Universal patterns in passenger flight departure delays.

Departure delays are a major cause of economic loss and inefficiency in the growing industry of passenger flights. A departure delay of a current flight is inevitably affected by the late arrival of the flight immediately preceding it with the same aircraft. We seek to understand the mechanisms of such propagated delays, and to obtain universal metrics by which to evaluate an airline’s operational effectiveness in delay alleviation. Here we use big data collected by the American Bureau of Transportation Statistics to design models of flight delays. Offering two dynamic models of delay propagation, we divided all carriers into two groups exhibiting a shifted power law or an exponentially truncated shifted power law delay distribution, revealing two universal delay propagation classes. Three model parameters, extracted directly from dual data mining, help characterize each airline’s operational efficiency in delay mitigation. Therefore, our modeling framework provides the crucially lacking evaluation indicators for airlines, potentially contributing to the mitigation of future departure delays.


GENERAL MESSAGE OF 14 MAIN AIRLINES IN 2014
Tables of operation message of domestic passenger flights of 14 airlines in 2014  CCDFs for airlines in Group 1 (Fig. S1 -Fig. S10)                 In the year 2014, airline VX appears in Group 2. Due to such jumps, temporal variation behaviors of β 1 , m and λ interrupt at corresponding years as shown in Fig. S69.
As we argued in last section, β 1 in general measures inversely the ability of an airline to absorb the delay in the case of DTPD. From Fig. S66 -Fig. S68 we see that negative relations similar to formula (11) cover airlines in Group 1 from 2009 to 2013 except 2011, while those similar to formula (12) cover Group 2 for all 20 years. No correlation in Group 1 can be found in each panel with SPL before 2005, since only a representing point of airline AA appears. In the L(β 1 ) graphs after 2004, points of HA keep staying at the positions with lower levels of both L and β 1 . Its CCDF less deviates from a power-law meaning heterogeneity. In 2005 and 2006, points for airline F9 stay leftward, exhibiting more heterogeneity of their CCDFs. In addition, points for AA shift rightward in 2007 and 2008, exhibiting more homogeneity of their CCDFs. Their behaviors ruin the deterministic negative correlation between L and β 1 (see Fig. S64,Fig. S65 and Fig. S67). Actually, airlines MQ and EV (before its jumping in 2008) varies less than F9 and AA with either L or β 1 in the same panels. Moreover, in view of little transposition of points of HA which keep being isolated from the main strain line of statistical fitting in other years, no obvious negative correlation of L(β 1 ) could be found in Group 1 during 2005 to 2008(also 2011). While the relations L(β 1 ) of Group 2 behave in a negatively linear dependence except airline WN which does not follow the main strain relation and keeps staying at lower L (around 10 minutes) during 15 years (1995 − 2009). This is reminiscent of us with distinctive feature of its point-to-point operation strategy(which is also true for airline HA when only domestic flights are concerned). Detail investigation of such cases would be expected. It is noticeable that airline WN joins the main strain of negatively linear correlations L(β 1 ) from 2010 ( Fig. S66 -Fig. S68, and Fig.3A in the text), keeping its low delay -absorbing level but with increased β 1 from 2010 to 2014 in Group 2. Therefore, for majority of the airlines and for most years, a negative linear correlation between the average delay -absorbing L and β 1 is valid. We should conclude that, the more homogeneity of an airlines's DD distribution, the weaker ability for it to absorb immediate preceding delay with the same aircraft, in general.
In observation of the temporal variations(see Fig. S69 (a) and (b)), β 1 around 2005−2008 has higher or lower bumps compared with the previous values except B6 and UA, which means decreased absorption of delays by other 6 airlines. Simultaneous appearance of bumps of β 1 by AA, MQ, F9 and HA, especially F9 approaching HA, corresponds to the invalidation of negative linear correlation between L and β 1 . After 2008, F9 ( Fig. S69(a)) and WN ( Fig. S69(b)) continue their essential increase of β 1 . While after 2013, a new round of β 1 -increase happen to almost all airlines (except the new comer VX), which could predict similar situation Group 2 faced during last bump of β 1 . In contradiction, we can rank UA the best since it keeps lowest β 1 hence highest L in almost all years.
For airlines in group 2, FL, AS, US and OO have larger values β 1 from 2005 to 2008, which means decreased ability to absorb DTPD. More importantly, due to quicker drop to zero than the k(l) assumed in Model 2, critical delay λ plays more important role than β 1 or β 2 in CCDFs. Usually, larger λ and smaller m enable higher precision of the phenomenological models (see (c), (d), (e) and (g) of Fig.2A and Fig.2B in the text, for examples). Large m strongly reduces the effect of DTPD assumed in formula (1). Airlines EV and B6 after 2008 (see panels (a) and (e) of Fig. S27  -Fig. S31, Fig. S55 -Fig. S60, UA andWN during 2005 -2008 (see (b) and (d) of Fig. S23 -Fig. S27, Fig. S52  -Fig. S56), and OO in 2008 and 2013 (see panels (g) of Fig. S26 -Fig. S31, Fig. S55 -Fig. S60, respectively, serve the worse examples than other airlines during the same years, and in comparison with their own performance in other years.
Comparing panel (b) with (c) in Fig. S69, one can easily see that the levels of λ are always higher than corresponding β 1 in Group 2, and λ sometime share similar oscillating-increase tendency of β 1 , say, airlines UA, WN, B6 US, OO, and EV, for examples. Observation of the λ -variation can further divide the operation behaviors of the airlines into 4 subgroups: AS, FL and DL often oscillating at higher level above 80; UA, US and WN having gentle oscillations and increases often below 80; B6, EV, and OO having continuous valleys; and finally VX, the isolated new comer. Common behaviors within individual subgroups and difference between them characterized by both β 1 and λ are expected to be understood by investigation to detailed operations.
In addition to the λ -valleys they have, airlines B6, EV and OO have higher levels of m(see Fig. S69(c) and (d)), which means more chances to be affected by NPF and ITPD. Besides, UA and WN both undergo m -bumps after 2005, then decrease essentially after 2008, which indicates the decrease of effect of NPF and ITPD and growing operation order by DTPD from 2009. Moreover, both WN and US display stable levels of m for a few years, which indicates the operational stability of the airlines. Finally, m of all airlines drop down simultaneously from 2013, indicating the growing tendency of more DTPD in the flight delay events. * These two authors contributed equally † To whom correspondence may be addressed. Email: oldpigman1234@126.com, hes@bu.edu