- Research
- Open Access
Framework for determining airport daily departure and arrival delay thresholds: statistical modelling approach
- Ronald Wesonga^{1}Email author and
- Fabian Nabugoomu^{2}
- Received: 23 November 2015
- Accepted: 17 June 2016
- Published: 8 July 2016
Abstract
The study derives a framework for assessing airport efficiency through evaluating optimal arrival and departure delay thresholds. Assumptions of airport efficiency measurements, though based upon minimum numeric values such as 15 min of turnaround time, cannot be extrapolated to determine proportions of delay-days of an airport. This study explored the concept of delay threshold to determine the proportion of delay-days as an expansion of the theory of delay and our previous work. Data-driven approach using statistical modelling was employed to a limited set of determinants of daily delay at an airport. For the purpose of testing the efficacy of the threshold levels, operational data for Entebbe International Airport were used as a case study. Findings show differences in the proportions of delay at departure (μ = 0.499; 95 % CI = 0.023) and arrival (μ = 0.363; 95 % CI = 0.022). Multivariate logistic model confirmed an optimal daily departure and arrival delay threshold of 60 % for the airport given the four probable thresholds {50, 60, 70, 80}. The decision for the threshold value was based on the number of significant determinants, the goodness of fit statistics based on the Wald test and the area under the receiver operating curves. These findings propose a modelling framework to generate relevant information for the Air Traffic Management relevant in planning and measurement of airport operational efficiency.
Keywords
- Airport
- Delay threshold
- Efficiency
- Statistical models
- Receiver operating curves
Background
Airport delay computations are often construed to suit different definitions (Madas and Zografos 2008). Some definitions include aircraft turn-round time, whereas others exclude it. The problem is even larger when one desires to assess daily efficiency of an airport. Many studies have been conducted with the purpose of assessing efficiencies of operations at an airport. In their study of the factors for delays at European airports relative to the airports of the United States of America (Santos and Robin 2014) found that while delays were higher at hub airports, hub airlines experienced lower delays than non-hub airlines. A similar study (Liu et al. 2014) found that there was 30 % greater traffic at airports in the United States of America airports than at European airports that explained more delay at such airports. However, none of the studies considered optimal delay thresholds and its effect on drawing such important conclusions about levels and differences between airports. In his recent study (Wesonga 2015) published the first study that attempted to analyse delay thresholds at airport.
This study introduces the concept of threshold to be employed so as to determine the minimum acceptable proportion above which a day is declared a delay-day at an airport. This study is based on our previous work (Wesonga et al. 2012).
In this paper, data modelling was performed through algorithm design to determine an acceptable threshold for airport delay day (Wong and Tsai 2013; Autey et al. 2013). Furthermore, data modelling was done to a limited set of determinants of delay at an airport for the purpose of testing the efficacy of the threshold levels (Wang et al. 2012; Agustin et al. 2012), using Entebbe International Airport as a case study.
Data and methodology
Daily data for aviation and meteorological study parameters for the period 2004 through 2008
Parameter no. | Parameter | Variable type | Daily aggregated data range | |
---|---|---|---|---|
Minimum value | Maximum value | |||
1 | Air temperature | Scale, continuous | 19 | 25 |
2 | Aircraft arriving on time (%) | Scale, discrete | 1 | 42 |
3 | Aircraft delaying arrival (%) | Scale, discrete | 0 | 93 |
4 | Aircraft delaying departure (%) | Scale, discrete | 10 | 89 |
5 | Aircraft on-time departure (%) | Scale, discrete | 0 | 81 |
6 | Chartered flights | Scale, discrete | 0 | 50 |
7 | Dew point temperature | Scale, continuous | 16 | 21 |
8 | Freighters | Scale, discrete | 0 | 12 |
9 | Non-commercial flights | Scale, discrete | 0 | 57 |
10 | Persons on board-in | Scale, discrete | 138 | 3128 |
11 | Persons on board-out | Scale, discrete | 130 | 3277 |
12 | Queen’s nautical height | Scale, continuous | 975 | 1098 |
13 | Scheduled flights | Scale, discrete | 5 | 55 |
14 | Visibility | Scale, continuous | 7558 | 9999 |
15 | Wind direction | Scale, discrete | 107 | 329 |
16 | Wind speed | Scale, discrete | 2 | 9 |
For each day at an airport, there are registered levels of delay. These vary in proportions over time and would be misleading if one performed analysis based on the consideration that any positively registered delay at an airport is actually a delay in its real sense. Some delays are meant to enable an aircraft perform more efficiently throughout its trajectory with minimum disturbances and distortions such as being re-routed through other airports or even being cancelled. Therefore, if not all delays are bad in the real sense, a question of what proportion of delay should be treated as a threshold for computational and modelling purpose became eminent and a subject for this study.
Statistical model framework
Modelling was premised on the fact that different levels of thresholds could dynamically affect the statistical significance of determinants for airport delay. The question of their levels of influence was studied using generalised linear models as demonstrated in Eqs. (1), (2) and (3).
Logistic regression model with dummies ‘0’ for airport’s daily on-time performance while and ‘1’ for daily airport delay, constituted the dependent variable (Konishi and Kitagawa 2007; Nerlove and Press 1973). Determining what threshold to apply in this generalised linear modelling was an area of interest for this study. An aircraft is said to have delayed if the difference between the actual and scheduled times of arrival or departure were positive. In this study, a value for the dependent variable change based on what threshold is applied. The threshold start point was a proportion of 1 % and the ultimate being 100 % which implied that on any given day for any reporting based on the chosen proportion (1 through 100 %) of delay, such a day would be classified as a delay-day (DD) otherwise not-delay-day (NDD). Note that the daily proportions of delay were obtained by dividing the number of aircrafts that delay their operation by the total number for such an operation multiplied by one hundred; the operations could be departures or arrivals.
Furthermore, a logistic regression model, known to estimate the probability with which a certain event would happen or the probability of a sample unit with certain characteristics expressed by the categories of the predictor variables, to have the property expressed by the value 1 representing an airport’s delay day was employed. Estimation of the probability was done by the logistic distribution as in Eq. (2), where β’s are the regression coefficients of the categories to which the sample unit belongs.
Note that the values \(0 \le \pi \left( {{\text{X}}_{\text{i}} } \right) \le 1\) represent the probability of delay-day based on a set of meteorological and aviation parameters as shown in Table 1.
Since the logistic regression model is known to exhibit a curve rather than a linear appearance, the logistic function implied that the rate of change in the odds \(\uppi ( {\text{X}}_{\text{i}} )\) per unit change in the explanatory variables \({\text{X}}_{\text{i}}\) varied according to the relation \(\frac{{\partial {\pi (}{\text{X}}_{\text{i}} \text{)}}}{{\partial \text{(}{\text{X}}_{\text{i}} \text{)}}} =\upbeta_{j} [{\pi (}{\text{X}}_{\text{i}} \text{)}]\,[1 - {\pi (}{\text{X}}_{\text{i}} \text{)}]\). For example, if the odds of the proportion of delay \(\uppi\left( {{\text{X}}_{\text{i}} } \right) = \frac{1}{2}\) and the coefficient of the number of ‘scheduled flights’ \(\upbeta = 0.46\), then the slope \(\frac{{\partial {\pi (}{\text{X}}_{\text{i}} \text{)}}}{{\partial \text{(}{\text{X}}_{\text{i}} \text{)}}} = 0.46 * \frac{1}{2} * \frac{1}{2} = 0.115\). The value 0.115 represents a change in the odds of departure delay, \(\uppi ( {\text{X}}_{\text{i}} )\) per unit change in the number of ‘scheduled flights’. In simpler terms, for every 100 scheduled flights at Entebbe International Airport, 11 delay to departure. The R platform for statistical computing scientists (Chambers 2008; Dalgaard 2008) was applied because of its known strengths in computing that include, but not restricted to: the most comprehensive statistical analysis package available because it incorporates all of the standard statistical tests, models and analyses, as well as provides a comprehensive language for managing and manipulating data.
Findings and discussions
Data structure
Descriptive statistics for the dependent dummy threshold levels
Descriptive statistics for candidate threshold dummy variables
Departure delay thresholds | Arrival delay thresholds | |||||||
---|---|---|---|---|---|---|---|---|
dT50 | dT60 | dT70 | dT80 | aT50 | aT60 | aT70 | aT80 | |
Mean | 0.945 | 0.499 | 0.267 | 0.051 | 0.829 | 0.363 | 0.182 | 0.044 |
Standard error | 0.005 | 0.012 | 0.010 | 0.005 | 0.009 | 0.011 | 0.009 | 0.005 |
Standard deviation | 0.229 | 0.500 | 0.442 | 0.221 | 0.377 | 0.481 | 0.386 | 0.206 |
Sample variance | 0.052 | 0.250 | 0.196 | 0.049 | 0.142 | 0.231 | 0.149 | 0.042 |
Kurtosis | 13.187 | −2.002 | −0.884 | 14.533 | 1.050 | −1.679 | 0.715 | 17.654 |
Skewness | −3.895 | 0.005 | 1.057 | 4.064 | −1.746 | 0.568 | 1.647 | 4.431 |
Sum | 1726 | 911 | 487 | 94 | 1514 | 664 | 333 | 81 |
Count | 1827 | 1827 | 1827 | 1827 | 1827 | 1827 | 1827 | 1827 |
Confidence level (95.0 %) | 0.010 | 0.023 | 0.020 | 0.010 | 0.017 | 0.022 | 0.018 | 0.009 |
From Table 2, examining the candidate thresholds for departure delay descriptive statistics, for one to get an unbiased threshold, it was desirable that the statistics point at the middle values as much as possible. In the event that there was no one candidate presenting the desired exact middle values, then the threshold candidate with values approximating the middle characteristics was preferred. Therefore, preliminary findings in this study based on the actual operational data at Entebbe International Airport both for departure (\(\bar{X} = 0.499; \,SE = 0.012)\) and arrival (\(\bar{X} = 0.363; \,SE = 0.011)\) delay thresholds propose for recommendation a delay thresholds of 60 % (Ivanov et al. 2012).
Algorithm for determination of thresholds for departure and arrival delays
Departure delay determinants
Model estimations based on four threshold levels for departure delay determinants
Characteristic | Adjusted odds ratio | |||
---|---|---|---|---|
Threshold 1 (50 %) | Threshold 2 (60 %) | Threshold 3 (70 %) | Threshold 4 (80 %) | |
Arrival threshold | 2.871** | 0.457** | 0.203** | 1.000 |
Arrival delay | 0.891** | 1.011 | 1.003 | 0.978 |
Aircraft operations | 0.541* | 1.288** | 1.810** | 4.815 |
Scheduled flights | 1.910* | 0.651** | 0.466** | 0.002 |
Chartered flights | 1.723* | 0.635** | 0.485** | 0.002 |
Freighters | 2.145** | 0.827** | 0.598** | 0.002 |
Non-commercial | 2.224** | 0.842** | 0.584** | 0.002* |
Persons outgoing | 0.999* | 1.000 | 1.001** | 1.002 |
Persons incoming | 1.001 | 1.000 | 1.000 | 0.999 |
Visibility | 0.999 | 0.999* | 1.000 | 1.000 |
Wind speed | 1.003 | 1.038 | 1.039 | 1.005 |
Constant | 4.794 | 62.914** | 1.710 | 0.339 |
Observations (N) | 1827 | 1827 | 1827 | 1827 |
Covariate patterns | 1827 | 1827 | 1827 | 1827 |
Pearson chi^{2} | 3312.400 | 1703.820 | 2000.380 | 1092.990 |
Prob > chi^{2} | 0.000 | 0.970 | 0.001 | 1.000 |
Area under ROC curve | 0.841 | 0.887 | 0.875 | 0.807 |
Arrival delay determinants model based on four threshold levels
Characteristic | Adjusted odds ratio | |||
---|---|---|---|---|
Threshold 1 (50 %) | Threshold 2 (60 %) | Threshold 3 (70 %) | Threshold 4 (80 %) | |
Departure threshold | 0.578 | 0.248** | 0.137** | 1.000 |
Departure delay | 1.012 | 1.087** | 1.028 | 0.936* |
Number of operations | 0.965 | 1.462** | 1.960** | 2.984 |
Scheduled flights | 1.035 | 0.577** | 0.000 | 0.000 |
Chartered flights | 1.052 | 0.650** | 0.001** | 0.000 |
Freighters | 0.883* | 0.534** | 0.001** | 0.000 |
Non-commercial | 1.057 | 0.660** | 0.001** | 0.000 |
Persons outgoing | 1.000 | 1.001 | 1.001* | 1.001 |
Persons incoming | 1.001* | 1.000* | 1.001** | 1.001 |
Visibility | 1.000 | 0.999* | 1.000 | |
Wind speed | 0.976 | 1.025 | ||
Constant | 1.097 | 21.430** | 4.618 | 0.880 |
No. of observations | 1827 | 1827 | 1827 | 1827 |
No. of covariate patterns | 1827 | 1827 | 1827 | 1827 |
Pearson chi^{2} | 1874.690 | 2648.530 | 1352.250 | 1119.900 |
Prob > chi^{2} | 0.161 | 0.000 | 1.000 | 1.000 |
Area under ROC curve | 0.679 | 0.882 | 0.802 | 0.844 |
Discussions and conclusions
We explored modelling approach premised on the binary logistic regression to determine a better level of delay threshold that optimally evaluates the dynamics of air traffic delay during departure and arrival at an airport (Santos and Robin 2010). Four different scenarios were evaluated for both cases of departures and arrivals. The study established that at Entebbe International Airport, departure delay threshold of air traffic flow operations of 60 % provided the best and stable model characteristics. Variations of levels of significance for parameters of delay were detected at different delay thresholds, thus generating different numbers of significant parameters. For example, in both Tables 4 and 5; sub-table (d) presented the worst levels of parameter sensitivity with the least number of significant variables while sub-table (b) provided more stable models in both cases (Wesonga and Nabugoomu 2014; Helmuth et al. 2011).
These findings are significance in two ways; first, to the air traffic flow managers that daily proportions of aircraft delay below the 60 % threshold level could be considered normal operations. Therefore, such daily delays may be attributed to normal airport operational such as the turn-around time before actual departures and arrivals. Secondly, to the other aviation stakeholders including air passengers, the higher threshold level would indicate inefficiency of traffic flows. Comparison of air traffic flow inefficiencies based on the findings for departures are in the threshold order of 60 %, then 70 % compared to arrival threshold of 60 % followed by 50 % indicating that traffic flow at arrival was less inefficient than that during departure since arrivals permitted lower threshold level than departures (Wesonga et al. 2013; Zheng et al. 2010).
Besides, comparing aircraft flow performance between daily departures and arrivals, this framework is candidate to providing methodology for assessment and ranking of airports based on their departure and arrival operational efficiency. Airports with derived higher delay thresholds would be assessed as operationally more inefficient than those with lower delay thresholds (Chou 2009; Wei et al. 2011). Therefore, a multi-airport analysis based on this framework is recommended as a possible area of further analysis and application of the derived framework of this study (Mukherjee and Hansen 2009; Bianco et al. 2001).
Declarations
Authors’ contributions
Both RW and FN conceived the idea and participated in the design. RW acquired the data, analysed and drafted the manuscript while FN revised and both RW and FN approved the final manuscript and agree to be accountable for all aspects of the work and jointly own the work. Both authors read and approved the final manuscript.
Acknowledgements
The authors are grateful to the management of the Civil Aviation Authority and Makerere University, School of Statistics and Planning for an enabling research environment.
Competing interests
Both authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Agustin A, Alonso-Ayuso A, Escudero LF, Pizarro C (2012) On air traffic flow management with rerouting. Part II: stochastic case. Eur J Oper Res 219(1):167–177. doi:10.1016/j.ejor.2011.12.032 View ArticleGoogle Scholar
- Autey J, Sayed T, El Esawey M (2013) Operational performance comparison of four unconventional intersection designs using micro-simulation. J Adv Transp 47(5):536–552. doi:10.1002/atr.181 View ArticleGoogle Scholar
- Bianco L, Dell’Olmo P, Odoni AR (2001) New concepts and methods in air traffic management. Springer, BerlinView ArticleGoogle Scholar
- Chambers JM (2008) Software for data analysis: programming with R. Springer, New YorkView ArticleGoogle Scholar
- Chou SH (2009) Computationally efficient characterization of standard cells for statistical static timing analysis. Thesis M. Eng., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer ScienceGoogle Scholar
- Dalgaard P (2008) Introductory statistics with R, 2nd edn. Springer, New YorkView ArticleGoogle Scholar
- Helmuth JA, Reboux S, Sbalzarini IF (2011) Exact stochastic simulations of intra-cellular transport by mechanically coupled molecular motors. J Comput Sci 2(4):324–334. doi:10.1016/j.jocs.2011.08.006 View ArticleGoogle Scholar
- Ivanov SV, Kosukhin SS, Kaluzhnaya AV, Boukhanovsky AV (2012) Simulation-based collaborative decision support for surge floods prevention in St. Petersburg. J Comput Sci 3(6):450–455. doi:10.1016/j.jocs.2012.08.005 View ArticleGoogle Scholar
- Konishi S, Kitagawa G (2007) Information criteria and statistical modeling. Springer series in statistics. Springer, New YorkGoogle Scholar
- Liu Y, Hansen M, Zou B (2014) Aircraft gauge differences between the US and Europe and their operational implications. J Air Transp Manag 29:1–10. doi:10.1016/j.jairtraman.2012.12.001 View ArticleGoogle Scholar
- Madas MA, Zografos KG (2008) Airport capacity vs. demand: mismatch or mismanagement? Transp Res Part A Policy Pract 42(1):203–226. doi:10.1016/j.tra.2007.08.002 View ArticleGoogle Scholar
- Mukherjee A, Hansen M (2009) A dynamic rerouting model for air traffic flow management. Transp Res Part B Methodol 43(1):159–171. doi:10.1016/j.trb.2008.05.011 View ArticleGoogle Scholar
- Nerlove M, Press J (1973) Univariate and multivariate loglinear logistic models. Rand, Santa MonicaGoogle Scholar
- Santos G, Robin M (2010) Determinants of delays at european airports. Transp Res Part B 44:392–403View ArticleGoogle Scholar
- Santos G, Robin M (2014) Determinants of delays at European airports. Transp Res Part B Methodol 44(3):392–403. doi:10.1016/j.trb.2009.10.007 View ArticleGoogle Scholar
- Wang S, Zhang Y, Zhang Z, Yu H (2012) Multi-objectives optimization on flights landing sequence at busy airport. J Transp Syst Eng Inf Technol 12(4):135–142. doi:10.1016/S1570-6672(11)60218-3 Google Scholar
- Wei Z, Kamgarpour M, Dengfeng S, Tomlin CJ (2011) A hierarchical flight planning framework for air traffic management. Proc IEEE 100(1):179–194. doi:10.1109/jproc.2011.2161243 Google Scholar
- Wesonga R (2015) Airport utility stochastic optimization models for air traffic flow management. Eur J Oper Res 242(3):999–1007. doi:10.1016/j.ejor.2014.10.042 View ArticleGoogle Scholar
- Wesonga R, Nabugoomu F (2014) Bayesian model averaging: an application to the determinants of airport departure delay in Uganda. Am J Theor Appl Stat 3(1):1–5. doi:10.11648/j.ajtas.20140301.11 View ArticleGoogle Scholar
- Wesonga R, Nabugoomu F, Jehopio P (2012) Parameterized framework for the analysis of probabilities of aircraft delay at an airport. J Air Transp Manag 23:1–4. doi:10.1016/j.jairtraman.2012.02.001 View ArticleGoogle Scholar
- Wesonga R, Nabugoomu F, Jehopio P, Mugisha X (2013) Modelling airport efficiency with distributions of the inefficient error term: an application of time series data for aircraft departure delay. Int J Sci Basic Appl Res (IJSBAR) 12(1):103–114Google Scholar
- Wong J-T, Tsai S-C (2013) A survival model for flight delay propagation. J Air Transp Manag 23:5–11. doi:10.1016/j.jairtraman.2012.01.016 View ArticleGoogle Scholar
- Zheng P, Hu S, Zhang C (2010) Chaotic phenomena and quantitative analysis of aftereffect delay spread time on flights. J Transp Syst Eng Inf Technol 10(4):68–72. doi:10.1016/S1570-6672(09)60054-4 Google Scholar