A survival model for flight delay propagation

A survival model for

flight delay propagation

Jinn-Tsai Wong

, Shy-Chang Tsai

a_{Institute of Traffic and Transportation, National Chiao Tung University, 4F, 118, Sec. 1, Chung-Hsiao W. Rd., Taipei 100, Taiwan}

b_{Department of Aviation Services and Management, China University of Science and Technology, 200, Zhonghua St., Hengshan, Hsinchu County 31241, Taiwan}

Keywords:

Flight delay propagation Survival analysis

Cox proportional hazards model

a b s t r a c t

This paper examinesflight delay propagation involving a Taiwanese domestic airline. The Cox propor-tional hazards model is used to develop departure and arrival delay models that show howflight delay propagation can be formulated through repeated chain effects in aircraft rotations. The hazard ratios obtained provide measures of the chances of recovering fromflight delays under a variety of situations and the effects that individual contributing factors offlight delays have on airline schedule reliability.

Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Airline passenger complaints concerning delays, cancellations, and denied boarding have prompted the US Congress to consider stronger measures to ensure passenger protection (US Government Accountability Office, 2011). Of the delay costs analyzed by Aus-trian Airlines, only 22% can be attributed directly to the effect of delays; 24% stem from permanent loss of passenger loyalty and 54% stem from induced knock-on delays in aircraft rotation schedules (Airline Business, 1999). United Airlines estimates that it saves approximately $1.6 million by using aflight delay projection model during the_{first quarter of 2004 (}Abdelghany et al., 2004). Both of these cases suggest that the consequences of delays and their propagation in the air transport system, including decrease of productivity of aircraft as well as loss of time and loyalty of passengers, cannot be neglected.

Optimizing aircraft utilization requires airlines have tight turn-around times betweenflights, but this can increase the likelihood of delays in subsequentflights. The typical approach to dealing with disruptions is to re-optimize the schedule, but a more proactive approach can be to build robustness into the schedule at the planning stage (Lan et al., 2006; Wu and Caves, 2002). One prerequisite to the development of tools for building more robust airline schedules is an understanding of the relationship between planned schedules and delay propagation.

A range of methodologies has been adopted to deal with the issue of airline delay propagation.Abdelghany et al. (2004)used a deter-ministic model to predict the propagated delays along aircraft routes

based on the concept of resource networks and shortest paths, while

Beatty et al. (1998)used a‘delay multiplier’ metric to estimate the scale of delay propagation. Schaefer and Millner (2001)used the ‘detailed policy assessment tool’ to model delay propagation in a network of airports when facing inclement weather conditions. To limit flight delays, considerable efforts have also been made to develop proactive schedule recovery models. Delays in airline schedules may be the result of many different causes but most attention has been paid to the technical aspects of optimizing airline schedules and failed to consider the role played by airline ground operations and other delay causes in contributing and controlling delays in daily operations;Wu and Wong (2007)being an exception. Although different methods have been used to investigateflight delay propagation, the stochastic effects offlight delay propagation resulting from various delay causes have not been thoroughly captured. Major shortcoming of using airline dependability statis-tics is that they are ex post measurements and only reveals the results of schedule delays without any investigating the deter-mining factors such as schedule design and airline operations (Wu and Caves, 2002). In addition, different causes offlight delays may have different effects on airline schedule reliability, and the effects of_{flight delays resulting from the same delay cause may not be the} same in all cases.

Here we develop a way to explore the problem offlight delay propagation in a dynamic operating environment by considering the stochastic characteristics of turnaround and block operations and clarifying the relationship between flight delays and the associated causes.

2. Flight delay mechanisms

Theflights assigned to the same aircraft during one cycle (usually one day for domestic operations and one week for international * Corresponding author. Department of Aviation Services and Management,

China University of Science and Technology, 200, Zhonghua St., Hengshan, Hsinchu County 31241, Taiwan. Tel.:þ886 3 593 5707; fax: þ886 3 593 3590.

E-mail address:[email protected](S.-C. Tsai).

Contents lists available atSciVerse ScienceDirect

Journal of Air Transport Management

j o u rn a l h o m e p a g e : w w w . e ls e v i e r . c o m / l o c a t e / j a i r t r a m a n

0969-6997/$e see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.jairtraman.2012.01.016

operations) form the ‘routes’ on which the aircraft is operated. Buffer times play an important role in the implementation of recovering schemes associated with irregular operations. Turn-around buffer time, which is the extra time scheduled beyond the time required for ground handling, is usually built-in to accom-modate potential delays from late inbound aircraft and aircraft turnaround operations. Scheduled block buffer time, on the other hand, is the extra time added to aflight’s scheduled arrival time to permit a degree of variability inflight operations between airports. Although a published airline schedule generally incorporates buffer time,flight delays can occur when accumulated delays exceed the buffer time.

2.1. Departure delay

Fig. 1 illustrates the relationships among flight delays in an airline schedule. The solid arrows represent the original schedule of departures and arrivals forflight legs i1 and i. STD and STA refer to scheduled time of departure and scheduled time of arrival, respectively, and STA_1refers to scheduled time of arrival offlight f_i1. The dotted arrows represent the actual departures and arrivals of theseflight legs. ATD and ATA refer to actual time of departure and actual time of arrival, respectively, and ATA_1refers to actual time of arrival of flight fi1. Eq. (1) describes the relationship

between the scheduled time of arrival offlight fi1(STA1), the

on-chock time, and the scheduled time of departure offlight fi(STD),

the off-chock time. The scheduled turnaround time of_{flight f}i(TS) is

the interval between the arrival offlight fi1 at the gate and the

time at which this aircraft departs for flight fi, comprising two

parts: the scheduled required ground handling time (gS) and scheduled turnaround buffer time (bsg) (Eq.(2)).

STD ¼ STA1þ TS (1)

TS _{¼ g}s_{þ b}sg ₍₂₎

If the delay in an aircraft’s arrival is shorter than the scheduled turnaround buffer time, the scheduled turnaround buffer time is

capable of absorbing it. A delay, however in arrival exceeding the scheduled turnaround buffer time might cause a delay in the departure of the nextflight. Given the interactions between fixed flight schedules and stochastic disruptions associated with turn-around operations, there might also be a ground delay forflight fi

(Gd). If the scheduled turnaround buffer time is incapable of absorbing this ground delay, it could lead to a delay in subsequent departure. Thus, the departure delay offlight fi(Dd) can be caused

by the arrival delay of the previousflight (Ad

1) and a ground delay

at the current airport (Gd). Eq.(3)shows that the scheduled turn-around buffer time (bsg) may be able to absorb these delays.

Dd ¼ maxn0; Ad

1þ Gd bsg

o

(3)

2.2. Arrival delay

The scheduled block time offlight fi(FS) includes the scheduled

required block operation time (fS), the minimum time required to complete the activities of taxi-out, airborne operation, and taxi-in, and the scheduled block buffer time (bsr) (Fig. 1). This buffer is expected to absorb any potential delays at the origin airport and in the block operations. Inboundflight fimight also have a block delay

(Rd) resulting from problems such as severe weather or air traffic control restrictions en-route or at destination airport. This would result in an arrival delay if the scheduled block buffer time (bsr) cannot absorb this block delay. Therefore, the arrival delay offlight fi(Ad) can be influenced by a departure delay at the origin airport

(Dd) and a block delay between airports (Rd), which might be absorbed by the scheduled block buffer time (bsr). Eq.(4)describes this relationship.

Ad ¼ maxn0; Dd_{þ R}d_{ b}sro ₍₄₎ By combining the mechanisms of departure and arrival delays, flight delay propagation in an airline network can be formulated through repeated chain effects. The challenge is to model the departure and arrival delays with their associated causes.

3. Data

We collected flight data from a Taiwanese domestic airline (‘Airline A’) to explore the effects of flight delay propagation that requires an extensiveflight network with a very high frequency of flights. After the launch of high-speed rail services along the west coast of Taiwan in January 2007, many passengers switched from traveling by airlines to high-speed rail, forcing airlines to reduce flight frequency and terminate some services. Our flight data was collected over twelve months in 2005, prior to the commencement of the high-speed rail service. It lists the state of the departing and arriving aircraft as they rotated through the airport system and indicates the causes for delays in the operations.

Because_{flight delays may have a wide range of causes and the} associated disturbances may result in various durations offlight delays, it is worthwhile investigating the relationship between delays and their causes.Table 1shows the delay causes of Airline A in nine categories.

Although the cause of a delay is recorded when an aircraft departs from an airport gate, an inboundflight can also be delayed during taxi-out, airborne operation, or taxi-in prior to arrival at an airport gate. The delay in arrival can be due to weather en-route or weather at the destination or at an alternative airport, which can also be the cause of departure delay if ground-holding policies are implemented (Vranas et al., 1994). Similarly, air traffic control restrictions en-route or at the destination airport can be the cause of arrival delay, or the cause of departure delay when ground-holding policies are imposed. Hence, of the delay causes of Airline A,‘weather’ and ‘air traffic control restrictions’ are the causes of arrival delay. On the other hand, departingflights can be disrupted by any of the causes listed inTable 1, which are therefore consid-ered as the causes of departure delay (Fricke and Schultz, 2009).

Because Airline A operates short-haul routes with many of its aircraftflying up to 10 consecutive segments in a day, delays in one segment could easily propagate to followingflights.Fig. 2shows that the distribution of delay time is“right-skewed,” indicating that the airline has more short delays and fewer long delays. Meanwhile, departure delays are closely related to arrival delays (Fig. 3).

Although airline companies normally schedule buffer times within turnaround operations at airports in addition to the ground handling time required, the information related to actual turn-around buffer time was unavailable in the dataset of Airline A. To obtain this information, the actual turnaround times of_{flights in} each route were ordered from the smallest to the largest. The 25th percentile (1st quartile) of the ordered actual turnaround times was selected as the required ground handling time, the minimum time required to complete all turnaround activities. Therefore, for every outbound aircraft,

actual turnaround buffer time ¼ scheduled time of departure  actual time of arrival  required ground handling time:

(5)

This means that after an inbound aircraft arrives at the gate, the difference between the actual time of arrival and the scheduled time of departure for the nextflight is the time available for the turnaround of the aircraft. The actual turnaround buffer time can be derived by subtracting the required ground handling time from the available turnaround time. Thus, the actual turnaround buffer time is positive if the available turnaround time exceeds the required ground handling time, and negative (generally resulting from a late flight arrival) if the available turnaround time is shorter than the required ground handling time.

Conversely, the block time includes block buffer time and required block operation time, which is the minimum time required to complete the activities of taxi-out, airborne operation, and taxi-in. However, the obtained dataset did not contain the information related to actual block buffer time. To derive this information, the actual block times offlights in each route were ordered from the smallest to the largest. The 25th percentile of the ordered actual block times was then selected as the required block operation time. Therefore, for every inbound aircraft,

actual block buffer time¼ scheduled time of arrival  actual time of departure  required block operation time:

(6)

In other words, after an outbound aircraft departs from the airport gate, the difference between the actual time of departure and the scheduled time of arrival represents the time available for the block operation of the aircraft. The actual block buffer time can be derived by deducting the required block operation time from the available block time. Thus, the actual block buffer time is positive if the avail-able block time exceeds the required block operation time, and negative (generally resulting from a late _{flight departure) if the} available block time is shorter than the required block operation time. 4. Delay modeling

4.1. Methodology

Survival analysis (Kleinbaum and Klein, 2005) is a method of analyzing survival data or failure time data. The outcome variable of Table 1

Categories of delay causes.

Category Description of Category Delay Cause 1 Airport facilities or

governmental authorities

Mandatory security, immigration, customs, health, airport facilities (e.g., air bridge, gate, parking space), etc.

2 Flight operations and crewing

Crew late in position, crew change, crew legality, etc.

3 Cargo and mail handling Late positioning of cargo/mail, readjustment of cargo/bag position or change of load plan, documentation, errors, etc.

4 Technical and aircraft equipment

Unable to release aircraft for maintenance reason, awaiting engineers/AOG spares, non-scheduled maintenance, etc. 5 Passenger and baggage

handling

Late check-in, check-in error, oversales, missing checked-in passenger, baggage processing, etc. 6 Reactionary Late arrival of aircraft.

7 Weather Ground handling impaired by

adverse weather conditions, weather at departure airport, weather en-route, weather at destination or alternative airport. 8 Air traffic control

restrictions

ATC restriction en-route or capacity, ATC restriction due to staff shortage or equipment failure en-route, ATC restriction at destination airport, ATC restriction due to weather at destination airport. 9 Miscellaneous Industrial action within own

airline, industrial action outside own airline, not covered by any of other defined codes, etc.

interest is ‘time to event,’ usually referred to as survival time or failure time. In aircraft rotation, the duration of a delay represents the period of time that the delay has survived before coming to an end. Therefore,‘survival time’ refers to the number of minutes from the beginning of an individual_{flight delay until an event occurs,} while an‘event’ means that the delay of an individual flight has come to an end. For inbound aircraft, the survival time of arrival delay ends when the aircraft arrives at an airport gate; for outbound aircraft, the survival time of departure delay ends when the aircraft departs from an airport gate. Because of the survival characteristics of flight delays, survival analysis is therefore an appropriate approach to explore the problem offlight delays and their associated causes and to analyze the distributions of delays propagated throughout an airline network.

In survival analysis, the object of primary interest is the survival function, which is defined as

SðtÞ ¼ PrðT>tÞ (7)

The survival function indicates the probability that aflight delay survives longer than specified time t. The survival distribution is plotted as a function that starts with the survival probability of 1 and descends down to the survival probabilities approaching zero for very long delays. Another key concept is the hazard function, which gives the instantaneous probability for an event to occur conditional on survival to time t and specifies the related survival function as well.

Our focus is to investigate the impact of delay contributing factors onflight delays. To examine the relationship between the survival distributions offlight delays and associated covariates, we

employ the Cox proportional hazards (PH) model (Eq. (8)), a method widely applied in survival analysis, to model_{flight delays} in a multiple-airport environment.

hðtjXÞ ¼ h0



t;

a

b

This model provides a hazard expression for aflight at delayed time t with a given speci_{fication of explanatory variables that is} being modeled to predict the hazard of aflight delay. Meanwhile, as shown in Eq.(9), one can easily evaluate the percentage change (increase or decrease) in the hazard offlight delay with a one-unit increase in xk, while other covariates remain unchanged.

^hðtjxkþ 1Þ  ^hðtjxkÞ

^hðtjxkÞ

¼ e^bk 1 ₍₉₎

Because airlines often assign different types of aircraft to various routes in aircraft daily operations, the distributions offlight delays may be influenced by aircraft type, route, peak/off-peak hour, and season, in addition to delay cause (Allan et al., 2001). Therefore, the survival curves offlight delay using the KaplaneMeier estimator are used to examine the possible impact of factors on delays in departure and arrival. Taking the survival curves for aircraft types as an example, it reveals that FK 50 aircraft tend to have delays with longer survival times than FK 100 aircraft for both departure and arrival delays (Fig. 4). Using the log-rank tests, the results inTable 2

indicate significant differences in the survival distributions of flight delays for the selected variables. Therefore, in addition to the variables influencing turnaround and block operations, aircraft type, route, delay cause, peak/off-peak hour, and season are also considered as the covariates for developing the departure delay and arrival delay models.

Due to the strong causal relationship between departure and arrival delays via aircraft routing,flight delay propagation can be investigated by recursively combining the departure delay and arrival delay models. Here,‘recursively’ means that the output of the departure delay model serves as the input of the arrival delay model, and the output of the arrival delay model serves as the input of the departure delay model. Accordingly, the development of models for departure and arrival delay is a prerequisite to exploring flight delay propagation.

4.2. Model specification

To formulate a departure delay model, the relationship between variables must be further clarified. First of all, there will be a longer Fig. 2. Histograms of departure delays and arrival delays.

buffer time in turnaround operations if ground handling activities are completed rapidly; conversely, there will be a shorter turn-around buffer time if more time is required to complete ground handling activities. In addition, the late arrival offlights also results in a reduction in turnaround buffer time. Therefore, to avoid any bias resulting from the highly correlated relationship with ‘turn-around buffer time’, ‘arrival delay’ and ‘ground handling time’ should be deleted from the model. Similarly, the routes of Airline A are operated using different types of aircraft, andflight delays are subject to the routes to which the aircraft are assigned. Accordingly, ‘route’ should also be removed from the model because the delays associated with a‘route’ are already reflected in the delays of the ‘aircraft type’, and a bias would be generated if both ‘route’ and ‘aircraft type’ are used as covariates. Furthermore, a delay caused by ‘late arrival of an aircraft’, recorded as ‘reactionary’ by airlines, is already counted as an‘arrival delay’ and reflected in ‘turnaround buffer time’ in the model. Consequently, ‘reactionary’ should also be deleted from the delay causes considered. Thus, the departure delay model is formulated as Eq.(10).

In aircraft rotations, a shortened taxi-out time, airborne time, and taxi-in time will result in a longer buffer time in block operations. By contrast, the block buffer time is shortened when taxi-out time, airborne time, and taxi-in time are longer. In addition, the late departure offlights also results in a reduction in block buffer time. Because the lengths of taxi-out time, airborne time, and taxi-in time depend on whether aircraft are operating in peak or off-peak hours, the delay information associated with these factors can also be obtained from‘peak/ off-peak hour_{’ in the model. Therefore, ‘departure delay’,} ‘taxi-out time’, ‘airborne time’, and ‘taxi-in time’ should be removed from the model to avoid an interdependent relationship between these covariates and the_{‘block buffer time’. Similarly,} ‘route’ should also be deleted to avoid simultaneously including both‘route’ and ‘aircraft type’, as discussed in the establishment of the departure delay model. The resulting arrival delay model is formulated as Eq.(11).

The variables used in the departure delay and arrival delay models for capturing the chain effects offlight delay propagation are listed inTable 3.

5. Results

5.1. Departure delay model

The results of the departure delay model are shown inTable 4; because ‘season’ and ‘peak/off-peak hour’ were not statistically significant, they were deleted. The higher the hazard is for an event to occur, the more likely theflight delay will end. Thus, for each 1-min increase in turnaround buffer time, which varies depending on arrival time or ground handling time, the chance of ending departure delays increases by only 0.4%. This reveals that departure delays may not be greatly improved though turnaround operations include built-in buffer time. Therefore, airlines may investigate other reasons behind the_{flight delays before taking the measure of} increasing buffer time. With respect to aircraft type, Fokker 50 aircraft have a 35.9% lower chance of ending departure delays than Fokker 100 aircraft.

Compared to the delays caused by‘airport facilities or govern-mental authorities’, departure delays resulting from ‘flight

opera-tions and crewing’, including crew arriving late for their position, crew change, crew legality, etc., have a 41.7% lower chance of recovery. As it is understood, an aircraft will be grounded unless problems associated with crew assignment are fixed. Departure delays caused by ‘cargo and mail handling’ and ‘passenger and baggage handling’, on the other hand, have 52.7% and 40.0% lower chances of recovery, respectively. In addition, departure delays resulting from‘technical and aircraft equipment’ have lower chance of recovery by 46.3%. Each of these significant contributing factors is related to airline operations, suggesting that potential improve-ments could be achieved through a suitable delay management program. Departure delays caused by ‘weather’ are with 43.7% lower chance of recovery and are beyond the control of airlines. Nevertheless, a well-designed contingency plan and useful management techniques could be helpful for airlines to alleviate the consequences of delays caused by weather. Air traffic control restrictions en-route or at the destination airport can be the cause

of departure delay when ground-holding policies are imposed. It reveals that departure delays caused by‘air traffic control restric-tions’ have a 25.4% lower chance of recovery.

hðdeparture delayjcovariatesÞ ¼ h0ðdeparture delayÞexp

0

@

_b

_b

i¼ 1

b

 category of delay causeiþ

b

X3 j¼ 1

b

1

A (10)

hðarrival delayjcovariatesÞ ¼ h0ðarrival delayÞexp

0

@

_b

_b

_b

þ

b

X3 j¼ 1

b

5.2. Arrival delay model

Table 5shows the results of the arrival delay model. Note that ‘aircraft type’, ‘season’, and ‘peak/off-peak hour’ were not statisti-cally significant and were therefore deleted from the model. The results indicate that the key contributing factors of arrival delays include‘block buffer time’ and ‘weather’. For each 1-min increase in block buffer time, which varies depending on departure time or block operation time, the chance for arrival delays to end increases by 6.8%. Most arrival delays are beyond the control of airlines except for delays that develop at departure airports. This implies that developing the means to prevent departure delays could be the key to reducing arrival delays from the origin.

Whilst outboundflights are subject to a wider range of diffi-culties leading to delays, inbound _{flights can be delayed by} ‘weather’ or ‘air traffic control restrictions’ en-route or at destina-tion airport. Compared to the delays caused by‘air traffic control restrictions’, arrival delays resulting from ‘weather’ have a 61.6% lower chance of recovery. As found in both the departure delay and Table 3

Variables used in departure and arrival delay models.

Variable Departure Delay Model Arrival Delay Model Dummy Code Description Turnaround buffer time U e e

Block buffer time U e e

Aircraft type U U 0 Fokker 100

1 Fokker 50

Category of delay cause

U 0 Airport facilities or

governmental authorities 1 Flight operations and

crewing, cargo and mail handling, technical and aircraft equipment, passenger and baggage handling, weather*, air traffic control restrictions, miscellaneous

U 0 Air traffic control

restrictions 1 Weather# Peak/off-peak hour U U 0 Peak hour 1 Off-peak hour Season U U 0 Spring

1 Summer, fall, winter Notes: *_{Including ground handling impaired by adverse weather conditions,} weather at departure airport, weather en-route, and weather at destination or alternative airport.#_{Including“only” weather en-route and weather at destination} or alternative airport. Dummy code 0: Base type.

Table 4

Results of departure delay model.

Factor b (eb1)  100%

Turnaround buffer time 0.004a _0.4%

Aircraft type 0.445a _35.9%

Category of delay cause

Flight operations and crewing 0.539 41.7%

Cargo and mail handling 0.749a _52.7%

Technical and aircraft equipment 0.622a _46.3%

Passenger and baggage handling 0.511a _40.0%

Weather 0.575a _43.7%

Air traffic control restrictions 0.294 25.4%

Miscellaneous 0.316 27.1%

a_{Statistically significant at 5% level.}

Table 5

Results of arrival delay model.

Factor b (eb1)  100%

Block buffer time 0.066 (0.000)a _6.8%

Category of delay cause

Weather 0.957 (0.000)a _61.6%

LR¼ 294.3, p-value ¼ 0.000.

a_{Statistically significant at significance levela¼ 0.05.} Table 2

Difference test of survival curves.

Factor Departure/Arrival delay Chi-square

Aircraft type Departure delay 158.2***

Arrival delay 107.4***

Route Departure delay 305.3***

Arrival delay 271.1***

Delay cause Departure delay 373.8***

Arrival delay 10.1***

Peak/off-peak hour Departure delay 11.8**

Arrival delay 12.3**

Season Departure delay 7.6***

Arrival delay 10.7***

Significance levels 0%***, 0.1%**, 1%*.

arrival delay models,‘weather’ is the cause of delays that tends to result in longer departure and arrival delays.

6. Conclusions

When irregularities occur, airlines might need to provide additional resources to resume normal operations, resulting in extra operating expenses. Planning a schedule control program that allows greater schedule flexibility and reliability against disruptions is a factor in reducing the problem offlight delays. Due to the stochastic characteristics of aircraft rotations, there has been a great deal of discussion on how to reduce flight delays while maximizing the utilization of aircraft with very tight connections between_{flights. If a flight schedule is, however, only} designed to absorb stochastic delays without addressing the root problem offlight delays, the schedule might not be adequately robust for future operations. Here we investigated the factors behind the mechanisms of departure and arrival delays to clarify the phases and activities involved in flight delays through an airline schedule.

The models for departure and arrival delay developed are able to capture the dynamic characteristics offlight delays and differ from the methods used in previous studies such as simulation models or statistical analyses. Cox regression analysis reveals that the key contributing factors of departure delays include‘turnaround buffer time’, ‘aircraft type’, ‘cargo and mail handling’, ‘technical and aircraft equipment_{’, ‘passenger and baggage handling’, and} ‘weather’, whilst the key contributing factors of arrival delays include ‘block buffer time’ and ‘weather’. The hazard ratios

obtained enable airlines to examine the chances of recovering from flight delays. This provides airlines the direction of how to allocate resources to maintain a well-designed schedule.

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