. eval- ACA uated and identifed the degree of congestion caused by aircraft entering the control sector with diferent headings using the minimum course change parameter of aircraft collision avoidance [17]. Various recent studies highlight the importance of considering network modeling to study the propagation of trafc congestion and fight delays [18–21]. Lin et al. propose TCA a new fight delay model to capture the impact of en-route congestion on fight delays over an air trafc network [22]. Wang et al. provide an algorithm, which using sliding correlation windows to extract the airport pairs with stable AS delay lags, to estimate statistically signifcant time lags be- tween airport delays from noisy, aggregate operational data [23]. Wu et al. developed an Airport-Sector Network Delays Figure 1: Hierarchical airspace structure. model which takes both airports and airspace capacities into account [24]. on the trafc congestion on ASs have mainly focused on the Other studies have focused on airspace sector capacity assessments [25–27] and aviation network modeling based queuing problem connecting the apron, taxiway, and run- way [1–3]; fight delays [4–7]; and taxiing management on complex networks [28–34]. Most studies that reveal the topology and dynamic behavior of air trafc network [8–10]. Yang et al. discussed the congestion of departing trafc fow but ignored the infuence of arriving trafc fow congestion are based on scale-free small-world network modeling in which airports and fights are represented by [11]. Two distinct phases are involved, namely, free-fow and congested phases, which are represented by a density- nodes and edges, respectively. However, this modeling speed-fow relationship [12]. When the AS congestion in- method only considers airport nodes; it ignores the root tensifes and continues for some time, trafc congestion cause of congestion in actual air trafc network fows. Te spreads to the airspace upstream of the trafc fow, causing entire process of civil aviation fight operation is under the congestion in the terminal area. Few studies have system- monitoring and command of the ATCS, which controls the atically examined the characteristics of trafc fow conges- AS, TCA, and ACA. A review of literature indicates that existing studies focus tion on ASs, especially concerning simultaneous taxiing after landing and before takeof. on the modeling and evaluation of air trafc systems without fully comprehending the inherent airspace fow dynamics Congestion in the TCA is another key manifestation of air trafc dynamics. Te trafc fow congestion propagation and rules that govern their coevolution. Moreover, reports within the TCA has motivated the formulation of several on the congestion propagation of air trafc at the universal ATF control models in recent years to predict the overall network system level are limited. impact on air trafc delays and support fow management in Te air trafc network fow system refers to the for- the TCA of hub airports. Inspired by the fundamental di- mation of an air trafc network among airports through air agram of trafc fow [13, 14], Zhang et al. proposed a cell routes. Te network and movement of aircraft in this net- transmission model-based terminal airspace fow model work generates ATF; the two comprises the air trafc net- with an assumed “fow density-velocity” relationship [15]. work fow system. When fow input is absent, that is, no aircraft fies in the airspace, the air trafc network is a static More detailed trafc fow phase transitions (free-fow, smooth, semistable, and congested) were identifed by Yang trafc network with only spatial characteristics. In contrast, when aircraft is fying in the airport and air route, the non- et al. [12]. In contrast to AS trafc congestion, which can be analyzed using a two-dimensional model, TCA congestion is negative fow endows the air trafc network with certain a time-dependent change process in a three-dimensional dynamic characteristics. With the support of the network space. Hence, a strong correlation between the TCA and AS transmission of fow, the operation of the air trafc network congestions is observed. Accordingly, most studies do not fow system also has certain dynamic characteristics. distinguish the mechanism of congestion between these two. Moreover, the quantitative characterization of the com- However, from the perspective of trafc fow, ground fow plexity of air trafc systems from the perspective of trafc fow and airspace interactions is lacking. To conduct such and ATF have diferent motion states, fow environments, and restrictive conditions. Terefore, the AS trafc fow and characterization at both the macroscopic and microscopic levels, a novel multilayer network approach is proposed. Te arrival/departure trafc fows in the TCA can be analyzed separately in terms of their motion states. multilayer network consists of two physical networks, namely, a topology-based airport-airspace network (TAAN) Another critical area of air trafc congestion is the ACA. In this space, aircraft operating in medium-altitude and and a fight trajectory network (FTN) and two hierarchical Journal of Advanced Transportation 3 networks, namely, a multilayer trafc fow network (MTFN) network modeling method can simplify the actual route and an interrelated trafc congestion propagation network network and abstract the complex route network distribu- tion into a simple network with node and edge connections. (ITCPN). To support such a multilayer network framework and reveal the internal ATF node congestion propagation For the airspace around and above an airport, the trafc dynamics, diferent classes of analytical metrics are de- fow movement space is divided into AS, TCA, and ACA veloped as follows: trafc fow metrics of (i) AS, (ii) TCA, according to the trafc fow rule and control range of the (iii) ACA, (iv) trafc congestion metrics, and (v) hierarchical ATCS. Te adjacent nodes are also connected with edges, as congestion propagation metrics. Te main contribution of shown in Figure 4. this study is the formulation of the multilayer coupled congestion propagation network model. It also contributes 2.2. FTN. Consider a Tianjin-Shanghai fight fow as an the following fndings obtained through the verifcation of example. When an aircraft departs from Tianjin Airport and the theoretical and empirical data of a representative region climbs to a certain altitude, the departure procedure ends, in China. Te content and fow of this study are shown in and the TCA ATC (Air Trafc Controller) transfers the fight Figure 2. to the ACC ATC. After climbing out of the ZBTJ (Tianjin) (1) Empirical fow diagram (FD) of AS, TCA, and ACA: TCA, the aircraft passes through ZBAA (Beijing) ACC, ZSJN At the level of the air trafc network based on MTFN, (Jinan) ACC, and ZSSS (Shanghai) ACC and fnally ap- AS, TCA, and ACA are used as network nodes for the proaches and lands through ZSSS (Shanghai) TCA. Te frst time, and a bidirectional fundamental FD based route passing the ACCs encounters three route interchanges, on empirical data is created. Tree distinct fow states which can be represented as ACA nodes, as shown in were identifed for the nodes of the three levels, Figure 5. Accordingly, the airspace traversed by the entire namely, free, smooth, and congested. fight process is represented in the form of nodes to simplify the movement path of ATF. (2) Time slot-based congestion propagation network: At the regional level, the layered coupled congestion propagation network model is verifed by analyzing 2.3. MTFN. To defne an MTFN, TAAN and FTN as well as the trafc fow data of the AS, TCA, and ACA on three types of nodes (AS, TCA, and ACA) are combined. a typical busy day. Te introduction should be Considering the coupling network after the fusion of the succinct, with no subheadings. Limited fgures may TAAN and FTN, an MTFN was established according to the be included only if they are truly introductory and trafc fow motion connectivity. Subnetwork A represents contain no new results. the AS layer, and the AS nodes are not connected. Sub- network B represents the TCA layer; if the terminals are 2. Multilayer Network for Congestion close to or contain each other, the adjacent TCA nodes are connected by edges. Subnetwork C represents the ACA Propagation Modeling layer. According to the sector distribution of low/high- In region-based operations, the complex ATF dynamics altitude airspaces, adjacent ACA nodes are connected by between AS, TCA, and ACA and the operational status edges. As shown in Figure 6, the nodes among the layers are changes in AS, TCA, and ACA arise from the interactions linked according to the connection relationship of between the regional airway distribution, air trafc, and trafc fow. ATCS. To analyze the trafc fow dynamics of AS-TCA comprehensively, a novel analytical framework based on 2.4.ITCPN. Te nodes in Figure 6 correspond to ASs, TCAs, a multilayer network is established. Te framework consists and ACAs, signifying the primary zones where air trafc of two physical networks, namely, TAAN and FTN, and two converges and becomes concentrated. When the volume of hierarchical networks, namely, MTFN and ITCPN. Tese air trafc within these zones reaches high levels, congestion networks are elaborated in the following sections. emerges. Connected air trafc convergence areas tend to facilitate congestion spread across multiple areas in a cas- cade efect. Tis congestion propagation follows a pattern 2.1. TAAN. Conventionally, airspace networks are modeled as graphs with waypoints as nodes and route segments similar to the transmission of an epidemic among in- dividuals residing in proximity. Consequently, the infectious (links) as edges. However, as presented in this section, a topology-based airspace network is constructed, utilizing disease model is an apt approach to illuminating the the dense interchanges of air routes in the airspace and transmission of air trafc congestion. surrounding airspace as airspace nodes to form the ACA In studies of trafc congestion propagation mechanisms, network. As shown in Figure 3, the nodes in the ACA infectious disease models such as susceptible-infectious- network represent major air route interchanges or air route susceptible (SIS), susceptible-infectious-removed (SIR), and interchanges formed by multiple interchanges. Te con- susceptible-exposed-infectious-removed (SEIR) are com- monly employed [5, 7, 35]. It is crucial to note that disease nectivity in the ACA network refects the adjacency of air routes between interchanges and dense air routes. High- propagation is irreversible, i.e., epidemiology spread is a unidirectional difusion that mirrors congestion propa- altitude air trafc congestion typically occurs during route interchanges. Terefore, the topology-based airspace gation. For this reason, traditional infectious disease models 4 Journal of Advanced Transportation Empirical Data Network Modeling Analytical Metrics Empirical Results AS Traffic Flow Metrics Airspace Airway-based Air traffic flow Configuration Airspace Network TCA Traffic Flow dynamic at network system level Metrics Fuse Flight Trajectory ACA Traffic Flow Flight Network Metrics Schedule Congestion Propagation Data- Driven Traffic Congestion Traffic Congestion Traffic Congestion Metrics propagation Propagation Radar mechanism at Network Multi-layer Trajectory SIR Model network system level Congestion Propagation Metrics Figure 2: Framework of proposed network-based air trafc congestion propagation analysis. ZBTJ ZSSS AS AS ZBTJ ZSSS TCA TCA ACA 1 ACA 3 Figure 3: Topology-based ACA network. ACA 2 Figure 5: FTN of Tianjin-Shanghai fight fow. MTFN to describe trafc congestion propagation both be- ACA tween nodes of each layer and nodes of diferent layers. Furthermore, trafc node states can be classifed into ACA free, congested, and removed states which correspond to susceptible, infected, and removed states in SIR model, TCA respectively. In Section 3, defnitions of free, congested, and removed states will be investigated based on node type, TCA connecting distribution patterns of nodes within, and AS among distinct layers. Te connection relationship between the nodes AS and layers in the subnetwork is constructed based on the TAAN and FTN. Te defnitions of parameters are shown in Table 1. To represent the probability that node A in subnet A Figure 4: Interlayer connection structure. is randomly selected to have j edges connected to B, P (j) is defned. Similarly, P (i, j, k) represents the probability that are deemed appropriate for explaining air trafc congestion a node is randomly selected in subnet B with i edges con- propagation caused by single source of congestion. nected to A, j edges to B, and k edges to C. Te probability of Nonetheless, congestion propagation between nodes is randomly selecting a node in subnet C that has j edges solely associated with node degree and congestion trans- connected to B and k edges connected to C is represented by mission rate within traditional infectious disease models, P (j, k). with no diferentiation between node types or direct con- According to the foregoing parameter defnitions, the gestion transmission rates of various node types. Te string- total number of nodes in the three subnetworks and the coupled three-layer network model [28] overcomes this number of nodes in the crowded, free, and removed states shortcoming in the SIR model and can be implemented in can be expressed as follows: MTFN Journal of Advanced Transportation 5 AS 13 AS 14 AS 12 AS 15 TCA 10 TCA 11 TCA 12 Layer A- AS Layer B- AS 11 ACA 8 TCA Layer C- TCA 9 ACA AS 10 TCA 8 ACA 9 AS 9 ACA 10 ACA 7 ACA 11 TCA 6 TCA 7 AS 8 ACA 2 AS 7 ACA 1 ACA 6 ACA 3 TCA 1 AS 1 ACA 5 TCA 2 AS 2 ACA 4 TCA 3 TCA 4 AS 3 TCA 5 AS 5 AS 4 AS 6 Figure 6: MTFN example. Table 1: Defnition of ITCPN parameters. Parameter Meaning N Number of nodes in layer A with degree (0, j) A A A I (or S or R ) Number of infected (or susceptible or removed) nodes in layer A with degree (0, j) j j j N Number of nodes in layer B with degree (i, j, k) i,j,k B B B I (or S or R ) Number of infected (or susceptible or removed) nodes in layer B with degree (i, j, k) i,j,k i,j,k i,j,k N Number of nodes in layer C with degree (j, k) j,k C C C I (or S or R ) Number of infected (or susceptible or removed) nodes in layer C with degree (j, k) j,k j,k j,k n Maximum degree value of nodes in layer A connecting to layer B n (or n or n ) Maximum degree value of nodes in layer B connecting to layer A (or B or C) 21 22 23 n (or n ) Maximum degree value of nodes in layer C connecting to layer B (or C) 32 33 ⟨k⟩ Average degree value of nodes in layer A connecting to layer B ⟨k⟩ (or ⟨k⟩ or ⟨k⟩ ) Average degree value of nodes in layer B connecting to layer A (or B or C) 21 22 23 ⟨k⟩ (or ⟨k⟩ ) Average degree value of nodes in layer C connecting to layer B (or C) 32 33 6 Journal of Advanced Transportation A(orBorC) A(orBorC) A(orBor C) A(orBorC) Table 2: Equations of ITCPN parameters. ⎧ ⎪ N � I + S + R , ⎪ n 12 Parameter Value A A A A A A I 􏼐or S or R 􏼑 � 􏽘 I 􏼐or S or R 􏼑, ⎪ A A j j j ⎪ P (j) N /N ⎪ A j ⎪ j�0 B B P (i, j, k) N /N n n n B i,j,k 21 22 23 (1) B B B B B B I 􏼐or S or R 􏼑 � 􏽘 􏽘 􏽘 I 􏼐or S or R 􏼑, C C i,j,k i,j,k i,j,k P (j, k) N /N C j,k i�0 j�0 ⎪ k�0 n n ⎪ 22 23 P (i, ·, ·) 􏽐 􏽐 P (i, j, k) ⎪ B j�0 B n n k�0 32 33 C C C C C C n n 21 23 I 􏼐or S or R 􏼑 � 􏽘 􏽘 I 􏼐or S or R 􏼑. P (·, j, ·) 􏽐 􏽐 P (i, j, k) ⎪ j,k j,k j,k B B i�0 k�0 j�0 k�0 n n 21 22 P (·, ·, k) 􏽐 􏽐 P (i, j, k) B i�0 j�0 B Te joint degree distribution, marginal degree distri- P (j, ·) 􏽐 P (j, k) C C k�0 bution, average degree values (φ �1), and second moment of P (·, k) 􏽐 P (j, k) degree (φ � 2) are calculated; the results are summarized in C j�0 C Table 2. φ 12 φ ⟨k ⟩ 􏽐 j P (·, j) 12 j�0 A Te probability that the AS node of layer A is trans- φ 21 φ ⟨k ⟩ 􏽐 i P (i, ·, ·) 21 i�0 B formed into a congested state by the infuence of the TCA φ 22 φ node of layer B (connected layer) is λ . Te probabilities that 21 ⟨k ⟩ 􏽐 j P (·, j, ·) 22 j�0 B the TCA node of layer B is transformed into a congested φ 23 φ ⟨k ⟩ 􏽐 k P (·, ·, k) 23 B k�0 state by the infuence of the AS node of layer A, TCA node of φ φ layer B, and ACA node of layer C (connected layers) are λ , ⟨k ⟩ 􏽐 j P (j, ·) 12 32 j�0 C λ , and λ , respectively. Te probabilities that the ACA 22 32 φ 33 φ ⟨k ⟩ 􏽐 k P (·, k) 33 C k�0 node in layer C is afected by the TCA node in layer B and ACA node in layer C in the congested state are λ and λ , 23 33 C C C C N , i � I /N are similarly defned. According to the respectively. In subnetworks A, B, and C, the transition ι,κ ι,κ ι,κ ι,κ previous assumption, we build a networked mean-feld probabilities of the congested state nodes to the removed spreading model, which is composed of [(n + 1) state are μ , μ , and μ , respectively. 1 2 3 12 +(n + 1)(n + 1)(n + 1) + (n + 1)(n + 1)]-dimensio Based on the foregoing assumptions, the SIR density 21 22 23 32 33 A A A A A A nal ordinary diferential equations as follows: model of ITCPN is established. Let s � S /N , i � I /N , ε ε ε ε ε ε which represent the corresponding densities. Te other B B B B B C B C densities s � S /N , i � I /N , s � S / ϕ,c,η ϕ,c,η ϕ,c,η ϕ,c,η ϕ,c,η ϕ,c,η ι,κ ι,κ ⎪ δs (τ) ⎪ ε � −λ εs (τ)Θ (τ), ⎪ 21 21 δτ δi (τ) ⎪ ε A A � λ εs (τ)Θ (τ) − μ i (τ), 21 ε 21 1 ε δτ δs (τ) ϕ,c,η Β Β Β � −λ ϕs (τ)Θ (τ) − λ cs (τ)Θ (τ) − λ ηs (τ)Θ (τ), ⎪ 12 ϕ,c,η 12 22 ϕ,c,η 22 32 ϕ,c,η 32 δτ (2) ⎪ δi (τ) ⎪ ϕ,c,η Β Β Β Β ⎪ � λ ϕs (τ)Θ (τ) + λ cs (τ)Θ (τ) + λ ηs (τ)Θ (τ) − μ i (τ), 12 ϕ,c,η 12 22 ϕ,c,η 22 32 ϕ,c,η 32 2 ϕ,c,η δτ ⎪ δs (τ) ι,κ C C � −λ ιs (τ)Θ (τ) − λ κs (τ)Θ (τ), ⎪ 23 23 33 33 ι,κ ι,κ ⎪ δτ δi (τ) ι,κ C C C � λ ιs (τ)Θ (τ) + λ κs (τ)Θ (τ) − μ i (τ), 23 ι,κ 23 33 ι,κ 33 3 ι,κ δτ where ε � 0, 1, ..., n ; ϕ � 0, 1, ..., n ; c � 0, 1, .., n ; η � 0, 1, In the traditional SIR model with average degree value 12 21 22 ..., n ; ι � 0, 1, ..., n ; κ � 0, 1, ..., n . equals ⟨k⟩, Θ (τ) is the probability that any adjacent contact 23 32 33 To simplify the model further, the coupled network is can be afected to transform into a crowded state. It can be assumed to be degree-independent, that is, the degree of any expressed as the average joint probability of connecting from node is independent of the degrees of its neighboring nodes. a node of degree α to a node of degree β, P(β | α)i (τ), β β Journal of Advanced Transportation 7 which is abbreviated as Θ(τ) � 􏽐 P(β | α)i (τ) � 1/ Te classical SIR model was used to analyze the prop- β β ⟨k⟩􏽐 kP(k)i (τ). agation of the crowded state of the node in the ITCPN. Te k k basic regeneration number, R In the string-coupled three-layer network model, the , is an important parameter formulas are expressed as follows: for revealing the propagation characteristics. In this study, R represents the number of nodes converted into a crowded n 0 ⎧ ⎪ A state; the conversion is due to the infuence of one congested ⎪ Θ (τ) � 􏽘 jP (j)i (τ), 12 A ⎪ j ⟨k⟩ ⎪ 12 node propagation in all the free-state network nodes. Te j�0 ⎪ fundamental reproduction number, R , can be calculated −1 ⎪ n n n using the next generation matrix, Γ � FV [35], where F is 21 22 23 ⎪ Θ (τ) � 􏽘 􏽘 􏽘 iP (i, j, k)i (τ), the rate of new occurring congestion, and V is the rate of 21 B ⎪ i,j,k ⟨k⟩ ⎪ 21 i�0 j�0 k�0 transferring congested nodes from the free nodes group. ⎪ Matrix Γ can be rewritten using similar transformations as n n n 21 22 23 follows: Θ (τ) � jP (i, j, k)i (τ), ⎪ 􏽘 􏽘 􏽘 ⎪ 22 B i,j,k ⟨k⟩ i�0 j�0 k�0 (3) n n n 21 22 23 ⎪ Θ (τ) � 􏽘 􏽘 􏽘 kP (i, j, k)i (τ), ⎪ 23 B i,j,k ⟨k⟩ ⎪ 23 i�0 j�0 k�0 ⎪ n n 32 33 ⎪ Θ (τ) � 􏽘 􏽘 jP (j, k)i (τ), 32 C ⎪ j,k ⟨k⟩ j�0 ⎪ k�0 ⎪ n n 32 33 ⎪ 1 Θ (τ) � 􏽘 􏽘 kP (j, k)i (τ). ⎩ 33 C j,k ⟨k⟩ j�0 k�0 λ ⟨k ⟩ λ λ 21 21 21 21 ⎡ ⎢ 0 ⟨k⟩ ⟨k⟩ 0 0 ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 22 23 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ μ ⟨k⟩ μ μ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 2 21 2 2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ λ ⟨k ⟩ ⎥ ⎢ ⎥ ⎢ 12 12 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 0 0 0 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ μ ⟨k⟩ ⎥ ⎢ ⎥ ⎢ 2 12 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ λ λ ⟨k ⟩ λ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 22 22 22 22 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 ⟨k⟩ ⟨k⟩ 0 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 21 23 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ μ μ ⟨k⟩ μ ⎥ ⎢ ⎥ ⎢ 2 2 22 2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ Γ � ⎢ ⎥. (4) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ λ ⟨k ⟩ λ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 32 32 32 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 0 0 0 ⟨k⟩ ⎥ ⎢ ⎥ ⎢ 33 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ μ ⟨k⟩ μ ⎥ ⎢ ⎥ ⎢ 3 32 3 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ λ λ λ ⟨k ⟩ ⎥ ⎢ ⎥ ⎢ 23 23 23 23 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 ⟨k⟩ ⟨k⟩ 0 0 ⎥ ⎢ ⎥ ⎢ 21 22 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ μ μ μ ⟨k⟩ ⎥ ⎢ ⎥ ⎢ 2 2 2 23 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ λ λ ⟨k ⟩ ⎦ 33 33 33 0 0 0 0 ⟨k⟩ μ μ ⟨k⟩ 3 3 33 Te fundamental reproduction number of the model is operating network, and region-level interrelated trafc � ρ (Γ), where ρ (Γ) is the spectral radius of matrix Γ. congestion propagation characteristics are thoroughly in- vestigated using diferent network representations of such 2.5. Summary. Te primary aim of this study is to explore complex systems. Te MTFN is a three-layer network representing the the complex dynamics of ATF and the operating states of airports and airspaces through which trafc fows; empirical connection between the AS, TCA, and ACA. As described in Section 2.3, to model the MTFN, generating the cross-layer and comprehensive approaches are used. To achieve this objective, the ATF trajectory, AS-TCA-ACA air trafc linkage between the TAAN and FTN is crucial. Tis 8 Journal of Advanced Transportation identifes the aircraft fying along each airspace sector and We set the length of the time window presented in derives the trafc fow metrics presented in Section 3. An Sections 3 to 30 min. Te nonlinear relationships between ITCPN is developed based on the integrated TAAN-FTN PretTTD, PoldTTD, normalized PretTS, and normalized and the propagation law of the node congestion state. Te PoldTS are shown in Figure 7(a). For a detailed analysis of propagation parameters of the node congestion state in the the AS taxiing trafc fow dynamics, the three phases are three-layer AS-TCA-ACA network are represented based on divided by dissecting the fundamental diagram (Figures 7(b) the characteristics of the classical SIR model in the layered and 7(d)) and temporal-spatial diagrams (Figure 8). coupled network. Tis provides a mathematical model for (1) Free phase corresponds to relatively low PretTTD simulating the parameters and verifying empirical data. and PoldTTD, adequate taxiway availability, and low interaction among aircraft. Generally, the entry of 3. Analytical Metrics aircraft into the parking stand is not restricted if no fight follows after its landing. Consequently, the 3.1. AS Trafc Flow Metrics. An FD that showed the phase PoldTS in the free-state is typically higher than the transition of trafc fow on a road section by describing the PretTS in the same state. Tis is because departing one-to-one relationship among the fundamental parameters fights have more limiting factors, such as fight time, of trafc fow was proposed by Daganzo [13]. Considering runway, and airspace. that the dynamic monitoring of AS trafc fow mainly in- cludes the taxiing trafc fow of aircraft about to take-of and (2) Smooth phase corresponds to relatively high has landed, the following are defned: PretTTD and PoldTTD with a distinct reduction in both PretTS and PoldTS compared with the free (1) Take-of Trafc Flow, Q . Number of aircraft taking To phase owing to the high fow density. At this stage, of from the airport within a specifed time. the trafc fow in the AS signifcantly increases, the (2) Landing Trafc Flow, Q . Number of aircraft Ld average pre-takeof taxiing speed of the aircraft landing at airports within a specifed time. considerably decreases, and the waiting time for (3) Pre-takeof Taxiing Trafc Density (PretTTD), ρ . takeof increases, although the operation is within To-t Total number of aircraft taxiing before take-of the normal range. within a specifed time. (3) Congested phase corresponds to the drop in both (4) Postlanding Taxiing Trafc Density (PoldTTD), ρ . the PretTTD and PretTS as PoldTTD and PoldTS Ld-t Total number of aircraft taxiing after landing within continue to increase as a consequence of the a specifed time. higher priority of aircraft landing than of take-of. At this stage, the postlanding trafc fow moves to (5) Average Pre-takeof Taxiing Speed (PretTS), V . Te To-t the AS at high density and velocity. Meanwhile, average taxiing speed of all aircraft ready to take-of the pretake-of trafc fow is heavy because nu- at the airport in the taxiing state within a specifed merous aircraft are in the waiting state owing to time; this can be regarded as the average velocity of resource limitations. Moreover, the density and the entire trafc fow in the pre-takeof taxiing state. velocity are both low. (6) Average Postlanding Taxiing Speed (PoldTS), V . Ld-t Te average taxiing speed of all aircraft in the taxiing As for the ITCPN model presented in Section 2, both the state after landing at the airport within a specifed free and smooth phases are defned as a free state (sus- time; this can be regarded as the average velocity of ceptible state), and the congested phase is defned as the entire trafc fow in the postlanding taxiing state. a congested state (infected state). When the AS changes from a relatively distinct congested phase to a smooth phase, Airports are a major source of initial delays in fight. Te both the PretTTD and PretTS rise back to a relatively high operation for taxiing on the ground and fying in air can be level, whereas PoldTTD and PoldTS drop to an average level. regarded as a complete closed-loop trafc fow. Te trafc Tis can be considered as the end of the congested state and fow within the AS mainly includes fights taxiing on the defned as the removed state. ground after landing and fights preparing to take-of after leaving the parking stand. Te congestion of AS mainly occurs in the movement 3.2. TCA Trafc Flow Metrics. Similar to the approach of range of aircraft on the ground, such as aprons, taxis, and defning parameters in the AS node, the spatial range is runways. Te fight congestion caused by the limited resources defned from an altitude of 1000 m to the upper limit of the of the airport is represented by fight departure delay. Con- airspace of the terminal area (generally 6000 m). Moreover, sidering that the taxiing route and taxiing time in diferent the latitude and longitude boundaries of the horizontal airports cannot be directly compared because of the diferent range of the terminal area are considered as nodes. To study confgurations of runways, taxiways, and parking stands, the the motion state of trafc fow in TCA nodes, the following taxiing speed must be normalized. If the total taxiing process is are defned. 1 and the taxiing time is x (in minutes), the average taxiing speed after normalization is 1/x. Te length of the time slot is (1) Departure Trafc Density (DTD), ρ . Number of Dpt 30 min. Figure 7 shows ρ , ρ , and the nonlinear re- aircraft departing from the terminal area within To-t Ld-t lationship between the normalized V and V . a specifed time. To-t Ld-t PretTTD (aircraf/30mins) Journal of Advanced Transportation 9 Frequency 1.0 Congested 0.9 0.8 20 0.7 0.6 Smooth 0.5 0.4 0.3 0.2 Free 0.1 22 0 5 10 15 20 25 14 PretTTD (aircraf/30 min) Normalized PretTS Normalized PoldTS (a) (b) 0.35 0.35 1.4 1.4 0.30 0.30 1.2 1.2 0.25 0.25 1.0 1.0 0.20 0.20 0.8 0.8 0.15 0.15 0.6 0.6 0.10 0.10 0.4 0.4 0.05 0.05 0.2 0.2 0.00 0.00 0.0 0.0 0 5 10 15 20 25 0 5 10 15 20 25 PretTTD (aircraf/30 min) PoldTTD (aircraf/30 min) Normalized PretTS Normalized PretTS Normalized PoldTS Normalized PoldTS (c) (d) Figure 7: FD in pudong AS: (a) FD; (b) PretTTD-PoldTTD-frequency relationship; (c) PretTTD-PretTS-PoldTS relationship; (d) PoldTTD-PretTS-PoldTS relationship. 0.30 0.20 0.25 0.16 Congested 0.20 0.12 Congested 0.15 0.08 0.10 5 0.04 0.05 Smooth Free Smooth 0 0.00 0 0.00 06:00 08:00 10:00 12:00 14:00 16:00 18:00 20:00 22:00 00:00 06:00 08:00 10:00 12:00 14:00 16:00 18:00 20:00 22:00 00:00 PretTTD PretTTD PoldTTD Normalized PretTS Normalized PretTS Normalized PoldTS (a) (b) Figure 8: Continued. PoldTTD (aircraf/30mins) PretTTD/PoldTTD (aircraf/30 min) Normalized PretTS Normalized PretTS/PoldTS Normalized PoldTS PretTTD (aircraf/30 min) Normalized PretTS PoldTTD (aircraf/30 min) Normalized PoldTS Normalized PretTS 10 Journal of Advanced Transportation 0.28 0.24 Congested 0.20 0.16 0.12 0.08 0.04 Smooth Smooth Free 0 0.00 06:00 08:00 10:00 12:00 14:00 16:00 18:00 20:00 22:00 00:00 PoldTTD Normalized PoldTS (c) Figure 8: Temporal density-speed diagram of taxiing trafc fow in AS of pudong on 12/15/2019: (a) PretTTD-PoldTTD-PretTS-PoldTS- time relationship; (b) PretTTD-PretTS-time relationship; (c) PoldTTD-PoldTS-time relationship. (2) Approach Trafc Density (ATD), ρ . Number of that in Section 3.1, the three phases were divided by scru- Arv aircraft approaching the terminal area within tinizing the fundamental diagram (Figures 9(b)– 9(d)) and a specifed time. temporal-spatial diagrams (Figure 10). (3) Average Departure Speed (ADS), V . Te average Dpt (1) Free Phase corresponds to relatively low DTD and departure speed of all departure fights in the ter- ATD, large available space, and negligible interaction minal area within a specifed time can be regarded as among aircraft. Te departure speed of an aircraft is the average departure speed of the entire departure generally extremely high if no congestion occurs in trafc fow. the subsequent route after takeof; hence, the ADS in the free-state is generally higher than the AAS in the (4) Average Approach Speed (AAS), V . Te average Arv approach speed of all departure fights in the ter- same state. Tis is because the approach procedure has more restrictive conditions than the departure minal area within a specifed time can be regarded as the average velocity of the entire approach procedure in the terminal area. trafc fow. (2) Smooth Phase corresponds to a relatively high DTD and ATD with a distinct reduction in ADS compared Te terminal area is the main section in which increased with the free phase owing to the high fow density. At fight congestion occurs. Owing to the reduction in airspace this stage, the trafc fow of the terminal airspace capacity caused by the weather, convergence of waypoints, signifcantly increases, and the AAS of the aircraft trafc congestion in the air section, and other reasons, the considerably decreases. Moreover, the ADS fuctu- trafc fow in this area may turn into a crowded state. ates within a certain range, although it is within the Previous studies have shown that terminal trafc fow has normal operation range. the basic properties of a fuid, and the trafc capacity be- tween approach and departure has the property of a convex (3) Congested Phase corresponds to a drop in the AAS function [36]. as the ATD continues to increase. At this stage, the Owing to the diferent fight paths of the approach and departing trafc fow moves at a relatively high departure procedures in the terminal area, the approach and density and speed in the terminal area. Furthermore, departure processes cannot be directly compared. Terefore, the approach trafc fow has numerous aircraft in the trafc fow speed in the terminal area is assumed to be waiting states at low speed owing to resource limitations. normalized regardless of the diference in the basic time of approach and departure caused by fight procedure difer- For the ITCPN model presented in Section 2, the free ences. In the terminal area, if the total fight process is 1 and and smooth phases are defned as a free state (susceptible the approach (departure) time is x (in minutes), the average state), and the congested phase as a congested state (infected normalized approach (departure) speed is 1/x. Te length of state). Te TCA may change from a relatively distinct the time slot is 30 min. Figure 9 shows ρ , ρ , and the Dpt Arv congested phase to a smooth phase in which the ATD drops nonlinear relationship between the normalized V Dpt to a normal level and the AAS rises back to a relatively and V . Arv average level, whereas the DTD and ADS return to a smooth Figure 9(a) shows the nonlinear relationship between the phase. Tis can be considered as the end of the congested DTD, ATD, normalized ADS, and normalized AAS. For state; this is defned as the removed state. a detailed analysis of TCA trafc fow dynamics, similar to PoldTTD (aircraf/30 min) Normalized PoldTS DTD (aircraf/30 mins) Journal of Advanced Transportation 11 Frequency Congested Smooth Free 10 25 20 0 0 5 1015202530 30 0 DTD (aircraf/30 min) Normalized ADS Normalized AAS (a) (b) 0.20 0.20 0.20 0.20 Free 0.15 0.15 0.15 0.15 0.10 0.10 0.10 0.10 0.05 0.05 0.05 0.05 0.00 0.00 0.00 0.00 0 5 1015202530 0 5 10 15 20 25 DTD (aircraf/30 min) ATD (aircraf/30 min) Normalized ADS Normalized ADS Normalized AAS Normalized AAS (c) (d) Figure 9: FD in Shanghai TCA. (a) Fundamental diagram; (b) DTD-ATD-frequency relationship; (c) DTD-ADS-AAS relationship; (d) ATD-ADS-AAS relationship. 40 40 0.14 0.12 30 0.12 0.10 20 0.10 Congested 0.08 Free Smooth Congested Smooth 10 0.08 0.06 0 0.06 06:00 08:00 10:00 12:00 14:00 16:00 18:00 20:00 22:00 00:00 06:00 08:00 10:00 12:00 14:00 16:00 18:00 20:00 22:00 00:00 DTD DTD ATD Normalized ADS Normalized ADS Normalized AAS (a) (b) Figure 10: Continued. 0.20 0.15 0.10 0.05 ATD (aircraf/30 mins) Normalized ADS DTD/ATD (aircraf/30 min) Normalized AAS Normalized ADS/AAS DTD (aircraf/30 min) Normalized ADS ATD (aircraf/30 min) Normalized ADS Normalized AAS 12 Journal of Advanced Transportation 40 0.070 30 0.065 20 0.060 Free Congested Smooth Smooth 10 0.055 0 0.050 06:00 08:00 10:00 12:00 14:00 16:00 18:00 20:00 22:00 00:00 ATD Normalized AAS (c) Figure 10: Temporal density-speed diagram of trafc fow in Shanghai TCA on 15/12/2019: (a) DTD-ATD-ADS-AAS-time relationship; (b) DT-ADS-time relationship; (c) ATD-AAS-time relationship. 3.3. ACA Trafc Flow Metrics. Te studies that have been 32.6 conducted on congestion propagation in the ACA are limited. In general, if congestion occurs at the TCA and lasts 32.4 for a long time, congestion also occurs at the upstream ACA. Tis can be manifested by the decrease in the horizontal or 32.2 vertical interval of the aircraft to a minimum, adjustment in the altitude or speed of the aircraft, aircraft circling, and 32.0 waiting at the designated waypoint, etc. In the high-altitude airspace, ATF congestion mainly occurs at the air route α 31.8 interchange. Te area where multiple air routes cross is prone to a shortage of airspace resources, leading to con- 31.6 gestion. To simplify the representation of the ACA nodes, as shown in Figure 11, the intersection where the main route 31.4 crosses is selected as the ACA node. To study the motion state of the trafc fow in ACA nodes, the following are defned: 31.2 (1) Cruising Trafc Density (CTD), ρ R . Number of C us 31.0 aircraft cruising across the ACA node area within 116.6 116.8 117.0 117.2 117.4 117.6 117.8 118.0 a specifed of time. Longitude (°E) (2) Average Cruising Speed (ACS), V R . Te average C us Figure 11: Trajectory data of a typical intersection of air routes in cruising speed of all cruising fights across the ACA China. node area within a specifed time can be regarded as the average cruising speed of the entire cruising Te motion state of trafc fow at the ACA node is trafc fow. studied using the analysis method of FD in the AS and TCA. (3) Average Angular Velocity (AAV), ω R . Te average C us Because the cruising trafc fows fying into and out of ACA angular velocity of all cruising fights across the ACA nodes have varying angles, fight distances, and fight paths, node area within a specifed time can be regarded as the status of all cruising fights passing through ACA nodes the average angular velocity of the entire cruising cannot be directly compared. Terefore, the geometric trafc fow. center where the ACA node trafc fow converges is selected In high-altitude airspaces, the air route intersection is the as the center of the circle. Te length that can contain the main area of increasing fight congestion. Owing to the main convergence area is considered as the radius. Te reduction in airspace capacity caused by public navigation circular area and its center (orange), which is set as the ACA stations and the limited range of available airspace, the trafc node, are shown in Figure 11. Te cruise trafc fow velocity fow in this area may enter a crowded state. of ACA nodes is assumed to be normalized ignoring the time ATD (aircraf/30 min) Latitude (°N) Normalized AAS Journal of Advanced Transportation 13 diference between entering and leaving the ACA node 4. Numerical Simulation Analysis area owing to fight path variations. In the ACA node A congestion propagation simulation analysis is imple- area, if the total fight process is 1, and the time from mented based on the ITCPN model presented in Section 2 entering the node airspace range to leaving this range is x and the analysis of the phase transition of the AS-TCA-ACA min, the average normalized cruise speed is 1/x. If the node discussed in Section 3. Te following equation rep- angle of the total fight process is α and the time from resents the average crowded state density of subnetworks A, entering the node airspace range to leaving this range is x B, and C at time t: (in minutes), the average normalized angular velocity is α/x. Te length of the time slot is 15 min. Te nonlinear n n n n 12 21 22 23 A A B relationship between ρ , V , and ω are shown in i (t) � 􏽘 P (j) · i (t), i (t) � 􏽘 􏽘 􏽘 P (i, j, k) Crus Crus Crus A j B Figure 12. j�0 i�0 j�0 k�0 (5) Figure 12(a) shows the relationship between the CTD, n n 32 33 B C C ACS, and AAV. Te fgure indicates that the AAV is · i (t), i (t) � 􏽘 􏽘 P (j, k) · i (t). i,j,k j,k j�0 fundamentally proportional to the ACS, whereas the CTD k�0 is negatively correlated with the relationship between the two. Sections 3.1 and 3.2 elaborate the analysis of ACA trafc fow dynamics. Te three phases are divided by 4.1. Fundamental Parameter Characteristics of ITCPN Model. scrutinizing the fundamental diagram (Figures 12(b) and Set N � N � 80. Tis indicates that the number of airports A B 12(d)) and temporal-spatial diagram (Figure 13). is equal to the number of terminal areas, ignoring the case in which adjacent airports share terminal areas. Te status of (1) Free Phase corresponds to a relatively low CTD, the airport node has no relationship with those of other large available cruise space, and low interaction airport nodes; hence, the node degree in layer A is zero. among aircraft. When the cruise fight route is not Some nodes in the terminal area are distant from adjacent congested, the aircraft speed is generally overly high. nodes according to the distribution law of trafc fow in the Moreover, the ACS and AAV in the free-state are terminal area in an actual airspace. Teir mutual infuence considerably high. can be ignored and represented as isolated nodes in this (2) Smooth Phase corresponds to a relatively high CTD layer. In a part of the terminal area, mainly the terminal area with considerable reductions in ACS and AAV corresponding to the airport group, nodes are clustered in compared with the free phase owing to high fow a small range; they are represented as circular nodes con- density. At this stage, the trafc fow at the air route nected to each other in pairs. Accordingly, the following can interchange signifcantly increases, and the ACS and be assumed: n � <k> �1, n � <k> � 1, n � 3, and 12 12 21 21 22 AAV of the aircraft decrease to a certain extent. As <k> � 2. shown in Figure 13, the CTD, ACS, and AAV all A route node network of a layer is formed by connecting fuctuate within a certain range over time but within adjacent nodes. If N � 30, a random route node network in the normal operating range. which n � 6 and <k> � 4.47 is formed, as shown in 33 33 (3) Congested Phase in this case difers from that in Figure 14. the AS and TCA. Here, the CTD does not rise but One-to-one correspondence does not exist between falls when the ACA is crowded. Tis is because if the nodes of the route and terminal area. A terminal area the intersection of air routes is congested, the node is connected to at least one route node, and a route aircraft is directed to circle around or wait; node can be connected to multiple terminal area nodes. In consequently, both the ACS and AAV drop to the simulated random network, n �1, n � 4, <k> �1, 23 32 23 a low value. In the congested phase, owing to the and <k> � 2.67. large airspace occupied by circling or waiting aircraft, the number of aircraft cruising across the 4.2.NodeStateDensity(i,s,r)-Time(T). Let’s assume that the intersection of air routes decreases. Furthermore, initial delay occurs in the terminal area (i.e., i (0) � 0.01) and the CTD decreases compared with that in the that the propagation probability of crowded nodes (λ) is 0.5. smooth phase. Te changes in the crowded, free, and removed state den- Regarding the ITCPN model presented in Section 2, the sities of the nodes at each layer over time are shown in free and smooth phases are defned as free states (suscep- Figure 15. tible states), and the congested phase is defned as a con- Figure 15 indicates that if the propagation probability of gested state (infected state). Te ACA can change from a crowded state among nodes is the same, the ACA node of a relatively distinct congested phase to a smooth phase in layer C is closely related to each node owing to its high average which the ACS and AAV rise back to a relatively average degree of nodes. Te congested state density of the ACA node level, whereas the CTD returns to a smooth phase. Tis may increases with time at the highest rate, followed by the TCA be considered as the end of the congested state, which is node; the AS node had the lowest propagation rate. Under this defned as the removed state. condition, the basic regeneration number, R , is 3.21. 0 CTD (aircraf/30 min) 14 Journal of Advanced Transportation 010 20 30 40 CTD (aircraf/30 min) (a) (b) 1200 25 510 15 20 25 010 20 30 40 AAV (°/min) CTD (aircraf/30 min) (c) (d) Figure 12: FD in typical intersection of air routes in China: (a) FD; (b) CTD-ACS relationship; (c) AAV-ACS relationship; (d) CTD-AAV relationship. 35 17 35 880 30 16 30 25 15 25 820 20 14 20 15 13 15 760 Congested Congested 10 12 10 Free Free Smooth Smooth Smooth Smooth 5 11 5 700 0 10 0 06:00 08:00 10:00 12:00 14:00 16:00 18:00 20:00 22:00 00:00 06:00 08:00 10:00 12:00 14:00 16:00 18:00 20:00 22:00 00:00 CTD CTD AAV ACS (a) (b) Figure 13: Temporal density-speed diagram of trafc fow in typical intersection of air routes in China on 12/15/2019: (a) CTD-ACS-AAV- time relationship and (b) DTD-ADS-time relationship. AAV (°/min) CTD (aircraf/30 min) ACS (km/h) ACS (km/h) AAV (°/min) CTD (aircraf/30 min) AAV (°/min) ACS (km/h) ACS (km/h) Journal of Advanced Transportation 15 Figure 14: Simulated network confguration of layer C. 1.0 0.8 0.6 0.4 0.2 0.0 0 2 4 6 8 10 12 14 16 18 20 iA sA rA iB sB rB iC sC rC Figure 15: Node state density i, s, r-time T with assumption; N � N � 80, N � 30, n � <k> �1, n � <k> � 1, n � 3, <k> � 2, A B C 12 12 21 21 22 22 n � 6, <k> � 4.47, n �1, n � 4, <k> �1, <k> � 2.67. 33 33 23 32 23 32 In a real aviation network, the congestion propagation As shown above, the cross-infection rates (λ , λ and 12 21 rate among ACA nodes is considerably less than that of TCA λ , λ ) have the same infuence on R . Tis shows that 23 32 0 nodes. Tis is because the trafc fow in the route can be the degree of infuence of congestion propagation be- alleviated by speed regulation and the adjustment of the tween the AS and TCA is the same in both directions. Tis horizontal and longitudinal spacings among aircraft. means that the probability of congestion in the TCA resulting from the congestion in the AS and the proba- bility of fight congestion in the AS resulting from the 4.3. Infection Rate (λ) and Crowded State Density (i). All congestion in the TCA have the same infuence on the infection rates in the network are set as 0.5, 0.4, and 0.3, and large-area trafc fow congestion throughout the system. the recovery rate is assumed to be 1. With a decrease in In this network model, an increase in λ can considerably infection rate, as shown in Figure 16, the peak value of the increase R because of the large average degree of layer congestion rate decreases, and the time point of the peak C. Tis can be explained by the complex ACA distri- value subsequently shifts. bution; the trafc fow congestion propagation rate in the ACA has a more signifcant impact on the large-scale congestion of the entire system. However, trafc con- 4.4. Infection Rate (λ) and Fundamental Reproduction gestion in actual operations can be alleviated though Number(R ). Te efect of infection rate on the fundamental various means, such as rerouting, regulating speed, and reproduction number (R ) considering diferent network converting the altitude level. Tis can lead to a low nu- structures is investigated. As presented in Figure 17, when merical value of the system (λ ) such that the congestion one of the infection rates changes, the other rates are fxed propagation in the network decreases. at 0.1. Density 16 Journal of Advanced Transportation 0.30 0.25 0.20 0.15 0.10 0.05 0.00 0 2 4 6 8 10 12 14 16 18 20 iA λ=0.5 iA λ=0.4 iA λ=0.3 iB λ=0.5 iB λ=0.4 iB λ=0.3 iC λ=0.5 iC λ=0.4 iC λ=0.3 Figure 16: Congested density-infection rate relationship. 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 0.2 0.4 0.6 0.8 1.0 λ λ 22 23 λ λ 12 32 λ λ 21 33 Figure 17: Congested density-infection rate relationship. corresponding names are listed in Table 3. To explore the 5. Empirical Results: Case Study in Central and congestion propagation process of air trafc in a large South-Eastern China space based on the multilayer network framework and analytical metrics, the fight trajectory data on a typical 5.1. Data Description. Te central and south-eastern parts of busy day in December 2019 were collected. Te time China have 12 airports. Because the air networks are densely resolution of the trajectory data is 15 s. Te sample tra- distributed, the trafc operational scope of the 12 airports, jectory data for the TCA and ACA are shown in Figure 18. including 6 of the top 15 airports in China, is selected in An MTFN integrated with TAAN and FTN is established terms of fight volume. Tese airports include the Shanghai according to the data analysis method presented in Sec- Hongqiao, Shanghai Pudong, Hangzhou Xiaoshan, Nanjing tion 3. Te node distribution and MTFN are shown in Lukou, Wuhan Tianhe, and Zhengzhou Xinzheng airports Figures 19 and 20, respectively. and fve medium-sized airports. Te nodes number and Density Journal of Advanced Transportation 17 Table 3: Node number and corresponding name. AS no. Airport identifer TCA no. Approach control 1 PVG 1 Shanghai 2 SHA 2 Hangzhou 3 WUX 3 Ningbo 4 NTG 4 Nanjing 5 HGH 5 Hefei 6 NGB 6 Wuhan 7 NKG 7 Zhengzhou 8 CZX 9 XUZ 10 HFE 11 WUH 12 CGO 35 35 34 34 33 33 32 32 31 31 30 30 29 29 112 114 116 118 120 122 112 114 116 118 120 122 Longitude (°E) Longitude (°E) (a) (b) 112 114 116 118 120 122 Longitude (°E) (c) Figure 18: Trajectory data in central and south-eastern China for one typical busy day: (a) trajectory data of (a) ASs, (b) TCAs, and (c) ACAs. 5.2. Model Verifcation. Using the data analytical metrics and gray grids represent the free, congested, and removed described in Section 3, aircraft taxiing, takeof, and landing states, respectively. as well as the fight trajectory data for a day are analyzed. A comparison of the crowded node density between the Moreover, the set time slot is every 30 min. Te states of each empirical and simulation data is shown in Figure 22. In the node in the ITCPN are established according to real data simulation, the TCA2 node (i.e., the node whose degree changes over time, as shown in Figure 21. Te light, dark, values in layer B connecting A, B, and C are 1, 3, and 1, Latitude (°N) Latitude (°N) Latitude (°N) 18 Journal of Advanced Transportation 35 35 34 34 33 33 32 8 4 32 5 31 31 30 30 29 29 112 114 116 118 120 122 112 114 116 118 120 122 Longitude (°E) Longitude (°E) (a) (b) 7 4 112 114 116 118 120 122 Longitude (°E) (c) Figure 19: MTFN inner layer node distribution: layers (a) A-AS. (b) B-TCA, and (c) C-ACA. 10 9 12 6 11 5 Layer A- AS Layer B- TCA Layer C- ACA Figure 20: MTFN-integrated TAAN and FTN. Latitude (°N) Latitude (°N) Latitude (°N) Journal of Advanced Transportation 19 Timeslot 6:00 6:30 7:00 7:30 8:00 8:30 9:00 9:30 10:00 10:30 11:00 11:30 12:00 12:30 13:00 13:30 14:00 14:30 15:00 15:30 16:00 16:30 17:00 17:30 18:00 18:30 19:00 19:30 20:00 20:30 21:00 21:30 22:00 22:30 23:00 23:30 Node State AS1 AS2 AS3 AS4 AS5 AS6 AS7 AS8 AS9 AS10 AS11 AS12 TCA1 TCA2 TCA3 TCA4 TCA5 TCA6 TCA7 ACA1 ACA2 ACA3 ACA4 ACA5 ACA6 ACA7 ACA8 ACA9 ACA10 ACA11 Figure 21: Node state of ITCPN based on empirical data. 0.4 0.3 0.2 0.1 0.0 15:00 16:00 17:00 18:00 19:00 20:00 21:00 22:00 23:00 ρA iA ρB iB ρC iC Figure 22: State density of node i based on simulation results (square) and empirical data (triangle). respectively) is initially in a crowded state. We assume that infuence of the congestion propagation between two ad- other nodes are in free-state at initial and there is only one jacent terminals is the highest. In contrast, λ , λ , and λ 12 21 23 trafc congestion cause. Te propagation probability values are relatively low, indicating that the possibility of trafc fow of the crowded state among the nodes are set as λ � 0.2, congestion at the AS and intersection trafc congestion in air λ � 0.1, λ � 0.8, λ � 0.2, λ � 0.4, and λ � 0.4; the re- routes caused by terminal area congestion is low. In the ACA 21 22 23 32 33 covery rates are set as μ � 0.3, μ � 0.3, and μ � 0.7. Te layer network, the nodes are connected by air routes, in- 1 2 3 maximum value of λ in the simulation indicates that the cluding long air routes that can bufer and absorb the Density 20 Journal of Advanced Transportation congested trafc fow at the intersection. Accordingly, the Tis study presents a comparison of the crowded node recovery rate of ACA nodes, μ , considerably exceeds those density based on empirical and simulation data in Section 5. of μ and μ . Results show that the congestion propagation between two 1 2 adjacent terminals has the highest infuence, while the 6. Conclusions possibility of congestion in the AS and intersection trafc congestion in air routes caused by terminal area congestion Te study of the state of complex ATF in the range of is low. Tis underscores the pivotal role played by the in- ground and air operations is essential to understand the terplay of trafc management within airport terminal areas nature of the interaction between the ATF, airport, and in instigating and exacerbating air trafc congestion. By airspace and to reveal the technical potential of advancing prioritizing the management of trafc fow within airport ATM. Among the various analytical approaches, network terminal areas, it is possible to mitigate the adverse con- analysis is an efective and intuitive method for examining sequences of large-scale trafc congestion. the behavior of complex systems with varying granular- In future research, testing the generality of the fndings ities. Te objective is to depict the characteristics of the and verifying the key characteristics of more complex avi- operational status of ATF and explain the internal ation networks and airport layouts are necessary. Tis model mechanism of trafc fow congestion state propagation in is useful for aviation network layout design, congestion diferent ground and air operation spaces. Accordingly, prediction, and stability evaluation. a comprehensive multilayer network is constructed to integrate and present factors, such as airports, terminal Nomenclature areas, air route networks, aircraft trajectories, trafc fow space states, and congestion state propagation laws. To ACA: Area-controlled airspace characterize the congestion propagation among diferent ACC: Area control center ATF regions further, an ITCFN is established. Te fol- AS: Airport surface lowing are implemented: ATC: Air trafc controller (1) Te node state parameters and crowded state ATCS: Air trafc control system ATF: Air trafc fow propagation parameters in the ITCPN network are ATM: Air trafc management modeled and analyzed. FTN: Flight trajectory network (2) Te AS and terminal area of Shanghai Pudong ITCPN: Interrelated trafc congestion propagation Airport and the typical air route crossing nodes network in central China are considered as examples. MTFN: Multilayer trafc fow network Te proposed network node state and analysis SIR: Susceptible-infectious-removed infectious disease index, including trafc fow dynamics and three models fow stages (free, congested, and removed), are TAAN: Topology-based airport-airspace network analyzed. TCA: Terminal controlled airspace. (3) Empirical analysis is performed on the fight data of 12 airports and 7 terminal areas and route networks Data Availability in central and south-eastern China on a typical busy day to verify the accuracy of the ITCPN network Te data used to support the fndings of this study are model parameters. available from the corresponding author upon request. Tis study uses general methods to study the fight data in some regions of China empirically. Te efectiveness of Conflicts of Interest using string-coupled three-layer network model to describe Te authors declare that they have no conficts of interest. congestion at the convergence of air trafc fows both on ground and in the airspace is verifed. Acknowledgments In the numerical simulation of Section 4, the study further reveals that cross-infection rates have the same in- Tis research was supported by the National Natural Science fuence on R , suggesting that the degree of infuence of Foundation of China (Grant no. U2233209) and the Fun- congestion propagation between the AS and TCA is the damental Research Funds for the Central Universities same in both directions. An increase in λ has a consider- (Grant no. 3122022053). able impact on R due to the complex ACA distribution, specifcally on the trafc fow congestion propagation rate in References the ACA, which has a more signifcant impact on the large- scale congestion of the entire system. Tus, it is imperative to [1] U. Benlic, A. E. I. Brownlee, and E. K. Burke, “Heuristic search devise a more coherent and efective layout of the air trafc for the coupled runway sequencing and taxiway routing network, while implementing more efcient methods of problem,” Transportation Research Part C: Emerging Tech- trafc planning and management, so as to curtail the nologies, vol. 71, pp. 333–355, 2016. widespread ramifcations of air trafc congestion within the [2] K. K. H. Ng, C. K. M. Lee, F. T. S. Chan, and Y. 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Published: May 5, 2023

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