Statistical Analysis of Intervals between Projected Airport Arrivals

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1 Statistical Analysis of Intervals between Projected Airport Arrivals Thomas R. Willemain Hui Fan Huaiyu Ma Department of Decision Sciences and Engineering Systems Rensselaer Polytechnic Institute Troy, NY USA Abstract The pattern of arrivals at an airport is a major influence on the efficiency of runway operations. The desired distribution of landing time intervals will depend on the runway configuration and the mix of aircraft types but will nevertheless be as regular as possible. A purely random arrival stream has an exponential distribution of intervals between arrivals, so the exponential distribution is a meaningful benchmark against which to measure the disorder of the stream. We examined data on all arrivals to nine major US airports during December 2003 for evidence of exponentiality in the distribution of estimated times of arrival as estimated when the aircraft were 100 miles from their destinations. Testing for exponentiality in projected interarrival times is complicated by the fact that arrival rates vary substantially over time. We analyzed the data two ways to neutralize this complication. First, we divided the data at each airport into hour-by-dayof-week subsets and computed two summary statistics related to the shape of the distribution: the coefficient of variation and the coefficient of skewness. Second, we developed a new methodology that allowed pooling of all interval data for estimation of the shape parameter of a Weibull distribution, of which the exponential is a special case with shape parameter unity. This method requires only the assumption that the mean interval changes relatively slowly, so that successive intervals can be regarded as having the same average value. Both these analyses confirmed the nearly-exponential character of the intervals between projected ETAs. This means that the combination of the schedules set by the airlines and the handling of aircraft by en route air traffic controllers results in an arrival stream that is close to random as the aircraft near their destination. This places a burden on the controllers who handle arrivals in the final 100 miles of their flights. 1

2 1. Introduction The pattern of arrivals at an airport is a major influence on the efficiency of runway operations. Considering the runway as a server and arriving aircraft as customers, classical queueing theory posits a tradeoff between runway utilization and airborne delay, which is aggravated if there is a greater degree of disorder in the timing of arrivals. When aircraft approach an airport, there are periods when more than one simultaneously desire access to the runway, alternating with periods when the runway sits idle. There are many sources for this unevenness in demand. Perhaps the most important is the schedules designed by the air carriers, who may independently plan landings at the same or nearly the same times. Even when the schedule does not result in simultaneous arrivals, various delays in the national airspace system (NAS) can scramble the schedule and create contention for access by the time aircraft actually arrive at their destination. The Federal Aviation Administration (FAA) is charged with the safe and expeditious movement of air traffic. Both en route and terminal area FAA air traffic controllers influence the flow of traffic into the destination airport. The FAA has an interest in creating more orderly flows, and deploys software systems, such as TMA (Traffic Management Advisor) to assist controllers with sequencing arrivals and assigning aircraft to runways when more than one is available. Recently, large portions of the FAA were reorganized into a performance based organization (PBO), called the Air Traffic Organization (ATO). Consistent with the philosophy of a PBO, the ATO is working to establish performance based incentives, with separate portions to reward efficiency in en route and terminal operations. At the same time, FAA staff charged with performance analysis realized that en route and 2

3 terminal operations interact. They decided that the linkage between en route and terminal operations should be documented, leading to the study of which this paper is a part. In this paper, we analyze FAA data on arrival times to statistically characterize the flow of traffic into selected airports. The arrival stream varies by day of week, time of day, and airport, so we consider these factors in our analysis. Of particular interest is whether the times between expected arrivals have an exponential distribution. If so, the positive implication is that it will be easy to generate simulated arrivals for queueing analyses of airborne delays and runway operations. The negative implication is that the air traffic control system must work to convert the arrival stream from one that is purely random to one that is much more orderly and easy to manage. Our technical approach follows two tracks. Both are responses to the fact that airport arrival rates vary widely over time. The first track subsets the data by hour of day and day of week, then plots summary statistics against an exponential reference. The second track develops a new method that compensates for the time variation in demand, allowing all the data for an airport to be pooled for a more powerful analysis that fits a Weibull distribution to the data, looking to see if the Weibull shape parameter is unity, which would indicate an exponential distribution. 2. Data We were given FAA data on operations at major US airports in December The data included information on the identification and origin of each arrival and its estimated time of arrival (ETA). The ETA was computed when the aircraft was 100 nautical miles from the destination airport by the ETMS (Enhanced Transportation Management System) 3

4 algorithm in the FAA Host Computer in Cambridge, MA. This figure was intended to estimate the unimpeded time to the runway, in order to facilitate queueing analysis of airport traffic. Because of the various maneuvers, spacing requirements, and delays that can occur in the final 100 miles of a flight, the ETA at 100 miles is not necessarily a good forecast of the actual arrival time (Venkatakrishnan et al. 1993, Ballin and Erzberger 1998). We selected for analysis data from nine airports: Atlanta (ATL), Boston (BOS), Dallas (DFW), Detroit (DTW), Los Angeles (LAX), New York (LGA), Chicago (ORD), Phoenix (PHX) and Seattle (SEA). This selection provided information on a range of airport sizes and locations. This is not a random sample of airports. ORD, ATL and DFW were chosen because they are the busiest airports. The other six airports were chosen because they were of interest to FAA staff with respect to other aspects of the larger project. 3. Methodology It is commonplace in analyses of the NAS to study arrival counts. For instance, Figure 1 shows hourly counts for flights arriving at ATL. Clearly hour of the day is a major determinant of average arrival rate. Day of week also has some influence, though the importance of day of week varies with choice of airport. Arrival demand information is available to the public in real time ( as shown in Figure 2. Our interest in this paper goes beyond estimating average arrival rates to understanding the details of the airport arrival stream as a stochastic process. For this, we turn to understanding the distribution of intervals between (predicted) arrivals, rather than 4

5 the distribution of counts of arrivals in fixed intervals. Queueing theory relates the behavior of queues to the distribution of intervals between arrivals: fixed intervals are the easiest to handle, exponentially distributed intervals represent a purely random arrival process, and some more variable distributions produce worse than random performance. Because it represents a purely random arrival process, the time-varying Poisson distribution, and its counterpart time-varying exponential distribution, have long been key baselines for the study of airport traffic (Gulliher and Wheeler 1958, Herbert and Dietz 1997, Rue and Rosenshine 1997). Of particular interest, therefore, is the shape of the distribution of intervals, and whether the shape is well-approximated by an exponential distribution, albeit one with a mean value that changes over time. Because arrivals to major airports combine streams from many origins and several directions, nature conspires to render the combined arrival stream random. It is well known (Cox and Lewis 1966, chapter 8) that combining many random streams, each of about the same intensity and all approximately independent, produces a combined stream that is asymptotically a Poisson process. 3.1 Analysis by hour of day To study the time-varying distribution of intervals for our nine airports, we first divided the data for December 2003 into subsets by hour of day and day of week. To keep reasonable cell counts, we included only arrivals during the busiest part of the day, from 8 am to 8 pm local time. By focusing on data from a single hour in a single day of the week, we go some way toward controlling for the complication that the mean interval between arrivals varies both within and across days. 5

6 The choice of a bin width of one hour is a compromise. Narrow bins track the varying arrival rate more accurately, but they make statistical estimation more difficult by reducing the number of intervals contained in each bin. Essentially, this is a tradeoff between bias reduction, which favors narrow bins, and variance reduction, which favors wider bins. An additional factor favoring wider bins is that the resulting analysis produces fewer numbers to understand. However, the large variations within 15 minute intervals that appear in Figure 2 suggest that bins of 15 minutes would be desirable if sufficient data were available. One approach to testing exponentiality is compare summary statistics describing distribution shape with the corresponding parameter values for an exponential distribution. For the intervals within each hourly bin, we computed two scale-free measures influenced by distribution shape: the coefficient of variation or CV (i.e., standard deviation divided by the mean) and the skewness. (We did not compute the kurtosis because of its great variability in small samples.) The CV measures the relative variability of the intervals, while the skewness measures the degree of asymmetry. As a reference, for an exponential distribution the CV is 1.0 and the skewness is 2.0. We plot the two summary statistics of shape against these reference values for each hour and day of the week at each airport and informally assess the match to any exponential distribution. 3.2 Pooled analysis Our second look at the data involved a new methodology that eliminates the need to divide the data into subsets. By using ratios of successive intervals, it produces a scalefree transformation of the data, thereby allowing data from all hours and days to be 6

7 pooled into a single large dataset for analysis. Large datasets allow much more powerful tests of the hypothesis that intervals have an exponential distribution. The new method assumes that intervals are independent samples from a Weibull distribution with a fixed shape parameter (equal to 1 for an exponential distribution) and a slowly varying scale parameter. Given that the mean interval in our datasets changes over a scale of tens of minutes but typical intervals are on the scale of minutes, it would appear that the FAA data satisfy the requirement that the scale is slowly varying. To explain the new method, denote the j th interval by X j, j = 1..m. For convenience, assume the number of intervals m is even. By hypothesis, X j is a random variable independent of other intervals and Weibull distributed with fixed shape parameter α > 0 and scale parameter β > 0. Form the series of ratios R i = X 2i /X 2i-1, i = 1..n = m/2. Note that successive ratios are based on nonoverlapping pairs of intervals, thus buttressing the assumption of independence. When the scale parameter is constant (or varies sufficiently slowly), both the numerator and denominator have (very nearly) the same mean, so the ratio is (very nearly) scale free. The method exploits a known result for the ratio of independent random variables (Kendall et al. 1987, p. 387). Let X 1 0 have density function f 1 () and distribution function F 1 (). Let X 2 0 have density function f 2 (). Then the transformation of variables R = X 2 /X 1, S = X 1 leads to the following expression for the distribution function of the ratio H(R): H(R) = F 1 (RS)f 2 (S)dS, S = 0.., R 0. (1) For a Weibull distribution F(x) = 1 exp(-(x/β) α ). (2) 7

8 If the first interval in a pair has scale parameter β 1 and the second has β 2, then substituting (2) into (1) yields H(R) = (β 2 /β 1 ) α R α /(1+(β 2 /β 1 ) α R α ), R 0. (3) In the special case of two identical Weibull distributions, (3) has the particularly simple scale-free form H(R) = R α /(1+R α ), R 0. (4) Working with the ratio of successive intervals converts the difficult problem of testing for exponentiality when the mean interval is slowly varying into a standard problem, e.g., one can compare (4) with the empirical distribution function of the ratios in a Kolmogorov-Smirnov (K-S) goodness of fit test. More generally, one can determine the value of the shape parameter α that maximizes the goodness of fit in the K-S test and then test whether the empirical distribution of the ratios is significantly different from the bestfitting Weibull distribution. In theory, the K-S test requires a fully specified null distribution, with known values of all parameters, but it is commonly used with estimated parameters when sample sizes are large, as they are in our air traffic application. The empirical distribution of the ratios is estimated by H est (R i ) = (# data values R i )/n (5) 4. Results We first present the comparison of the summary statistics against the exponential norms for data divided by airport by hour by day of week. Then we present the results of pooling the data at each airport to estimate the Weibull shape parameter and compare it to the value for an exponential distribution. 8

9 4.1 Results for data by hour of day Table 1 shows, for each airport, the average, minimum and maximum count of intervals across hour of day and day of week. There were 13 hours x 7 days = 91 subsets of data for each airport. Typically, from 20 to 100 intervals were included in each subset. Figures 3 11 plot the coefficients of variation and skewness across the 91 subsets for each of the nine airports. There was no obvious temporal pattern to the coefficients (e.g., weekdays versus weekends, early versus late). There was a good deal of variability across hour-by-day-of-week subsets, reflecting some combination of natural variability and relatively small sample sizes. Averaging across the subsets, Table 2 summarizes the mean values of the two coefficients. Only PHX had mean coefficients indistinguishable from those of an exponential distribution. For the most part, both coefficients fell somewhat below the reference values for an exponential distribution. This means that the actual intervals were close to exponential, but somewhat more regular, i.e., less random, in character. The only exception was DTW, which had mean coefficients suggesting even greater variability than the exponential. We suspect that the anomaly for DTW might be an artifact of the mixture of distributions whose mean changes significantly over the course of one hour (see Figure 2) and put greater trust in the pooled analysis described next. 4.2 Results for pooled data Table 3 summarizes the results of fitting a Weibull distribution to the pooled ratio data at each airport distribution. The fitted shape parameter values were those that minimized the maximum deviation between the theoretical distribution function (4) and the empirical distribution (5). 9

10 For all nine airports, the estimated shape parameter was slightly larger than the value of 1 defining an exponential distribution. The empirical distributions were not significantly different from the fitted Weibull distributions for any of the airports at the 0.01 level of significance, and for eight of the nine airports at the 0.05 level. The distributions closest to exponential in shape were at ATL, DTW and ORD (α = 1.03), while the least exponential distributions were at BOS (α = 1.17) and LGA (α = 1.15). In all cases, there was enough data to distinguish the distributions from the exponential benchmark, but the deviations from the exponential shape were relatively subtle. 5. Summary and Conclusions We examined intervals between estimated times of arrival (ETAs) computed when arriving aircraft are 100 nautical miles from their destination. The data included all arrivals to nine major US airports during December Essentially, this distribution represents the raw material with which the final en route controller and the approach controller must work to shape a more orderly arrival flow. The ETAs are determined by the airlines schedules and by gate, taxi-out and en route delays encountered before reaching a distance of 100 miles from the destination airport. If the traffic stream arriving at an airport is purely random in time, then the air traffic control system must work hard to transform the flow into a more orderly sequence. Otherwise, arriving aircraft will not make efficient use of the destination runway(s) and will endure avoidable airborne delay. A purely random arrival stream has an exponential distribution of intervals between arrivals. The desired distribution of ultimate landing time intervals will depend on the current runway configuration and the mix of aircraft types but will nevertheless be 10

11 much more regular than the exponential distribution. (Venkatakrishnan et al.1993). A purely random arrival stream is therefore a meaningful benchmark against which to measure the disorder of the stream. Testing for exponentiality in the distribution of projected interarrival times is complicated by the fact that arrival rates vary substantially by hour of day and, to a lesser extent, by day of week. We analyzed the data two ways to neutralize this complication. First, we divided the data at each airport into subsets by hour of day and day of week and computed two summary statistics related to the shape of the distribution: the coefficients of variation and skewness. We found no obvious temporal patterns in the coefficients. We did find that the mean values of the coefficients were generally consistent with intervals slightly less random than the exponential reference distribution. Second, we developed a new methodology that allowed pooling of all interval data and fitting to a Weibull distribution, which includes the exponential as a special case. This method requires only the assumption that the mean interval change relatively slowly, so that successive intervals can be regarded as having the same average value. These analyses confirmed that the distributions of intervals are not quite exponential. Clearly, it would ease the work of the final air traffic controllers if the distribution of arrivals were more orderly. However, the burden of moving toward a more regular flow must be shared by the airlines, who design the original schedules, and by the en route controllers who bring the aircraft to the 100 mile mark. Further research might focus on what point along the flight path holds the greatest potential for improvement. But one must also take account of the limits created by the fundamental phenomenon that 11

12 superposition of several streams tends to produce a combined stream that is purely random. Acknowledgements This work was supported by the U. S. Federal Aviation Administration. The authors gratefully acknowledge the data, direction and insight received from Dave Knorr and Ed Meyer of the FAA and Dr. James Bonn of CNA Corporation. Professor Pasqualle Sullo provided useful guidance on the new pooling method for testing exponentiality. All statements in this paper are the responsibility of the authors and do not represent official FAA positions. 12

13 References Ballin, M. and Erzberger, H Potential benefits of terminal airspace traffic automation for arrivals. Journal of Guidance Control and Dynamics, 21, Cox, D. and Lewis, P The Statistical Analysis of Series of Events. Chapman and Hall. Gulliher, H. and Wheeler, R Nonstationary queueing probabilities for landing congestion of aircraft. Operations Research, 6, Herbert, J. and Dietez, D Modeling and analysis of an airport departure process. Journal of Aircraft, 34, Kendall, M., Stuart, A., and Ord, J Kendall s Advanced Theory of Statistics, Vol. 1. Oxford. Rue, R. and Rosenshine, M The application of semi-markov decision processes to queueing of aircraft for landing at an airport. Organization Science, 8, Venkatakrishnan, C., Barnett, A., and Odoni, A Landings at Logan Airport: Describing and increasing airport capacity. Transportation Science, 27,

14 120 Mean Arrival Rate vs Hour by Day of Week, ATL, Dec Aircraft per Hour Mon Tue Wed Thu Fri Sat Sun Figure 1: Mean arrival rate varies by hour of day and day of week Figure 2: Real time data on airport arrivals at DTW, 22Jun04, by 15 minute intervals 14

15 Figure 3: Summary statistics by hour of day and day of week, December 2003, ATL 15

16 Figure 4: Summary statistics by hour of day and day of week, December 2003, BOS 16

17 Figure 5: Summary statistics by hour of day and day of week, December 2003, DFW 17

18 Figure 6: Summary statistics by hour of day and day of week, December 2003, DTW 18

19 Figure 7: Summary statistics by hour of day and day of week, December 2003, LAX 19

20 Figure 8: Summary statistics by hour of day and day of week, December 2003, LGA 20

21 Figure 9: Summary statistics by hour of day and day of week, December 2003, ORD 21

22 Figure 10: Summary statistics by hour of day and day of week, December 2003, PHX 22

23 Figure 11: Summary statistics by hour of day and day of week, December 2003, SEA 23

24 Table 1: Summary statistics for counts of intervals per hour by airport Airport Mean Minimum Maximum ATL BOS DFW DTW LAX LGA ORD PHX SEA Table 2. Mean values of coefficients of variation and skewness by airport Airport Mean of CV 95% LCL 95% UCL Mean of Skewness 95% LCL 95% UCL ATL BOS DFW DTW LAX LGA ORD PHX SEA Table 3: Fitted Weibull shape parameters and significance of test for Weibull distribution with estimated shape Airport # ratios Shape Significance ATL 19, p > 0.05 BOS 3, p > 0.05 DFW 16, < p < 0.05 DTW 9, p > 0.05 LAX 12, p > 0.05 LGA 7, p > 0.05 ORD 19, p > 0.05 PHX 10, p > 0.05 SEA 6, p >