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Operational Reliability of the Airport System: Monitoring and Forecasting
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XIII Conference on Transport Engineering, CIT2018 XIII Conference on Transport Engineering, CIT2018
Operational Operational Reliability Reliability of of the the Airport Airport System: System: Monitoring Monitoring and and Forecasting Forecasting
Álvaro Rodríguez-Sanz a,a, **, Beatriz Rubio Fernández aa, Fernando Gómez Comendador aa, Álvaro Rodríguez-Sanz a , Beatriz Rubio Fernández , Fernando Gómez Comendador , Rosa Arnaldo Valdés a, José Manuel Cordero García bb and Margarita Bagamanova cc Rosa Arnaldo Valdés , José Manuel Cordero García and Margarita Bagamanova b b
a Universidad Politécnica de Madrid (UPM), Plaza Cardenal Cisneros N3, Madrid 28040, Spain. a Universidad Politécnica de Madrid (UPM), Plaza Cardenal Cisneros N3, Madrid 28040,Allende, Spain. Madrid 28022, Spain. CRIDA (Reference Center for Research, Development and Innovation in ATM), Avenida de Aragón 402 Edificio CRIDA (Reference Centerc Universitat for Research, Development and Innovation ATM),deAvenida Aragón 402 08202, EdificioSpain. Allende, Madrid 28022, Spain. Autònoma de Barcelona (UAB), in Carrer EmpriusdeN2, Sabadell c Universitat Autònoma de Barcelona (UAB), Carrer de Emprius N2, Sabadell 08202, Spain.
Abstract Abstract Airports are intermodal nodes that can act as disruption drivers throughout the entire transport system. To ensure the robustness of Airports are intermodal nodesmanagers that can act disruption drivers the entire transport system.and To delve ensureinto theits robustness of the transport network, airport andaspolicy makers need throughout means to assess operational reliability precursors. the transport network, airport managers and policy makers need means to assess operational reliability and delve into its precursors. This paper develops a model to evaluate potential malfunctions of an airport, through the definition of proactive performance This paper The develops potential malfunctions of an airport, through the definition of of proactive indicators. modelaismodel based to onevaluate the Multi-State Systems (MSS) reliability theory, as a natural extension classicalperformance binary-state indicators. The modelpresent is based on the Multi-State Systems reliability theory, as awith natural extension of on classical binary-state evaluation: airports different performance levels (MSS) and several failure modes various effects the entire system evaluation: airports present different performance levels and several failure modes with various effects on the entire system performance (degradation). The operational reliability assessment is achieved with random processes (Markov) methods. The performance (degradation). The operational reliability assessment is achieved with random processes (Markov) methods. The analysis is focused on the airspace-airside integrated infrastructure, using a dynamic spatial boundary associated with the Extended analysis is focused on the airspace-airside integrated infrastructure, using a dynamic spatial boundary associated with the Extended Terminal Maneuvering Area (E-TMA) concept. The study evaluates the ‘visit’ of an aircraft to the E-TMA, which consists of three Terminalsections: Maneuvering Areaapproach; (E-TMA) (b) concept. The study evaluates the initial ‘visit’ of an aircraft to the which consists of threea separate (a) final turnaround process; and (c) climb segment. TheE-TMA, reliability model represents separate sections: (a) final approach; (b) turnaround process; and (c) initial climb segment. The reliability model represents framework to test different ‘what-if’ scenarios and to reduce uncertainty by categorizing different behavior patterns. Therefore, ita framework test different ‘what-if’ scenarios and to to enter reducea degraded uncertainty by categorizing different behaviorthrough patterns.a case Therefore, it allows us totopredict how probable is for the system state. The methodology is validated study at allows us to predict how probable is for the system to enter a degraded state. The methodology is validated through a case study at Madrid Airport (LEMD): a collection of nearly 34,000 aircraft turnarounds is used to statistically determine the system operational Madrid Airport The (LEMD): a collection of 34,000 turnarounds is used to statistically determine the system operational characteristics. main contribution ofnearly this paper is toaircraft provide a mechanism to monitor and forecast the system’s state, as a way characteristics. The main contribution of this paper is to provide a mechanism to monitor and forecast the system’s state, as a way to proactively assess the operational reliability of airports. to proactively assess the operational reliability of airports. © © 2018 2018 The The Authors. Authors. Published Published by by Elsevier Elsevier Ltd. Ltd. © 2018 The Authors. by Elsevier Ltd. This is an open accessPublished article under under the the CC BY-NC-ND BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) This is an and openpeer-review access articleunder underresponsibility the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection of the scientific committee of the XIII Conference on Transport Engineering, Selection and peer-review under responsibility of the scientific committee of the XIII Conference on Transport Engineering, CIT2018. CIT2018. CIT2018. Keywords: Airport operations; Reliability analysis; Multi-state systems; Markov processes; Performance assessment. Keywords: Airport operations; Reliability analysis; Multi-state systems; Markov processes; Performance assessment.
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Corresponding author. Tel.: +34 626574829. Corresponding Tel.: +34 626574829. E-mail address:author. [email protected]. E-mail address: [email protected].
2352-1465 © 2018 The Authors. Published by Elsevier Ltd. 2352-1465 © 2018 Thearticle Authors. Published by Elsevier Ltd. This is an open access under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) This is an and openpeer-review access article under the CC BY-NC-ND licensecommittee (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection under responsibility of the scientific of the XIII Conference on Transport Engineering, Selection and peer-review under responsibility of the scientific committee of the XIII Conference on Transport Engineering, CIT2018. CIT2018. 2352-1465 2018 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection and peer-review under responsibility of the scientific committee of the XIII Conference on Transport Engineering, CIT2018. 10.1016/j.trpro.2018.11.002
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1. Introduction and problem statement An efficient management of airport related operations is a key element when assessing the air transport system performance (Wu, 2012). Mutual interdependencies between airports result in system-wide effects for the air traffic network (Pyrgiotis et al., 2013). Therefore, a change to an air-to-air perspective is necessary, with a specific focus on the aircraft ground operations as major driver for airline operational efficiency. In fact, airport operations (including Air Traffic Flow and Capacity Management -ATFCM- associated issues) were the second highest cause in the share of primary delays (initial delays that are not inherited from the aircraft previous leg operation) (EUROCONTROL, 2017). Currently, wider boundaries are given to the turnaround operational scope: it encompasses the processes occurring at the airport’s Extended Terminal Maneuvering Area (E-TMA) (Rodríguez-Sanz et al., 2018). The concept of Airport Transit View (ATV) describes the “visit” of an aircraft to the airport and connects inbound and outbound flights (SESAR, 2014). It is divided into three sections: the final approach and inbound ground section of the arrival flight, the turnaround process section in which the inbound and the outbound flights are linked and the outbound ground section and the initial climb segment of the departure flight. Reliability engineering is the discipline devoted to ensure that a system will be reliable when operated under certain conditions (Lisnianski and Levitin, 2003). Through Multi-State Systems (MSS) reliability models, such as Markov-chains, more realistic and more precise representation of engineering systems are provided (Levitin and Xing, 2017). However, they are much more complex and present major difficulties in system definition and performance evaluation (Gu and Li, 2012). For this study, we have used information about nearly 34,000 airport transit view operations at Adolfo Suárez Madrid Barajas Airport (LEMD) during the months of July and August of 2016. All the information related with the operations at the airport is divided into four blocks: delay, capacity, environment and complexity. Then, we propose a Markov chain model that represents the global state of the airport with respect to the four blocks. Finally, we study some reliability functionalities and performance indicators of Markov-chains to model the system potential degradation and predict its behavior. The main contribution of this paper is to apply the MSS theory to the ATV, including the study of reliability features of Markov-chains. These features determine the status of the system and, through the predictive applications of the proposed model, allow us to forecast the behavior of the system for the next operations. 2. Materials and methods Traditionally, systems have been modelled in a binary way, thus each element has only two possible states: perfect functioning or complete failure (Lisnianski and Levitin, 2003). However, the majority of real systems can be in more than two states. The so-called Multi-State Systems (MSS) are the systems that present a finite number of states. Usually, a multi-state system is composed of elements that can also be in different states (Lisnianski et al., 2010). If a multi-state system is composed of n elements, its performance level is determined in an unambiguous way by the performance levels of the elements that constitute it. At each moment, the system elements have a level of performance that corresponds to their current state. The status of the entire system is determined by the states of its elements. Therefore, the definition of a multi-state model must include the stochastic process performance for each element of the system 𝑖𝑖 : Gi (t)(𝑖𝑖 = 1, … 𝑛𝑛) and the structure of system operation that causes the stochastic process corresponding to the output of the entire multi-state system: 𝐺𝐺(𝑡𝑡) = 𝜑𝜑(𝐺𝐺𝑖𝑖 (𝑡𝑡), … , 𝐺𝐺𝑛𝑛 (𝑡𝑡)). Markov-chains are a special type of discrete stochastic process in which the probability of an event only depends of the previous state of the system. This type of systems are memoryless thus satisfy the Markov property (Lisnianski et al., 2010). The probability of transition between state 𝑋𝑋𝑛𝑛−1 = 𝑖𝑖 and 𝑋𝑋𝑛𝑛 = 𝑗𝑗 is given by 𝛾𝛾𝑖𝑖,𝑗𝑗 , where 𝑛𝑛 is the number of transitions. The matrix 𝑷𝑷 collectively represents these transition probabilities (Sigman and Notes, 2009). The vector 𝜋𝜋𝑛𝑛𝑇𝑇 defines the probability of finding the system in a specific state on the 𝑛𝑛 -th transition. The probabilities of the state for each transition are determined iteratively multiplying the state vector by the transition matrix. The stationary distribution 𝜋𝜋 (steady state or long-term,𝑛𝑛 → ∞), does not change over time; i.e.: 𝛾𝛾1,1 𝛾𝛾2,1 𝜋𝜋 = 𝜋𝜋𝑷𝑷 𝑤𝑤𝑤𝑤𝑤𝑤ℎ 𝑷𝑷 = ( ⋮ 𝛾𝛾𝑘𝑘,1
𝛾𝛾1,2 𝛾𝛾2,2 ⋮ 𝛾𝛾𝑘𝑘,2
⋯ ⋯ ⋯ ⋯
𝛾𝛾1,𝑘𝑘 𝛾𝛾2,𝑘𝑘 ⋮ ) 𝛾𝛾𝑘𝑘,𝑘𝑘
(1)
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3. Results 3.1. State vector and blocks definition To simplify the analysis, the airport model is divided into four blocks: delay, capacity, environment and complexity. A number of parameters or elements, which have different performance rates, defines each block. To avoid an excessive number of states that will complicate the main model definition, the state of each block is defined in a conservative way, i.e., the state of the delay block will be assessed by the most adverse state of the parameters of the block. Once the blocks are defined, the main model states are determined as the number blocks that have failed. 3.1.1. Capacity and complexity The block of capacity relates the number of operations taking place during an hour to the airport capacity. It has two indicators: adaptation and relation. The metric ‘adaptation’ describes the relationship between the total number of operations (arrivals and departures) of each hour and the declared capacity of the airport. The runway “declared” (or practical) capacity is the maximum number of aircraft operations during a given period of time (i.e., one hour), when the average delay imposed on each aircraft movement does not exceed a tolerable level prescribed in advance (Horonjeff et al., 2010). The metric ‘relation’ connects the number of departures with the number of arrivals. It is calculated as the division of the number of departures by the number of arrivals. This block helps us to understand the overall situation of the airport; i.e., its level of workload. Table 1.illustrates the states the three possible states of the parameters. The objective of the complexity block is to consider the complexity of airport operations. Complexity can be represented by parameters such as meteorological conditions or runway configuration. In our case of study, the database has information about operations that were registered in the months of July and August, thus some meteorological conditions (visibility, amount of clouds) are always in excellent conditions and do not provide any relevant information about how meteorological variables affect the airport operational complexity. Future works will be oriented to improve this limitation. The states for the complexity block are summarized in Table 2. Table 1: Capacity block state intervals.
Table 2: Complexity block states intervals.
Adaptation
Relation
Target value
80% < x < 100%
0.1 < x < 1.1
Correct value
x ≤ 80%
0.2
Incorrect values
x ≥ 100%
Other values
Runway configuration
Wind
Normal operation
North (the airport’s preferential configuration)
Head
Complex operation
South
Cross or tail
3.1.2. Delay and environment The delay block is defined by two parameters: off-block time delay to represent the punctuality in departures and in-block time delay to represent punctuality in arrivals. The off-block time delay is the difference between the scheduled off-block time (SOBT) and the actual off-block time (AOBT) (EUROCONTROL et al., 2012). The inblock time is the difference between the scheduled in-block time (SIBT) and the actual in-block time (AIBT) (EUROCONTROL et al., 2012). The three possible states of the parameters of the delay block are illustrated in Table 3. As mentioned before, delay refers to the time difference between the actual time and the schedule time of a timestamp or process. Therefore, delays can be positive or negative, meaning early departure or early arrival of a flight. “Negative” delays (early arrivals) occur when the schedule is running close to plans and can cause issues for airport operations; e.g., disrupting the sequencing of flights and the allocation of resources (gates, handling equipment), especially during peak hours at busy airports (Wu, 2012). “Positive” flight delays often cause significant problems for all the involved stakeholders; e.g., they affect operational and financial performance of airports and airlines, schedule adherence and use of resources, passenger experience and satisfaction, and system reliability (Belobaba et al., 2015; Wu, 2012). Delay thresholds are defined according to operational targets: the ±3 minutes threshold for punctuality set by SESAR’s (target state) (Single European Sky ATM Research) performance metrics
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(Cook et al., 2013) and the 15 minutes threshold for defining delay (correct state) that has historically been common to both Europe and the US (Jetzki, 2009). Table 3: Delay and environment blocks states intervals.
Thresholds
Target time
Correct time
Incorrect time
-3 min < d < 3 min
-15 min < d ≤ -3 min or 3 min ≤ d < 15 min
d ≤ -15min or d ≥ 15 min
The environment block aims to include the metrics that reflect the emissions of particulates and gases (CO2, water vapor, hydrocarbons, carbon monoxide, nitrogen oxides, sulfur oxides, lead and black carbon) nearby and within the airport infrastructure. It can be represented by the extra time in the aircraft processes related to approach and onground operations. That is why this block is modelled by the additional taxi-in time, the additional taxi-out time, the additional Arrival Sequencing and Metering Area (ASMA) 40 NM time and the additional ASMA 60 NM time. The ASMA is defined as a virtual cylinder with a 40 NM or 60 NM radius around the airport (SESAR, 2014). One of the appraised timestamps in airport operations is the additional time the flight spends in ASMA 60 NM. This time is calculated as the difference between the actual time the aircraft needed to pass through this volume of airspace and the ideal time it should have needed, depending on the space congestion situation at that time. Therefore, the additional ASMA time is an indicator for the average arrival runway queuing time on the inbound traffic flow, during congestion periods at airports (SESAR, 2014). Just as the additional ASMA 60 NM time was defined for a 60 NM radius cylinder, a similar indicator can defined for the next volume of airspace the aircraft goes through, which is the ASMA 40 NM. For our case study is particularly important to study both ASMA 40 NM and ASMA 60 NM additional times. This is due to the fact that holdings at Adolfo Suarez Madrid-Barajas (LEMD) are located beyond a radius of 40 NM, as depicted in Fig. 1.
Fig. 1: ASMA 40 NM and ASMA 60 NM for Adolfo Suarez Madrid Barajas-Airport (LEMD).
Additional taxi-in time is the difference between an estimated time to perform taxi-in procedures (depending on the stand and the runway in use) and the actual process duration. Therefore, ‘additional taxi-in time’ will collect the inefficiencies in the airport taxiways due to traffic congestion or potential incidents. Conversely, taxi-in procedures could be carried out in a shorter time than expected (during periods of low inflow traffic), with the operation being able to absorb delays generated in other processes. The additional taxi-out time is appraised likewise. Note that unimpeded taxi times can only be as short as the physics of the process allows but can grow large in the event of a slow taxi operation (Simaiakis and Balakrishnan, 2015). The environment block states are defined in the same way as the delay block states (Table 3).
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3.1.3. Global model states When the contributions of the components to the cumulative system performance rate are different, the number of possible MSS states grows significantly; i.e., different combinations of k available units can provide a large number of performance rates for the entire system (Lisnianski et al., 2010). This is the reason why the global model only considers total failures of the blocks regardless of which block has failed. Otherwise, the number of global states would be unmanageable in practical applications. Therefore, in our study, the number of blocks that have failed measures the global performance of the system; i.e., the global model has a total of five possible states (𝐺𝐺1 , 𝐺𝐺2 , 𝐺𝐺3 , 𝐺𝐺4 and 𝐺𝐺5 ) according to the number of failed blocks. The performance rate of each state is defined as the relative number of blocks that have failed. That way 𝐺𝐺1 has a performance rate of 100%, 𝐺𝐺2 of 75%, 𝐺𝐺3 of 50%, 𝐺𝐺4 of 25% and 𝐺𝐺5 of 0%. 3.2. Markov Chain Once the system’s states are established, the next step is to obtain the transition matrix of the model. A Matlab © program (MathWorks, 2017) is implemented using the data for the almost 34,000 airport transit views at Adolfo Suarez Madrid-Barajas Airport (LEMD). This program calculates the probability of transitioning from one state to another from real operations. Once the transition matrices are obtained, the steady state can be appraised by solving Eq. 2 and knowing that the sum of the components of state vector 𝜋𝜋 must be equal to one. For the model construction and testing, data is divided into two partial datasets. The first one is used to train and validate the model, with 80% of the total number of observations (building and cross-validation sample); while the remaining 20% (test sample) is used to test the generalization of the model. Therefore, the process is as follows: 1) Randomly split the initial dataset into construction/building and testing sets: 80% construction and 20% test. 2) Perform cross-validation on the construction set to fit the model (k-fold with k=10) (Larose, 2005). 3) Test if results are generalizable, using a test set, which is completely separated from model development. Both the train and the error scores presents an average value of 10%, i.e., our model predicts new observations as well as it fits the original dataset. Therefore, we are not overfitting the model and results can be generalizable. Table 4: Global transition matrix. 𝑮𝑮𝟏𝟏
𝑮𝑮𝟐𝟐
𝑮𝑮𝟑𝟑
𝑮𝑮𝟒𝟒
𝑮𝑮𝟓𝟓
𝑮𝑮𝟏𝟏
0,483
0,439
0,074
0,004
0,000
𝑮𝑮𝟐𝟐
0,081
0,491
0,389
0,038
0,001
𝑮𝑮𝟑𝟑
0,006
0,192
0,514
0,280
0,008
𝑮𝑮𝟒𝟒
0,001
0,033
0,489
0,457
0,020
𝑮𝑮𝟓𝟓
0,000
0,022
0,351
0,470
0,157
The transition matrix among the five states of the global model is presented in Table 4. The transition matrix shows that total failures do not last long. The probability of being in state 𝐺𝐺5 (total failure) and staying in that same state is only 15.7%, which means that the system is not supposed to stay in that state for a large number of operations. It is more probable that the system transitions to state 𝐺𝐺4 through a partial repair. Moreover, the system tends to stay in the same state or suffer partial repairs or failures. This can be appraised by analyzing the diagonal of the matrix, which is the most likely set of transition probabilities, except for the case in which the system is in state 𝐺𝐺5 (as mentioned before, when it tends to move to state 𝐺𝐺4 ). Furthermore, total recoveries and total failures are not possible, the probability that the system is in state 𝐺𝐺5 for operation 𝑛𝑛 and in state 𝐺𝐺1 for operation 𝑛𝑛 + 1 is zero. The system neither can go from state 𝐺𝐺1 (perfect functioning) to state 𝐺𝐺5 (total failure) in just one-step. The system’s Markov-chain is ergodic, thus it has a unique stationary distribution (Roger A. Horn and Johnson, 2002). When the system reaches the stationary distribution, the more probable state is 𝐺𝐺3 (45.94%), this means that the system is working with a 50% performance rate or that is working with two failed blocks. Perfect functioning and total failure are very unlikely to happen (4% and 1 % respectively).
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3.3. Functionalities To study the system’s performance, three main indicators are evaluated: mean instantaneous performance, mean instantaneous deficiency and mean instantaneous availability. After that, three scenarios are presented: the first one studies the evolution of the system when the initial state is the perfect functioning (𝐺𝐺1 ), the second one studies the evolution of the system when it starts from total failure (𝐺𝐺5 ) and for the third one, the system starts in an intermediate state (𝐺𝐺3 ), which is the most probable. To characterize the average MSS output performance, we can use the performance expectation. The mean value of MSS instantaneous output performance at operation 𝑛𝑛 (En ) is defined as (Lisnianski and Levitin, 2003): 𝐸𝐸𝑛𝑛 = ∑𝑁𝑁 𝑘𝑘=1 𝑔𝑔𝑘𝑘 𝑝𝑝𝑘𝑘 (𝑛𝑛)
(2)
𝐷𝐷𝑛𝑛 = ∑𝑁𝑁 𝑖𝑖=1 𝑝𝑝𝑖𝑖 (𝑛𝑛) 𝑚𝑚𝑚𝑚𝑚𝑚(𝑤𝑤 − 𝑔𝑔𝑖𝑖 ; 0)
(3)
𝐴𝐴𝑛𝑛 = ∑𝐾𝐾 𝑖𝑖=1 𝑝𝑝𝑖𝑖 (𝑛𝑛)
(4)
Where 𝑁𝑁 is the total number of states, 𝑔𝑔𝑘𝑘 is the performance rate associated with state 𝑘𝑘 and 𝑝𝑝𝑘𝑘 (𝑛𝑛) is the probability that the system is in state 𝑘𝑘 at operation 𝑛𝑛. The mean instantaneous deficiency or deviation (Dn ) is defined as a weighted average between the system probability to be found in each state and the service levels associated to these states. A weighted average of the value of a random variable where the probability function provides weights can be understood as the expected value (Lisnianski and Levitin, 2003). In case the difference is negative, the average is weighted with a zero. That is because in those cases the system is meeting the expected demand and the aim of the index is to assess the cases when the system is not fulfilling the demand.
Where 𝑝𝑝𝑖𝑖 (𝑛𝑛) is the probability that the system is in state 𝑖𝑖 at 𝑛𝑛-th operation, 𝑤𝑤 is the expected demand and 𝑔𝑔𝑖𝑖 is the level of performance associated to state 𝑖𝑖. Another parameter to evaluate the performance of the system is the mean instantaneous availability (An ). It is defined as the sum of the probabilities of the acceptable states (Lisnianski and Levitin, 2003):
Where 𝐾𝐾 is the number of acceptable states and 𝑝𝑝𝑖𝑖 (𝑛𝑛) is the probability that the system is in state 𝑖𝑖 at operation 𝑛𝑛. First, it is necessary to establish the number of acceptable states and the expected level of demand. This study contemplates that an acceptable situation for the airport system includes the states 𝐺𝐺1 , 𝐺𝐺2 and 𝐺𝐺3 . The expected level of demand (𝑤𝑤) is at least 50%. For the stationary distribution, the mean instantaneous availability is 72.65%, the mean instantaneous deficiency is 7.1087 and the mean instantaneous performance is 50.59%. Fig. 2 (a), (b) and (c) show the evolution of the three performance indicators when the system starts at state 𝐺𝐺1 , with initial state vector 𝑝𝑝0 = [1 0 0 0 0]. The system is initially functioning perfectly (𝐸𝐸𝑛𝑛 = 100%, 𝐷𝐷𝑛𝑛 = 0 and 𝐴𝐴𝑛𝑛 = 100%). The system’s performance is degraded until it reaches the stationary distribution. The number of steps spent by the system to reach the stationary distribution is 20 operations (with an error of the estimation of 0.1010%). The second studied case is completely opposite of the previous one. The system is initially in complete failure (𝐺𝐺5 , 𝑝𝑝0 = [0 0 0 0 1]) and improves its performance rate by reaching the stationary distribution in 17 operations (with a maximum error of the estimation of 0.1403%). Fig. 2 (d), (e) and (f) represent this evolution, which is faster than the case before; i.e. the system recovers itself faster than it evolves towards a degraded state. To study the system evolution from a non-extreme initial situation (total failure or perfect functioning), a third case is appraised in which the system is initially in the most probable state (𝐺𝐺3 ). The initial state vector is 𝑝𝑝0 = [0 0 1 0 0] and the system takes 15 operations to reach the stationary distribution (with a maximum error in the estimation of 0.1060%). Now the system requires less time to reach the stationary distribution (only 15 steps) because the initial state vector is similar to the stationary state vector. Fig. 2 (g), (h) and (i) describe the temporary evolution of the three performance indicators for the third case. The evolution can be divided into two stages. In the first one, the system suffers a degradation, arriving at a minimum of mean instantaneous performance (47.6896%) in 2 steps, a minimum
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of mean instantaneous availability (70.57%) in 3 steps and a maximum of mean instantaneous deficiency (7.6340) in 3 steps. Then, the system recovers itself in the second stage of the evolution, arriving at the stationary distribution in 15 steps.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
Fig. 2: Performance indicators: (a) Mean instantaneous availability for case 1, (b) Mean instantaneous deficiency for case 1, (c) Mean instantaneous performance for case 1, (d) Mean instantaneous availability for case 2, (e) Mean instantaneous deficiency for case 2, (f) Mean instantaneous performance for case 2, (g) Mean instantaneous availability for case 3, (h) Mean instantaneous deficiency for case 3, (i) Mean instantaneous performance for case 3.
4. Conclusion This paper develops a reliability analysis of the airport transit view process. In this analysis, we use a multi-state system approach, through a Markov-Chain model. Four main elements of the system are used to appraise its evolution: delay, capacity, environment and complexity. From these partial elements, we evaluate the failure behavior of the system. The model application to a case of study of nearly 34,000 operations shows that the airport system tends to a stationary distribution for a sufficient number of operations. In this stationary distribution, the most probable state is the one with a performance rate of 50%, which is equivalent to the system being working with two failed blocks. The analysis of the values obtained for the performance indicators (mean instantaneous performance, mean instantaneous availability and mean instantaneous deficiency) allows us to represent the current behavior of the
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system’s dynamics. Moreover, we can obtain a prediction of its most likely evolution through the Markov-chain functionalities. Three cases were evaluated to understand the system behavior, with the system starting in different initial states: total failure, perfect functioning and the most probable state. These three situations show a fast evolution (indeed, the system reaches the stationary distribution in less than 5 operations) and prove that the system is repairable, thus it can recover from a failure. However, the system cannot suffer neither a total failure nor a total recovery. This is interesting for future research regarding recovery mechanisms for the system. The model and the indicators also enable airport operators and policy makers to forecast the behavior of the system given a certain number of operations. Starting at an initial state, we can predict what may happen in the next operations until the system reaches the stationary state/distribution. Future work will be focused on improving the accuracy and reliability of the model (e.g., with adverse meteorological data) and on studying the importance of each one of the blocks in relation with system behavior. We also need to analyze the potential measures that helps the system to recover from a failure. References Belobaba, P., Odoni, A., Barnhart, C., 2015. The Global Airline Industry, 2nd ed. New York. https://doi.org/10.1002/9780470744734 Cook, A., Tanner, G., Zannin, M., 2013. Towards superior air transport performance metrics – imperatives and methods. J. Aerosp. Oper. 2, 3– 19. https://doi.org/10.3233/AOP-130032 EUROCONTROL, 2017. CODA (Central Office for Delay Analysis) Digest: All-Causes Delay and Cancellations to Air Transport in Europe 2016. Brussels: European Organisation for the Safety of Air Navigation. EUROCONTROL, ACI, IATA, 2012. Airport CDM Implementation: The Manual. Brussels: European Organisation for the Safety of Air Navigation. Gu, Y., Li, J., 2012. Multi-state system reliability: A new and systematic review. Procedia Eng. 29, 531–536. https://doi.org/10.1016/j.proeng.2011.12.756 Horonjeff, R.M., McKelvey, F.X., Sproule, W.J., Young, S., 2010. Planning and Design of Airports, 5th Editio. ed. McGraw-Hill, New York. Jetzki, M., 2009. The propagation of air transport delays in Europe. PhD Thesis in the Department of Airport and Air Transportation Research. RWTH Aachen University. Larose, D.T., 2005. Model Evaluation Techniques, in: Discovering Knowledge in Data: An Introduction to Data Mining. Wiley & Sons, p. 222. https://doi.org/https://doi.org/10.1002/0471687545.ch11 Levitin, G., Xing, L., 2017. Multi-state systems. Reliab. Eng. Syst. Saf. 166, 1–2. https://doi.org/10.1016/j.ress.2017.06.008 Lisnianski, A., Frenkel, I., Ding, Y., 2010. Multi-state system reliability analysis and optimization for engineers and industrial managers, MultiState System Reliability Analysis and Optimization for Engineers and Industrial Managers. https://doi.org/10.1007/978-1-84996-320-6 Lisnianski, A., Levitin, G., 2003. Multi-State System Reliability: Assessment, Optimization and Applications, 1st ed. World Scientific, Singapore. MathWorks, 2017. MATLAB. www.mathworks.com/products/matlab. Pyrgiotis, N., Malone, K.M., Odoni, A., 2013. Modelling delay propagation within an airport network. Transp. Res. Part C Emerg. Technol. 27, 60–75. https://doi.org/10.1016/j.trc.2011.05.017 Rodríguez-Sanz, Á., Comendador, F.G., Valdés, R.A., Pérez-Castán, J.A., 2018. Characterization and prediction of the airport operational saturation. J. Air Transp. Manag. 69, 147–172. https://doi.org/10.1016/j.jairtraman.2018.03.002 Roger A. Horn, Johnson, C.R., 2002. Matrix Analysis, Cambridge University Press. https://doi.org/10.1017/CBO9781139020411 SESAR, 2014. SESAR Concept of Operations Step 2 Edition 2014 (Ed. 01.01.00). Brussels: SESAR Joint Undertaking. Sigman, K., Notes, L., 2009. Limiting distribution for a Markov chain Recurrence and transience 1–12. Simaiakis, I., Balakrishnan, H., 2015. A Queuing Model of the Airport Departure Process. Transp. Sci. trsc.2015.0603. https://doi.org/10.1287/trsc.2015.0603 Wu, C.L., 2012. Airline operations and delay management: Insights from airline economics, networks and strategic schedule planning, 1st ed. Ashgate Publishing, Surrey, England.
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XIII Conference on Transport Engineering, CIT2018 XIII Conference on Transport Engineering, CIT2018
Operational Operational Reliability Reliability of of the the Airport Airport System: System: Monitoring Monitoring and and Forecasting Forecasting
Álvaro Rodríguez-Sanz a,a, **, Beatriz Rubio Fernández aa, Fernando Gómez Comendador aa, Álvaro Rodríguez-Sanz a , Beatriz Rubio Fernández , Fernando Gómez Comendador , Rosa Arnaldo Valdés a, José Manuel Cordero García bb and Margarita Bagamanova cc Rosa Arnaldo Valdés , José Manuel Cordero García and Margarita Bagamanova b b
a Universidad Politécnica de Madrid (UPM), Plaza Cardenal Cisneros N3, Madrid 28040, Spain. a Universidad Politécnica de Madrid (UPM), Plaza Cardenal Cisneros N3, Madrid 28040,Allende, Spain. Madrid 28022, Spain. CRIDA (Reference Center for Research, Development and Innovation in ATM), Avenida de Aragón 402 Edificio CRIDA (Reference Centerc Universitat for Research, Development and Innovation ATM),deAvenida Aragón 402 08202, EdificioSpain. Allende, Madrid 28022, Spain. Autònoma de Barcelona (UAB), in Carrer EmpriusdeN2, Sabadell c Universitat Autònoma de Barcelona (UAB), Carrer de Emprius N2, Sabadell 08202, Spain.
Abstract Abstract Airports are intermodal nodes that can act as disruption drivers throughout the entire transport system. To ensure the robustness of Airports are intermodal nodesmanagers that can act disruption drivers the entire transport system.and To delve ensureinto theits robustness of the transport network, airport andaspolicy makers need throughout means to assess operational reliability precursors. the transport network, airport managers and policy makers need means to assess operational reliability and delve into its precursors. This paper develops a model to evaluate potential malfunctions of an airport, through the definition of proactive performance This paper The develops potential malfunctions of an airport, through the definition of of proactive indicators. modelaismodel based to onevaluate the Multi-State Systems (MSS) reliability theory, as a natural extension classicalperformance binary-state indicators. The modelpresent is based on the Multi-State Systems reliability theory, as awith natural extension of on classical binary-state evaluation: airports different performance levels (MSS) and several failure modes various effects the entire system evaluation: airports present different performance levels and several failure modes with various effects on the entire system performance (degradation). The operational reliability assessment is achieved with random processes (Markov) methods. The performance (degradation). The operational reliability assessment is achieved with random processes (Markov) methods. The analysis is focused on the airspace-airside integrated infrastructure, using a dynamic spatial boundary associated with the Extended analysis is focused on the airspace-airside integrated infrastructure, using a dynamic spatial boundary associated with the Extended Terminal Maneuvering Area (E-TMA) concept. The study evaluates the ‘visit’ of an aircraft to the E-TMA, which consists of three Terminalsections: Maneuvering Areaapproach; (E-TMA) (b) concept. The study evaluates the initial ‘visit’ of an aircraft to the which consists of threea separate (a) final turnaround process; and (c) climb segment. TheE-TMA, reliability model represents separate sections: (a) final approach; (b) turnaround process; and (c) initial climb segment. The reliability model represents framework to test different ‘what-if’ scenarios and to reduce uncertainty by categorizing different behavior patterns. Therefore, ita framework test different ‘what-if’ scenarios and to to enter reducea degraded uncertainty by categorizing different behaviorthrough patterns.a case Therefore, it allows us totopredict how probable is for the system state. The methodology is validated study at allows us to predict how probable is for the system to enter a degraded state. The methodology is validated through a case study at Madrid Airport (LEMD): a collection of nearly 34,000 aircraft turnarounds is used to statistically determine the system operational Madrid Airport The (LEMD): a collection of 34,000 turnarounds is used to statistically determine the system operational characteristics. main contribution ofnearly this paper is toaircraft provide a mechanism to monitor and forecast the system’s state, as a way characteristics. The main contribution of this paper is to provide a mechanism to monitor and forecast the system’s state, as a way to proactively assess the operational reliability of airports. to proactively assess the operational reliability of airports. © © 2018 2018 The The Authors. Authors. Published Published by by Elsevier Elsevier Ltd. Ltd. © 2018 The Authors. by Elsevier Ltd. This is an open accessPublished article under under the the CC BY-NC-ND BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) This is an and openpeer-review access articleunder underresponsibility the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection of the scientific committee of the XIII Conference on Transport Engineering, Selection and peer-review under responsibility of the scientific committee of the XIII Conference on Transport Engineering, CIT2018. CIT2018. CIT2018. Keywords: Airport operations; Reliability analysis; Multi-state systems; Markov processes; Performance assessment. Keywords: Airport operations; Reliability analysis; Multi-state systems; Markov processes; Performance assessment.
* *
Corresponding author. Tel.: +34 626574829. Corresponding Tel.: +34 626574829. E-mail address:author. [email protected]. E-mail address: [email protected].
2352-1465 © 2018 The Authors. Published by Elsevier Ltd. 2352-1465 © 2018 Thearticle Authors. Published by Elsevier Ltd. This is an open access under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) This is an and openpeer-review access article under the CC BY-NC-ND licensecommittee (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection under responsibility of the scientific of the XIII Conference on Transport Engineering, Selection and peer-review under responsibility of the scientific committee of the XIII Conference on Transport Engineering, CIT2018. CIT2018. 2352-1465 2018 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection and peer-review under responsibility of the scientific committee of the XIII Conference on Transport Engineering, CIT2018. 10.1016/j.trpro.2018.11.002
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1. Introduction and problem statement An efficient management of airport related operations is a key element when assessing the air transport system performance (Wu, 2012). Mutual interdependencies between airports result in system-wide effects for the air traffic network (Pyrgiotis et al., 2013). Therefore, a change to an air-to-air perspective is necessary, with a specific focus on the aircraft ground operations as major driver for airline operational efficiency. In fact, airport operations (including Air Traffic Flow and Capacity Management -ATFCM- associated issues) were the second highest cause in the share of primary delays (initial delays that are not inherited from the aircraft previous leg operation) (EUROCONTROL, 2017). Currently, wider boundaries are given to the turnaround operational scope: it encompasses the processes occurring at the airport’s Extended Terminal Maneuvering Area (E-TMA) (Rodríguez-Sanz et al., 2018). The concept of Airport Transit View (ATV) describes the “visit” of an aircraft to the airport and connects inbound and outbound flights (SESAR, 2014). It is divided into three sections: the final approach and inbound ground section of the arrival flight, the turnaround process section in which the inbound and the outbound flights are linked and the outbound ground section and the initial climb segment of the departure flight. Reliability engineering is the discipline devoted to ensure that a system will be reliable when operated under certain conditions (Lisnianski and Levitin, 2003). Through Multi-State Systems (MSS) reliability models, such as Markov-chains, more realistic and more precise representation of engineering systems are provided (Levitin and Xing, 2017). However, they are much more complex and present major difficulties in system definition and performance evaluation (Gu and Li, 2012). For this study, we have used information about nearly 34,000 airport transit view operations at Adolfo Suárez Madrid Barajas Airport (LEMD) during the months of July and August of 2016. All the information related with the operations at the airport is divided into four blocks: delay, capacity, environment and complexity. Then, we propose a Markov chain model that represents the global state of the airport with respect to the four blocks. Finally, we study some reliability functionalities and performance indicators of Markov-chains to model the system potential degradation and predict its behavior. The main contribution of this paper is to apply the MSS theory to the ATV, including the study of reliability features of Markov-chains. These features determine the status of the system and, through the predictive applications of the proposed model, allow us to forecast the behavior of the system for the next operations. 2. Materials and methods Traditionally, systems have been modelled in a binary way, thus each element has only two possible states: perfect functioning or complete failure (Lisnianski and Levitin, 2003). However, the majority of real systems can be in more than two states. The so-called Multi-State Systems (MSS) are the systems that present a finite number of states. Usually, a multi-state system is composed of elements that can also be in different states (Lisnianski et al., 2010). If a multi-state system is composed of n elements, its performance level is determined in an unambiguous way by the performance levels of the elements that constitute it. At each moment, the system elements have a level of performance that corresponds to their current state. The status of the entire system is determined by the states of its elements. Therefore, the definition of a multi-state model must include the stochastic process performance for each element of the system 𝑖𝑖 : Gi (t)(𝑖𝑖 = 1, … 𝑛𝑛) and the structure of system operation that causes the stochastic process corresponding to the output of the entire multi-state system: 𝐺𝐺(𝑡𝑡) = 𝜑𝜑(𝐺𝐺𝑖𝑖 (𝑡𝑡), … , 𝐺𝐺𝑛𝑛 (𝑡𝑡)). Markov-chains are a special type of discrete stochastic process in which the probability of an event only depends of the previous state of the system. This type of systems are memoryless thus satisfy the Markov property (Lisnianski et al., 2010). The probability of transition between state 𝑋𝑋𝑛𝑛−1 = 𝑖𝑖 and 𝑋𝑋𝑛𝑛 = 𝑗𝑗 is given by 𝛾𝛾𝑖𝑖,𝑗𝑗 , where 𝑛𝑛 is the number of transitions. The matrix 𝑷𝑷 collectively represents these transition probabilities (Sigman and Notes, 2009). The vector 𝜋𝜋𝑛𝑛𝑇𝑇 defines the probability of finding the system in a specific state on the 𝑛𝑛 -th transition. The probabilities of the state for each transition are determined iteratively multiplying the state vector by the transition matrix. The stationary distribution 𝜋𝜋 (steady state or long-term,𝑛𝑛 → ∞), does not change over time; i.e.: 𝛾𝛾1,1 𝛾𝛾2,1 𝜋𝜋 = 𝜋𝜋𝑷𝑷 𝑤𝑤𝑤𝑤𝑤𝑤ℎ 𝑷𝑷 = ( ⋮ 𝛾𝛾𝑘𝑘,1
𝛾𝛾1,2 𝛾𝛾2,2 ⋮ 𝛾𝛾𝑘𝑘,2
⋯ ⋯ ⋯ ⋯
𝛾𝛾1,𝑘𝑘 𝛾𝛾2,𝑘𝑘 ⋮ ) 𝛾𝛾𝑘𝑘,𝑘𝑘
(1)
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3. Results 3.1. State vector and blocks definition To simplify the analysis, the airport model is divided into four blocks: delay, capacity, environment and complexity. A number of parameters or elements, which have different performance rates, defines each block. To avoid an excessive number of states that will complicate the main model definition, the state of each block is defined in a conservative way, i.e., the state of the delay block will be assessed by the most adverse state of the parameters of the block. Once the blocks are defined, the main model states are determined as the number blocks that have failed. 3.1.1. Capacity and complexity The block of capacity relates the number of operations taking place during an hour to the airport capacity. It has two indicators: adaptation and relation. The metric ‘adaptation’ describes the relationship between the total number of operations (arrivals and departures) of each hour and the declared capacity of the airport. The runway “declared” (or practical) capacity is the maximum number of aircraft operations during a given period of time (i.e., one hour), when the average delay imposed on each aircraft movement does not exceed a tolerable level prescribed in advance (Horonjeff et al., 2010). The metric ‘relation’ connects the number of departures with the number of arrivals. It is calculated as the division of the number of departures by the number of arrivals. This block helps us to understand the overall situation of the airport; i.e., its level of workload. Table 1.illustrates the states the three possible states of the parameters. The objective of the complexity block is to consider the complexity of airport operations. Complexity can be represented by parameters such as meteorological conditions or runway configuration. In our case of study, the database has information about operations that were registered in the months of July and August, thus some meteorological conditions (visibility, amount of clouds) are always in excellent conditions and do not provide any relevant information about how meteorological variables affect the airport operational complexity. Future works will be oriented to improve this limitation. The states for the complexity block are summarized in Table 2. Table 1: Capacity block state intervals.
Table 2: Complexity block states intervals.
Adaptation
Relation
Target value
80% < x < 100%
0.1 < x < 1.1
Correct value
x ≤ 80%
0.2
Incorrect values
x ≥ 100%
Other values
Runway configuration
Wind
Normal operation
North (the airport’s preferential configuration)
Head
Complex operation
South
Cross or tail
3.1.2. Delay and environment The delay block is defined by two parameters: off-block time delay to represent the punctuality in departures and in-block time delay to represent punctuality in arrivals. The off-block time delay is the difference between the scheduled off-block time (SOBT) and the actual off-block time (AOBT) (EUROCONTROL et al., 2012). The inblock time is the difference between the scheduled in-block time (SIBT) and the actual in-block time (AIBT) (EUROCONTROL et al., 2012). The three possible states of the parameters of the delay block are illustrated in Table 3. As mentioned before, delay refers to the time difference between the actual time and the schedule time of a timestamp or process. Therefore, delays can be positive or negative, meaning early departure or early arrival of a flight. “Negative” delays (early arrivals) occur when the schedule is running close to plans and can cause issues for airport operations; e.g., disrupting the sequencing of flights and the allocation of resources (gates, handling equipment), especially during peak hours at busy airports (Wu, 2012). “Positive” flight delays often cause significant problems for all the involved stakeholders; e.g., they affect operational and financial performance of airports and airlines, schedule adherence and use of resources, passenger experience and satisfaction, and system reliability (Belobaba et al., 2015; Wu, 2012). Delay thresholds are defined according to operational targets: the ±3 minutes threshold for punctuality set by SESAR’s (target state) (Single European Sky ATM Research) performance metrics
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(Cook et al., 2013) and the 15 minutes threshold for defining delay (correct state) that has historically been common to both Europe and the US (Jetzki, 2009). Table 3: Delay and environment blocks states intervals.
Thresholds
Target time
Correct time
Incorrect time
-3 min < d < 3 min
-15 min < d ≤ -3 min or 3 min ≤ d < 15 min
d ≤ -15min or d ≥ 15 min
The environment block aims to include the metrics that reflect the emissions of particulates and gases (CO2, water vapor, hydrocarbons, carbon monoxide, nitrogen oxides, sulfur oxides, lead and black carbon) nearby and within the airport infrastructure. It can be represented by the extra time in the aircraft processes related to approach and onground operations. That is why this block is modelled by the additional taxi-in time, the additional taxi-out time, the additional Arrival Sequencing and Metering Area (ASMA) 40 NM time and the additional ASMA 60 NM time. The ASMA is defined as a virtual cylinder with a 40 NM or 60 NM radius around the airport (SESAR, 2014). One of the appraised timestamps in airport operations is the additional time the flight spends in ASMA 60 NM. This time is calculated as the difference between the actual time the aircraft needed to pass through this volume of airspace and the ideal time it should have needed, depending on the space congestion situation at that time. Therefore, the additional ASMA time is an indicator for the average arrival runway queuing time on the inbound traffic flow, during congestion periods at airports (SESAR, 2014). Just as the additional ASMA 60 NM time was defined for a 60 NM radius cylinder, a similar indicator can defined for the next volume of airspace the aircraft goes through, which is the ASMA 40 NM. For our case study is particularly important to study both ASMA 40 NM and ASMA 60 NM additional times. This is due to the fact that holdings at Adolfo Suarez Madrid-Barajas (LEMD) are located beyond a radius of 40 NM, as depicted in Fig. 1.
Fig. 1: ASMA 40 NM and ASMA 60 NM for Adolfo Suarez Madrid Barajas-Airport (LEMD).
Additional taxi-in time is the difference between an estimated time to perform taxi-in procedures (depending on the stand and the runway in use) and the actual process duration. Therefore, ‘additional taxi-in time’ will collect the inefficiencies in the airport taxiways due to traffic congestion or potential incidents. Conversely, taxi-in procedures could be carried out in a shorter time than expected (during periods of low inflow traffic), with the operation being able to absorb delays generated in other processes. The additional taxi-out time is appraised likewise. Note that unimpeded taxi times can only be as short as the physics of the process allows but can grow large in the event of a slow taxi operation (Simaiakis and Balakrishnan, 2015). The environment block states are defined in the same way as the delay block states (Table 3).
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3.1.3. Global model states When the contributions of the components to the cumulative system performance rate are different, the number of possible MSS states grows significantly; i.e., different combinations of k available units can provide a large number of performance rates for the entire system (Lisnianski et al., 2010). This is the reason why the global model only considers total failures of the blocks regardless of which block has failed. Otherwise, the number of global states would be unmanageable in practical applications. Therefore, in our study, the number of blocks that have failed measures the global performance of the system; i.e., the global model has a total of five possible states (𝐺𝐺1 , 𝐺𝐺2 , 𝐺𝐺3 , 𝐺𝐺4 and 𝐺𝐺5 ) according to the number of failed blocks. The performance rate of each state is defined as the relative number of blocks that have failed. That way 𝐺𝐺1 has a performance rate of 100%, 𝐺𝐺2 of 75%, 𝐺𝐺3 of 50%, 𝐺𝐺4 of 25% and 𝐺𝐺5 of 0%. 3.2. Markov Chain Once the system’s states are established, the next step is to obtain the transition matrix of the model. A Matlab © program (MathWorks, 2017) is implemented using the data for the almost 34,000 airport transit views at Adolfo Suarez Madrid-Barajas Airport (LEMD). This program calculates the probability of transitioning from one state to another from real operations. Once the transition matrices are obtained, the steady state can be appraised by solving Eq. 2 and knowing that the sum of the components of state vector 𝜋𝜋 must be equal to one. For the model construction and testing, data is divided into two partial datasets. The first one is used to train and validate the model, with 80% of the total number of observations (building and cross-validation sample); while the remaining 20% (test sample) is used to test the generalization of the model. Therefore, the process is as follows: 1) Randomly split the initial dataset into construction/building and testing sets: 80% construction and 20% test. 2) Perform cross-validation on the construction set to fit the model (k-fold with k=10) (Larose, 2005). 3) Test if results are generalizable, using a test set, which is completely separated from model development. Both the train and the error scores presents an average value of 10%, i.e., our model predicts new observations as well as it fits the original dataset. Therefore, we are not overfitting the model and results can be generalizable. Table 4: Global transition matrix. 𝑮𝑮𝟏𝟏
𝑮𝑮𝟐𝟐
𝑮𝑮𝟑𝟑
𝑮𝑮𝟒𝟒
𝑮𝑮𝟓𝟓
𝑮𝑮𝟏𝟏
0,483
0,439
0,074
0,004
0,000
𝑮𝑮𝟐𝟐
0,081
0,491
0,389
0,038
0,001
𝑮𝑮𝟑𝟑
0,006
0,192
0,514
0,280
0,008
𝑮𝑮𝟒𝟒
0,001
0,033
0,489
0,457
0,020
𝑮𝑮𝟓𝟓
0,000
0,022
0,351
0,470
0,157
The transition matrix among the five states of the global model is presented in Table 4. The transition matrix shows that total failures do not last long. The probability of being in state 𝐺𝐺5 (total failure) and staying in that same state is only 15.7%, which means that the system is not supposed to stay in that state for a large number of operations. It is more probable that the system transitions to state 𝐺𝐺4 through a partial repair. Moreover, the system tends to stay in the same state or suffer partial repairs or failures. This can be appraised by analyzing the diagonal of the matrix, which is the most likely set of transition probabilities, except for the case in which the system is in state 𝐺𝐺5 (as mentioned before, when it tends to move to state 𝐺𝐺4 ). Furthermore, total recoveries and total failures are not possible, the probability that the system is in state 𝐺𝐺5 for operation 𝑛𝑛 and in state 𝐺𝐺1 for operation 𝑛𝑛 + 1 is zero. The system neither can go from state 𝐺𝐺1 (perfect functioning) to state 𝐺𝐺5 (total failure) in just one-step. The system’s Markov-chain is ergodic, thus it has a unique stationary distribution (Roger A. Horn and Johnson, 2002). When the system reaches the stationary distribution, the more probable state is 𝐺𝐺3 (45.94%), this means that the system is working with a 50% performance rate or that is working with two failed blocks. Perfect functioning and total failure are very unlikely to happen (4% and 1 % respectively).
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3.3. Functionalities To study the system’s performance, three main indicators are evaluated: mean instantaneous performance, mean instantaneous deficiency and mean instantaneous availability. After that, three scenarios are presented: the first one studies the evolution of the system when the initial state is the perfect functioning (𝐺𝐺1 ), the second one studies the evolution of the system when it starts from total failure (𝐺𝐺5 ) and for the third one, the system starts in an intermediate state (𝐺𝐺3 ), which is the most probable. To characterize the average MSS output performance, we can use the performance expectation. The mean value of MSS instantaneous output performance at operation 𝑛𝑛 (En ) is defined as (Lisnianski and Levitin, 2003): 𝐸𝐸𝑛𝑛 = ∑𝑁𝑁 𝑘𝑘=1 𝑔𝑔𝑘𝑘 𝑝𝑝𝑘𝑘 (𝑛𝑛)
(2)
𝐷𝐷𝑛𝑛 = ∑𝑁𝑁 𝑖𝑖=1 𝑝𝑝𝑖𝑖 (𝑛𝑛) 𝑚𝑚𝑚𝑚𝑚𝑚(𝑤𝑤 − 𝑔𝑔𝑖𝑖 ; 0)
(3)
𝐴𝐴𝑛𝑛 = ∑𝐾𝐾 𝑖𝑖=1 𝑝𝑝𝑖𝑖 (𝑛𝑛)
(4)
Where 𝑁𝑁 is the total number of states, 𝑔𝑔𝑘𝑘 is the performance rate associated with state 𝑘𝑘 and 𝑝𝑝𝑘𝑘 (𝑛𝑛) is the probability that the system is in state 𝑘𝑘 at operation 𝑛𝑛. The mean instantaneous deficiency or deviation (Dn ) is defined as a weighted average between the system probability to be found in each state and the service levels associated to these states. A weighted average of the value of a random variable where the probability function provides weights can be understood as the expected value (Lisnianski and Levitin, 2003). In case the difference is negative, the average is weighted with a zero. That is because in those cases the system is meeting the expected demand and the aim of the index is to assess the cases when the system is not fulfilling the demand.
Where 𝑝𝑝𝑖𝑖 (𝑛𝑛) is the probability that the system is in state 𝑖𝑖 at 𝑛𝑛-th operation, 𝑤𝑤 is the expected demand and 𝑔𝑔𝑖𝑖 is the level of performance associated to state 𝑖𝑖. Another parameter to evaluate the performance of the system is the mean instantaneous availability (An ). It is defined as the sum of the probabilities of the acceptable states (Lisnianski and Levitin, 2003):
Where 𝐾𝐾 is the number of acceptable states and 𝑝𝑝𝑖𝑖 (𝑛𝑛) is the probability that the system is in state 𝑖𝑖 at operation 𝑛𝑛. First, it is necessary to establish the number of acceptable states and the expected level of demand. This study contemplates that an acceptable situation for the airport system includes the states 𝐺𝐺1 , 𝐺𝐺2 and 𝐺𝐺3 . The expected level of demand (𝑤𝑤) is at least 50%. For the stationary distribution, the mean instantaneous availability is 72.65%, the mean instantaneous deficiency is 7.1087 and the mean instantaneous performance is 50.59%. Fig. 2 (a), (b) and (c) show the evolution of the three performance indicators when the system starts at state 𝐺𝐺1 , with initial state vector 𝑝𝑝0 = [1 0 0 0 0]. The system is initially functioning perfectly (𝐸𝐸𝑛𝑛 = 100%, 𝐷𝐷𝑛𝑛 = 0 and 𝐴𝐴𝑛𝑛 = 100%). The system’s performance is degraded until it reaches the stationary distribution. The number of steps spent by the system to reach the stationary distribution is 20 operations (with an error of the estimation of 0.1010%). The second studied case is completely opposite of the previous one. The system is initially in complete failure (𝐺𝐺5 , 𝑝𝑝0 = [0 0 0 0 1]) and improves its performance rate by reaching the stationary distribution in 17 operations (with a maximum error of the estimation of 0.1403%). Fig. 2 (d), (e) and (f) represent this evolution, which is faster than the case before; i.e. the system recovers itself faster than it evolves towards a degraded state. To study the system evolution from a non-extreme initial situation (total failure or perfect functioning), a third case is appraised in which the system is initially in the most probable state (𝐺𝐺3 ). The initial state vector is 𝑝𝑝0 = [0 0 1 0 0] and the system takes 15 operations to reach the stationary distribution (with a maximum error in the estimation of 0.1060%). Now the system requires less time to reach the stationary distribution (only 15 steps) because the initial state vector is similar to the stationary state vector. Fig. 2 (g), (h) and (i) describe the temporary evolution of the three performance indicators for the third case. The evolution can be divided into two stages. In the first one, the system suffers a degradation, arriving at a minimum of mean instantaneous performance (47.6896%) in 2 steps, a minimum
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of mean instantaneous availability (70.57%) in 3 steps and a maximum of mean instantaneous deficiency (7.6340) in 3 steps. Then, the system recovers itself in the second stage of the evolution, arriving at the stationary distribution in 15 steps.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
Fig. 2: Performance indicators: (a) Mean instantaneous availability for case 1, (b) Mean instantaneous deficiency for case 1, (c) Mean instantaneous performance for case 1, (d) Mean instantaneous availability for case 2, (e) Mean instantaneous deficiency for case 2, (f) Mean instantaneous performance for case 2, (g) Mean instantaneous availability for case 3, (h) Mean instantaneous deficiency for case 3, (i) Mean instantaneous performance for case 3.
4. Conclusion This paper develops a reliability analysis of the airport transit view process. In this analysis, we use a multi-state system approach, through a Markov-Chain model. Four main elements of the system are used to appraise its evolution: delay, capacity, environment and complexity. From these partial elements, we evaluate the failure behavior of the system. The model application to a case of study of nearly 34,000 operations shows that the airport system tends to a stationary distribution for a sufficient number of operations. In this stationary distribution, the most probable state is the one with a performance rate of 50%, which is equivalent to the system being working with two failed blocks. The analysis of the values obtained for the performance indicators (mean instantaneous performance, mean instantaneous availability and mean instantaneous deficiency) allows us to represent the current behavior of the
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system’s dynamics. Moreover, we can obtain a prediction of its most likely evolution through the Markov-chain functionalities. Three cases were evaluated to understand the system behavior, with the system starting in different initial states: total failure, perfect functioning and the most probable state. These three situations show a fast evolution (indeed, the system reaches the stationary distribution in less than 5 operations) and prove that the system is repairable, thus it can recover from a failure. However, the system cannot suffer neither a total failure nor a total recovery. This is interesting for future research regarding recovery mechanisms for the system. The model and the indicators also enable airport operators and policy makers to forecast the behavior of the system given a certain number of operations. Starting at an initial state, we can predict what may happen in the next operations until the system reaches the stationary state/distribution. Future work will be focused on improving the accuracy and reliability of the model (e.g., with adverse meteorological data) and on studying the importance of each one of the blocks in relation with system behavior. We also need to analyze the potential measures that helps the system to recover from a failure. References Belobaba, P., Odoni, A., Barnhart, C., 2015. The Global Airline Industry, 2nd ed. New York. https://doi.org/10.1002/9780470744734 Cook, A., Tanner, G., Zannin, M., 2013. Towards superior air transport performance metrics – imperatives and methods. J. Aerosp. Oper. 2, 3– 19. https://doi.org/10.3233/AOP-130032 EUROCONTROL, 2017. CODA (Central Office for Delay Analysis) Digest: All-Causes Delay and Cancellations to Air Transport in Europe 2016. Brussels: European Organisation for the Safety of Air Navigation. EUROCONTROL, ACI, IATA, 2012. Airport CDM Implementation: The Manual. Brussels: European Organisation for the Safety of Air Navigation. Gu, Y., Li, J., 2012. Multi-state system reliability: A new and systematic review. Procedia Eng. 29, 531–536. https://doi.org/10.1016/j.proeng.2011.12.756 Horonjeff, R.M., McKelvey, F.X., Sproule, W.J., Young, S., 2010. Planning and Design of Airports, 5th Editio. ed. McGraw-Hill, New York. Jetzki, M., 2009. The propagation of air transport delays in Europe. PhD Thesis in the Department of Airport and Air Transportation Research. RWTH Aachen University. Larose, D.T., 2005. Model Evaluation Techniques, in: Discovering Knowledge in Data: An Introduction to Data Mining. Wiley & Sons, p. 222. https://doi.org/https://doi.org/10.1002/0471687545.ch11 Levitin, G., Xing, L., 2017. Multi-state systems. Reliab. Eng. Syst. Saf. 166, 1–2. https://doi.org/10.1016/j.ress.2017.06.008 Lisnianski, A., Frenkel, I., Ding, Y., 2010. Multi-state system reliability analysis and optimization for engineers and industrial managers, MultiState System Reliability Analysis and Optimization for Engineers and Industrial Managers. https://doi.org/10.1007/978-1-84996-320-6 Lisnianski, A., Levitin, G., 2003. Multi-State System Reliability: Assessment, Optimization and Applications, 1st ed. World Scientific, Singapore. MathWorks, 2017. MATLAB. www.mathworks.com/products/matlab. Pyrgiotis, N., Malone, K.M., Odoni, A., 2013. Modelling delay propagation within an airport network. Transp. Res. Part C Emerg. Technol. 27, 60–75. https://doi.org/10.1016/j.trc.2011.05.017 Rodríguez-Sanz, Á., Comendador, F.G., Valdés, R.A., Pérez-Castán, J.A., 2018. Characterization and prediction of the airport operational saturation. J. Air Transp. Manag. 69, 147–172. https://doi.org/10.1016/j.jairtraman.2018.03.002 Roger A. Horn, Johnson, C.R., 2002. Matrix Analysis, Cambridge University Press. https://doi.org/10.1017/CBO9781139020411 SESAR, 2014. SESAR Concept of Operations Step 2 Edition 2014 (Ed. 01.01.00). Brussels: SESAR Joint Undertaking. Sigman, K., Notes, L., 2009. Limiting distribution for a Markov chain Recurrence and transience 1–12. Simaiakis, I., Balakrishnan, H., 2015. A Queuing Model of the Airport Departure Process. Transp. Sci. trsc.2015.0603. https://doi.org/10.1287/trsc.2015.0603 Wu, C.L., 2012. Airline operations and delay management: Insights from airline economics, networks and strategic schedule planning, 1st ed. Ashgate Publishing, Surrey, England.