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Airline Mitigation of Propagated Delays via Schedule Buffers: Theory and Empirics

Abstract This paper presents an extensive theoretical and empirical analysis of the choice of schedule buffers by airlines. With airline delays a continuing problem around the world, such an undertaking is valuable, and its lessons extend to other passenger transportation sectors. One useful lesson from the theoretical analysis of a two-flight model is that the mitigation of delay propagation is done entirely by the ground buffer and the second flight’s buffer. The first flight’s buffer plays no role because the ground buffer is a perfect, while nondistorting, substitute. In addition, the apportionment of mitigation responsibility between the ground buffer and the second flight's buffer is shown to depend on the relationship between the costs of ground- and flight-buffer time. The empirical results show the connection between buffer magnitudes and a host of explanatory variables, including the variability of flight times, which simulations of the model identify as an important determining factor.

arrival, it also makes an early arrival more likely, and while passengers dislike delays, they also do not want flights to routinely arrive early, an outcome that leaves time gaps that must be filled. Longer scheduled flight times also raise the airline's (planned) operating costs, such as the cost of crew time. In setting its flight schedules, an airline will take all three factors (disutilities from lateness and earliness as well as operating costs) into account. Flight buffers are typically positive, reflecting a greater concern about late as opposed to early arrivals on the part of the carrier (following passenger preferences). Note that these same three elements also affect scheduling by other transportation providers, such as passenger railroads and intercity bus lines.
Delays depend on more than just the operating time of a particular flight. If the incoming aircraft arrives late, then the outbound flight is likely to depart late, possibly leading to its late arrival even if operating time is normal. Late-arriving aircraft are in fact the major source of flight delays, as seen in Figure 1, accounting for more delays than mechanical and crew-related delays ("air carrier delays") or weather. This type of delay is known as "propagated delay" since it propagates from one flight to another, and it is also present in the railroad and bus contexts.
Propagated delay can be addressed through a long flight buffer, which reduces the chance of a late-arriving aircraft, but another tool is the "ground buffer." This buffer is defined as the difference between the scheduled ground time and the minimum feasible aircraft turnaround time. A long ground buffer can absorb a late arrival of the inbound aircraft, allowing the next flight to depart on time despite this disruption. The flight buffer for the outbound flight can also address delay propagation, allowing the flight to arrive on time even if it departs late.
Lengthening this flight buffer is costly, however, and longer ground times are also costly since they require more gate space. 1 The purpose of the present paper is to analyze an airline's choice of flight and ground buffers, both theoretically and empirically. Our theoretical framework differs from most other models because it is stylized, rather than fully realistic, and starts from first principles, treating propagated delay in a setting with just two flights. Thus, the airline chooses two flight buffers and one ground buffer, taking passenger distuilities from lateness and earliness into account along with buffer costs. The analysis yields a number of insights. A principal result is that the flight buffer for flight 1 plays no role in mitigating delay propagation, which is instead handled by the ground buffer and flight 2's buffer. The reason is that, while flight 1's buffer and the ground buffer can both address delay propagation, the former distorts flight 1's scheduled The paper is related to several strands of previous work. Earlier papers in operations research, including Deshpande and Arikan (2012) and Arikan, Deshpande and Sohoni (2013), present models of the choice of flight buffers, noting the similarity to the classic newsvendor problem of Whitin (1955) (a flight's early (late) arrival is analogous to a vendor ending up with a surplus (shortage) of newspapers). Deshpande and Arikan (2012) consider US airlines and use the newsvendor approach to estimate the airlines' ratios of earliness to lateness costs.
They show that flight buffer choices depend on carrier types, route market shares, and route characteristics. Arikan, Deshpande and Sohoni (2013) develop schedule robustness measures 3 for airline networks and use them to show how US carriers use flight and ground buffers to absorb delay propagation. Zhang, Salant and Van Mieghem (2018) present an analysis related to those of Deshpande and Arikan (2012) and Arikan, Deshpande and Sohoni (2013). They show that the historical evolution of flight durations cannot explain increases in scheduled ground and flight times in the US and conclude that these increases instead have strategic motivations. Kafle and Zou (2016) theoretically decompose delays into propagated and newlyformed delays, and their empirical work investigates the determinants of propagated delays. 2 In the economics literature, Wang (2015) offers a different theoretical approach to the choice of ground buffers, relating this choice to the level of competition while providing empirical evidence of this link. In other empirical work by economists, Forbes, Lederman and Yuan (2019) use a much larger dataset to confirm the earlier finding of Shumsky (1993) showing that US carriers have added schedule padding (buffer time) over the years to improve their on-time performance. Forbes, Lederman and Wither (2019) show that this effect is stronger when airlines are large enough for required reporting of on-time performance, a criterion that excludes many regional carriers. 3 See also Hao and Hansen (2014), Kang and Hansen (2017), and Sohoni, Lee and Klabjan (2011) for related studies. Other studies by economists, which are not directly linked to our work, show the connection between market structure (mainly competition) and on-time performance (see, for example, Mazzeo (2003) and Prince and Simon (2015)).
The plan of the paper is as follows. Section 2 analyzes choice of the flight buffer in an introductory model with just one flight, where delay propagation is not an issue, and section 3 analyzes the two-flight model. Section 4 presents numerical examples, while section 5 offers a model extension that incorporates connecting passengers. Section 6 describes the empirical setup, and section 7 presents the regression results. Section 8 offers conclusions.

Buffer choice for a single flight
The analysis starts by considering the buffer choice for a situation where the airline operates only a single flight, denoted flight 1. After this analysis is complete, the focus turns to the case where two flights are operated. While delay propagation is not an issue in the single-flight case, it is a crucial factor when two flights are operated. It should be noted that the models we analyze are highly stylized. They are designed to expose the economic incentives faced by airlines in the choice of schedule buffers, without being fully realistic.
Flight 1 departs at time 0 and has an uncertain duration. If no outside influences affect the flight's operation, its duration is given by m 1 . Outside influences such as weather, mechanical issues, and unanticipated congestion can cause the flight duration to differ from m 1 , with the actual duration equal to m 1 + 1 , where 1 is a continuous random variable with support [ 1 , 1 ].
Bad weather would lead to a positive value for 1 , while a strong tail wind would lead to a negative value. Note that while the congestion effect is partly random, appearing in 1 , a persistently high level of airport congestion would lead to a large value for m 1 , the expected flight duration.
In scheduling the flight's arrival time, the presence of this random term will lead the airline to include a flight buffer, denoted b 1 , which is added to m 1 . The scheduled arrival time of the flight is thus t a1 = m 1 + b 1 , whereas the actual arrival time, which captures the random effect, is t a1 = m 1 + 1 . The flight is late in arriving if t a1 > t a1 , or if 1 > b 1 , and it is early if 1 < b 1 . Passengers dislike being late or early, valuing a minute of late time by the amount x and valuing a minute of early time by the amount y. Therefore, if the flight is late, it generates disutility equal to x( 1 − b 1 ), whereas disutility is y(b 1 − 1 ) if the flight is early.
The probability that flight 1 is late is equal to the probability that 1 exceeds b 1 , which is given by where f 1 is the density and F 1 the cumulative distribution function of 1 . Conversely, the probability of an early arrival is F 1 (b 1 ). The passenger's expected disutility from earliness or 5 lateness is equal to where the first integral captures lateness and the second captures earliness. Passenger disutility from late and early arrivals affects fare revenue, and assuming that the airline seeks to maximize profit, it takes into account this disutility. 4 But the airline also incurs planned operating costs from scheduled flight time and ground time. Scheduled flight costs include expenditures on fuel and crew salaries, while ground costs consist mainly of gate rental costs. To facilitate comparison with the two-flight model, suppose that the airline has leased the aircraft for a fixed period T , with scheduled flight time equal to m 1 + b 1 and ground time equal to T − (m 1 + b 1 ) > 0. Thus, the aircraft lease period more than covers the flight time, a situation we denote as "excess capacity" (see below for use of an alternate assumption). The leasing cost is fixed, but letting c f denote the cost per minute of scheduled flight time and c g denote the cost of ground time, total operating costs are c f (m 1 + b 1 ) + c g (T − (m 1 + b 1 )). The goal of the profit-maximizing airline is to minimize the sum of this expression and Ω 1 by choice of b 1 , which (ignoring constant terms) means minimizing Ω 1 from (2) plus b 1 (c f − c g ).
The first-order condition for this minimization problem is and the second-order condition, which requires (x + y)f 1 (b 1 ) > 0, is satisfied. Rearranging (3), 4 Letting G denote the benefit from travel, the net benefit from a flight, denoted V , is equal to G minus the distutility expression in (2). Letting C denote expenditure on a nontravel good, overall passenger utility equals U = C + V = Y − P + V, where Y is income and P is the fare for travel. The airline sets the fare at a level that makes the consumer indifferent between traveling and not traveling, which yields utility Y (in this case V − P = 0). The fare for a flight is then equal to V , or G minus (2), and assuming that the passenger count is normalized to 1, fare revenue equals this expression. the optimal buffer, denoted b * 1 , satisfies a formula similar to one derived by Deshpande and Arikan (2012). Differentiation of (4) yields with a greater disutility from lateness raising the buffer (thus reducing the chance of lateness) and a greater disutility from earliness reducing it. Similarly, a higher c f (c g ) reduces (increases) b 1 .
If the distribution of 1 is symmetric around zero, then F 1 (0) = 1/2. As a result, if (x + c g − c f )/(x + y) = 1/2, then b * 1 = 0. Rearranging, b * 1 > (<) 0 will then hold as (x − y)/2 > (<) c f − c g in this symmetric case. If x is much larger than y, as seems realistic, and c f is close to c g , then b * 1 is positive. This conclusion makes sense given that a much greater disutility from a late arrival will lead the airline to lessen the chance of this outcome via a positive buffer.

The setup
The airline is now assumed to operate two flights using the same aircraft, so that a delay for flight 1 can cause lateness of flight 2. While the existence of two flights introduces the possibility that some passengers connect from one flight to another, we assume initially that connections are absent, showing later in the paper how they affect the analysis.
Following the single-flight assumptions, m 2 denotes the undisrupted flight time for flight 2, with the actual flight time given by m 2 + 2 . The random term 2 has density f 2 , cumulative distribution function F 2 , and support [ 2 , 2 ]. The flight buffers are b 1 and b 2 , so that the scheduled arrival times of flights 1 and 2 are t a1 = m 1 + b 1 and t a2 = t d2 + m 2 + b 2 , where t d2 is flight 2's scheduled departure time. The scheduled aircraft ground time is denoted t g , and the ground buffer, given by b g = t g − t g , is the excess of ground time over the minimum feasible 7 aircraft turnaround time, denoted t g . The size of the ground buffer is thus set by choice of t g .
The scheduled departure time of flight 2 is then t d2 = t a1 + t g = m 1 + b 1 + t g , and the flight's scheduled arrival time is As before, the actual arrival time of flight 1 is t a1 = m 1 + 1 . The actual arrival time of flight 2 equals t a2 = t d2 + m 2 + 2 , where t d2 is the actual departure time of flight 2. To find t d2 , note that if flight 1 is late in arriving, then the ground time will be reduced below the scheduled time t g in an attempt to prevent delay propagation via late departure (and possible late arrival) of flight 2. However, ground time cannot be reduced below the minimum feasible time, equal to t g . Therefore, the actual departure time of flight 2 is given by Flight 2 departs on time if 1 + t g < b 1 + t g or Note that satisfaction of (8) is ensured if flight 1 is early, or if 1 < b 1 . But the inequality can also be satisfied when flight 1 is late provided that it is not too late. Using (8), the probability of an on-time departure for flight 2 is F 1 (b 1 + b g ), with late departure (which reverses the inequality in (8)) having probability 1 − F 1 (b 1 + b g ).
The formula for t d2 allows derivation of conditions for late arrival of flight 2. Using (6), flight 2 is late in arriving when If flight 2 departs on time, so that t d2 = m 1 + b 1 + t g from (7), then (substituting in (9)), it arrives late when 8 an outcome that has probability 1 − F 2 (b 2 ) (early arrival occurs when (10) is reversed and has probability F 2 (b 2 )). If flight 2 departs late, so that t d2 = m 1 + 1 + t g , then (substituting in (9)), it arrives late when With late departure, flight 2 arrives early when (11) is reversed. depends on the sum b 1 +b 2 +b g . Therefore, the ground buffer affects delay by altering this sum, conditional on a late departure for flight 2. However, b g also affects whether a late departure occurs via the direction of the inequality in the second column of the table. As will be seen, the overall impact of b g on the airline's objective function operates through both these channels.
Using (11) and the reverse of inequality (8), the probability of late departure and late arrival for flight 2 is given by Similarly, using the reverse of inequality (11) and the reverse of (8), the probability of late departure and early arrival for flight 2 is Combining all this information, the probability of a late arrival for flight 2 is given by Note that a conditional probability can be used in the first half of (14) because whether flight 2 is late conditional on an on-time departure (which depends only on 2 ) is independent of whether the departure is on-time (which depends only on 1 ). The absence of this kind of independence when flight 2 is late requires the different kind of expression in the last half of (14).
Similarly, the probability of early arrival for flight 2 is given by

The airline's objective function
The disutility for passengers of flight 1 is again given by in Ω 1 in (2). The expected disutility from late arrival of flight 2 is given by The first half of (16) captures disutility from a late arrival when flight 2 departs on time (note (14) is replaced by the integral). The rest of (16) captures late disutility when the departure is late (note that the bracketed term is added inside the integrand in (14)).
Similarly, the expected disutility from early arrival of flight 2 is The passenger-disutility portion of the airline's objective function, denoted by Ω, is given by the sum of (2), (16), and (17): In contrast to the single-flight case, our airline cost expression assumes the absence of excess capacity in the two-flight case, with the airline adjusting its aircraft lease to exactly cover the scheduled usage of the plane (see below for an alternate assumption). The airline's costs are then given by c f b 1 + c f b 2 + c g b g plus a constant involving the m i s, with the carrier paying for only the ground time allotted to the ground buffer and not for any time beyond the termination of flight 2. The airline minimizes the sum of Ω and this cost expression, and the derivatives of Ω with respect to b g , b 1 and b 2 are computed in the appendix. Adding the relevant buffer cost to these derivatives and setting the resulting expressions equal to zero yields the following first-order conditions: Note in (20) and (21) that ∂Ω/∂b 1 and ∂Ω/∂b 2 are equal to ∂Ω/∂b g plus the second term in the relevant equation.

Characterizing the optimum
The immediate implication of the first-order conditions is that the flight buffer b 1 has the same value as in the single-flight model. This conclusion can seen by using (19)  The lack of a delay-propagation role for flight 1's buffer makes sense. Even though b 1 and the ground buffer are, in effect, perfect substitutes in addressing delay propagation, use of the ground buffer instead of the flight buffer does not distort the balance between late and early disutilities for flight 1, making it the preferred tool. However, suppose the ground buffer were somehow constrained below its optimal value, due to a shortage of airport gate capacity, for example, which could be caused by hoarding of gates by a dominant airline (see Ciliberto and Williams (2010) for empirical evidence). In this case, b 1 would help to address delay propagation along with the other buffers. Such a constraint would make ∂Ω/∂b g in (20) less than −c g , causing b 1 to rise above its single-flight value, thus addressing delay propagation.
To see how the delay-propagation responsibility is apportioned between flight 2's buffer and the ground buffer, it is instructive to first consider the unrealistic case where c g = 0, with ground time being costless. When c g = 0, F 1 (b 1 + b g ) = 1 must hold at the optimum, so that the probability of late departure for flight 2 (which requires 1 > b 1 + b g ) equals zero. In effect, the lower limit of integration in (19) must be above the upper limit in order to make the integral zero. To establish this point, suppose to the contrary that F 1 (b 1 + b g ) < 1 is satisfied along with the first-order conditions. Since F 2 (b 1 + b 2 + b g − 1 ) ≤ F 2 (b 2 ) holds over the range of integration in (19), which is nonempty given the maintained assumption, it follows that where the last inequality follows because F 2 (b 2 ) − [x/(x + y)] < 0 by (21) (using ∂Ω/∂b g = 0 from (19)).

12
The inequalities in (22) contradict the assumption that (19) equals zero, ruling out the premise that F 1 (b 1 + b g ) < 1. Therefore, the ground buffer b g is set at a value large enough to eliminate the chance of late departure for flight 2, so that F 1 (b 1 + b g ) = 1. 5 Then, setting F 1 (b 1 + b g ) in (21) equal to 1, (21) matches the optimality condition (3) for the single-flight model when c g = 0, implying that the optimal b 2 equals the single-flight value.
Thus, when c g = 0, the flight buffer for flight 2 is set as if the flight were operating in isolation, with delay propagation not an issue. Since the same conclusion has already been established for flight 1, we can state Proposition 2. When ground time is costless, the ground buffer does all the work in mitigating delay propagation, fully eliminating it, with no contribution from flight 2's buffer.
Even though this is a natural conclusion, some work is required to derive it from the model, as seen in the previous discussion.
Now consider the realistic case where the cost of ground time is positive, with c g > 0.
Setting (19) equal to zero now implies that the integral must be negative, which means that the lower limit of integration cannot exceed the upper limit as before, leading to a zero value for the integral. In other words, F (b 1 + b g ) must now be less than rather than equal to 1, indicating that there is a chance of late departure for flight 2.
In this case, the mitigation of delay propagation is apportioned between the ground buffer and flight 2's buffer, with the exact apportionment depending on the relationship between c f and c g . Suppose first that c f < c g , so that the cost of the flight buffer is less than that of the ground buffer. Letting ** denote optimal values in the two flight model, it follows from (20) It should be noted that F 1 (b 1 + b g ) = 1 does not yield a unique solution for b g given that any b g value satisfying b 1 + b g ≥ 1 makes the equality true. However, replacing b 1 by b * 1 , the airline might be assumed to choose the smallest b g satisfying the equality, so that b * 1 + b g = 1 , yielding a unique solution given by With the last line of (23) thus positive, it follows that b * * 2 is larger than b * 2 , the single-flight value of b 2 , which makes the second line equal to zero. Since flight 2's buffer is thus larger than the value it would assume if the flight were operating in isolation, it follows that flight 2's buffer assists the ground buffer in addressing delay propagation. This conclusion is due to the relative cheapness of the flight buffer, which encourages its use in addressing propagation.
In the reverse case where c f > c g , reversal of the above argument yields b * * 2 < b * 2 , so that flight 2's buffer is less than its single-flight value. Now, relative cheapness of the ground buffer means that it takes extra responsibility in addressing delay propagation, causing flight 2's buffer to be reduced below its single-flight value. This adjustment means that the flight buffer now actually contributes to delay propagation, but this effect is offset by the greater role of the ground buffer. Finally, when c f = c g , flight 2's buffer equals its single-flight value (b * * 2 = b * 2 ), so that it neither helps nor offsets the ground buffer in mitigating delay propagation. With the buffers equally costly, the nondistorting ground buffer is thus set to do all the work in addressing delay propagation, although the chance of propagation is not reduced to zero given the costliness of the buffer. Summarizing yields Proposition 3. When c g > 0, mitigation of delay propagation is apportioned between the ground buffer and flight 2's buffer. When c f = c g , the ground buffer alone addresses delay propagation (while not fully eliminating it), with flight 2's buffer set at its singleflight value. When c f < (>) c g flight 2's buffer contributes to (partly offsets) the ground buffer's mitigation of delay propagation, taking a value above (below) its single-flight value.
The results yield a further implication in the case where the distributions of the random flight-duration terms are equal, allowing the 1 and 2 subscripts to be removed from the F functions. With common F 's, the single-flight values of b 1 and b 2 are the same. Then, the fact that b 2 is greater than (less than) the common single-flight value as c f < (>) c g means that flight 2's buffer is greater than (less than) flight 1's buffer, which always equals the common single-flight value, when c f < (>) c g . In other words, b * * 2 > (<) b * * 1 holds as c f < (>) c g . The reason is that flight 1's buffer plays no role in addressing delay propagation, while flight 2's buffer contributes to (borrows from) the ground buffer's delay-mitigation effect as c f < (>) c g .
The second-order conditions for the optimization problem have not been mentioned so far.
In the appendix, it is shown that, that if c f ≥ c g , then Ω is strictly convex in b 1 and b 2 at the optimum, so that conditional on b g , flight buffers satisfying the first-order conditions based on (20) and (21) yield a local minimum of Ω. However, it is not possible to establish convexity of Ω in all three buffers, a condition that must be assumed to hold.
With b 1 always equal to its single-flight value, comparative-static results for this flight buffer are given by (5). Comparative statics for b 2 and b g are generally unavailable, however, because they require total differentiation of the equation system (19)- (21), which leads to ambiguous results.

The effects of adding or removing excess capacity
As seen above, the single-flight model assumes that excess capacity is present, while it is absent in the two-flight model. These assumptions affect the comparison between flight buffers in the two models, making it useful to gauge the effect of alternate assumptions.
First, suppose that excess capacity is present in the two flight model, in which case the previous airline cost expression is replaced by where T is the total time covered by the aircraft lease. Ground time now includes the ground buffer and additional ground time following the termination of flight 2. It is easy to see that, following this change, c g disappears from the b g first-order condition (19), whereas c f in (20) and (21) is replaced by c f − c g . Arguments parallel to those leading to Proposition 2 can then be applied, yielding the same conclusions as in the proposition, namely, sole usage of the ground buffer in addressing delay propagation. Intuitively, when it comes to choice of the ground buffer, excess capacity is just like a zero c g , accounting for the identical conclusions. Note, however, that a corner solution emerges if T is not large enough to accommodate a sufficiently long ground buffer. 6 Alternatively, suppose that excess capacity is absent in both the single-and two-flight models. Then, the single-flight cost expression becomes c f b 1 , and −c g vanishes from (3). This change makes the expression in (3) larger, which requires a decline in the value of b 1 in order to maintain the zero equality. With the single-flight value falling and the flight buffers in the two-flight model unchanged, these latter buffers are now larger than the single-flight value, in contrast to the results of section 3.3. Being larger than the single flight-value, flight 1's buffer therefore now assists the ground buffer in mitigating delay propagation, and flight 2's buffer does the same regardless of relative buffer costs.
While these conclusion are not unreasonable, we believe that the results in Propositions 1, 2, and 3, which are based on excess capacity in the single-flight model, are more natural and intuitive. Moreover, when only one flight is operated, the existence of excess capacity is a plausible assumption. Since some of the comparative-static effects of x and y and of c f and c g have been derived

Numerical examples
analytically, the analysis focuses on the effects of greater variability in the random terms 1 and 2 . To generate the results, 1 and 2 are assumed to follow independent normal distributions with mean zero. Their standard deviations start out equal, satisfying σ 1 = σ 2 = 0.0. Then, each of the σ's increases up to 1.5 holding the other σ fixed, allowing the effect of greater flight-time variability for flights 1 and 2 to be appraised separately. Next, the σ's are set at a common value and increased simultaneously from 0.0 up to 1.5, allowing the effects of greater overall flight-time variability to be appraised.
Among the other parameters, the minimum turnaround time t g is set at 0.5 hours, and the lateness and earliness disutilities are set at x = 1.0 and y = 0.1, with y realistically much smaller than x. The buffer costs are initially set at c f = 0.05 and c g = 0.01. For computational reasons, the infinite upper and lower limits of the normal distributions are replaced by 10 and −10 respectively. With σ's taking the values mentioned above, the probability that 1 or 2 lies outside this range is virtually zero, making this restriction inconsequential. Figure 2 shows the effect of increasing σ 1 from 0.0 to 1.5 with σ 2 fixed at 0.5. As can be seen, the flight buffer b 1 rises as σ 1 increases, a natural finding, while the ground buffer b g also rises. The buffer b 2 for flight 2 appears to be constant in the figure, but it increases very slightly with σ 1 . The conclusion, therefore, is that b 1 and b g alone do almost all the work in absorbing the greater chance of an arrival delay and subsequent delay propagation that follows from an increase in flight-time variability for flight 1. Figure 3 shows the effect of increasing σ 2 from 0.0 to 1.5 with σ 1 fixed at 0.5. Now b 2 rises, while b 1 is constant (note that the b 1 solution from (25) is independent of σ 2 ). However, the ground buffer b g is decreasing in σ 2 , apparently because greater flight-time variability for flight 2 makes the ground buffer less effective at preventing a late arrival. 8 Figure 4 shows the effect of simultaneously increasing σ 1 and σ 2 from 0.0 to 1.5. Both flight buffers naturally increase with the common σ value, and although the figure makes the buffers look equal in size, b 1 is slightly larger than b 2 , as predicted. In addition, b g is increasing in the common σ value. However, Figure 5 shows that the behavior of the ground buffer is reversed when c f = 0.08 and c g = 0.03, falling as the common σ value increases (matching the outcome in Figure 2). Figure 5 shows an additional point that does not arise in the other cases. In particular, b g becomes negative as σ increases, showing that the ground time is set below the minimum feasible turnaround time t g . Nothing in the model prevents this outcome, which need not lead to late departure for flight 2 if flight 1's buffer is sufficiently large. In the data discussed below, however, the outcome is exceedingly rare, accounting for only 0.2% of the observations.
The implication is that the effect of flight-time variability on the ground buffer could be positive or negative depending on the magnitudes of the other parameters. If the ground buffer is sufficiently cheap compared to the flight buffers (c g = 0.01 vs. c f = 0.05), it is used along with the flight buffers to address the greater threat of delay propagation resulting from higher flight-time variability (Figure 4). But when the ground-buffer cost is larger as a proportion of c f (c g = 0.03 vs. c f = 0.08), then the flight buffers partly supplant the ground buffer as the threat of delay propagation rises, with b g falling ( Figure 5).

Adding passenger connections
So far the analysis has suppressed the possibility that some passengers connect from flight 1 to flight 2. These passengers would travel from the origin city of flight 1 to flight 2's destination city, making a connecting trip in the absence of nonstop service between the two cities. This type of connecting travel, however, has little effect on the model. Assuming that a share α of passengers on both flights are traveling nonstop while 1 − α are connecting, the airline's objective function would be altered in a straightforward way. In (18), Ω would be multiplied by α and then added to the term (1 − α)(Ω 2,late + Ω 2,early ), which represents the late and early disutilities for connecting passengers, who care only about their arrival time at flight 2's destination. Adding the two expressions, the disutility portion of the objective function reduces to (18)  The portion of the airline's objective function applying to nonstop trips and same-plane connecting trips is given by adding the Ω's for the A and B flights, with the previous α mod-ification incorporated. 10 The part of the objective function that applies to the remaining connecting passengers makes use of the probability of a missed connection. For 1A-2B connections, this probability is P AB ≡ P r(m 1 + 1A > t d2A ), using (7) and adding an A subscript, while for 1B-2A connections, the probability is given by the analogous expression P BA . In the previous expression, m 1 + 1A is the arrival time of flight 1, and a missed connection occurs when it is greater than flight 2's departure time.
Using P AB and P BA , the remaining (disutility) part of the objective function is given by . (24) The first and third terms in (24) give disutilities from missed connections, while disutilities for connecting passengers who make, rather miss, their connection are given by the second and fourth terms. Note that these early-late disutilities the same as for nonstop passengers on either flight 2A or 2B, being given by the terms multiplying 1 − P AB and 1 − P BA .
These second and fourth multiplicative terms in (24)  A second prediction is that buffer costs affect the levels of all three buffers, although the effect is unambiguous only in the case of b 1 , which is decreasing in c f and increasing in c g . We use airport characteristics (mainly hub status) as a proxy for c g and aircraft size as a proxy for c f .
A third prediction is that the buffers depend on the disutilities x and y for lateness and earliness, with unambiguous predictions only for available for b 1 , which is increasing in x and decreasing in y. We use the managerial employment share of the origin city as a proxy for x on the belief that such employees are highly averse to lateness.
A fourth prediction is that the position of a flight in the day's flight sequence affects the flight buffer. Recall that flight 2 has a longer buffer than flight 1 when c f < c g , with the relationship reversed when c f > c g . To test for such a sequencing effect, we include time-ofday departure variables (morning, evening, etc.) along with variables that measure a flight's exact position in the sequence of an aircraft's daily flights.
A fifth prediction is that a constraint that keeps the ground buffer below its optimal value 20 will lead to longer flight buffer for the previous flight, thus generating a negative correlation between the magnitudes of flight and ground buffers. This prediction is tested via simple correlation analysis.
A sixth prediction is that the share of connecting passengers affects the levels of the buffers.
No explicit predictions are available, but we expect the ground buffer to increase as the connecting share on the subsequent flight rises. The connecting share is computed at the routeairline-quarter level for the prior year using U.S. Department of Transportation (DOT) data.

Data collection and buffer measurement
The sample is obtained from the DOT and covers the US domestic airline market for the year 2018. We mainly rely on the 'Marketing Carrier On-Time Performance' dataset, which for each aircraft, uniquely identified by its tail number, contains information on the carrier operating the flight, the origin and destination, the departure date, the scheduled departure and arrival times, the actual departure and arrival times, and the taxi-in and taxi-out times. 12 We exclude international flights, flights that are canceled or diverted, and flights from/to US Commonwealth areas and Territories.
The aim of our empirical analysis is to investigate the determinants of the two choice The novelty of this dataset relative to the 'Reporting Carrier On-Time Performance' dataset, previously used in Forbes (2008) and other studies, is that it distinguishes whether the flight is operated by the reporting carrier or by its regional codeshare partners. Both datasets are downloadable at www.transtats.bts.gov/Tables.asp ?DB ID=120&DB Name=Airline%20On-Time%%20Performance%20Data&DB Short Name=On-Time.
To calculate the ground buffer, we first compute the actual ground time that separates flight k and flight k + 1 in the sequence of flights operated by the observed aircraft during the day: t kg = t d,k+1 − t ak . Then we calculate the minimum turnaround time as the shortest actual ground time observed across all aircraft of a given type using the turnaround airport.
The ground buffer for an aircraft turnaround is obtained by subtracting this minimum feasible ground time from the scheduled ground time.
We initially restrict the analysis to those aircraft that only operate two flights a day, as in the model, using what we refer to as the "two-flight sample". This restriction predominantly limits the focus to coast-to-coast flights or flights to/from Alaska. However, we later present results using an expanded sample that includes aircraft flying as many as 8 flights in a day, referred to as the "unrestricted sample." Negative values of the flight and ground buffers are possible. For the ground buffer, negativity occurs in the rare cases when the shortest actual ground time is larger than the scheduled ground time. Negative flight buffers are more frequent because they are obtained by subtracting an average value. In order to limit this phenomenon, we require that the number of observations used to obtain the flight and ground buffers must be at least equal to 30.

Empirical approach
Our empirical analysis is based on two sets of regressions: one for flight buffers and another for ground buffers. The standard errors are clustered by route and month in order to allow the residuals of different aircraft (possibly of different carriers) flying on the same route, during the same month, to be correlated.
The general equation to be estimated is: where Buffer is either the flight buffer or the ground buffer. The subscript j identifies the aircraft tail number, c the carrier, o the origin airport, d the destination airport (or the turnaround airport in the ground-buffer regression), and t the month. The η c , φ o , ρ d , and λ t terms are carrier, origin, destination, and month fixed effects 13 and u jcodt is the regression error, assumed i.i.d. with zero mean. The other independent variables are denoted by X, and they control for different aspects of the buffer decision. Table 2 shows definitions and summary statistics for the variables in the two-flight and unrestricted samples. In addition to the buffer, time-variability, and connecting-share variables discussed above, the table shows the month and airline dummies from (25). Other variables included in the set of X variables from (25) are hub origin and hub destination dummies (replaced by a hub turnaround variable for the ground-buffer regressions), congestion measures for the origin and destination (or turnaround) airports, a slot-control dummy for the turnaround airport, dummies for regional and low-cost carriers, a distance measure, time-of-day and weekend dummies, two variables indicating a flight's position in the aircraft's daily sequence, a large-aircraft dummy, and the managerial-share variable (for the origin or turnaround airport).
In the unrestricted sample, the mean flight buffer is about 5 minutes, indicating that carriers set scheduled flight times to be 5 minutes longer than average (or expected) flight times. The mean ground buffer shows that airlines set ground times to be 38 minutes above minimum turnaround times. The mean connecting share is 45%, 14 36% of flights have a hub origin or destination, 31% of flights are on regional carriers, 29% are on low-cost carriers, routes have one competitor on average, average distance is a bit over 800 miles, 27% of flights are on weekends, 3% of flights use heavy aircraft, and the managerial share of the origin work force averages about 5%. Differences in means between the samples are mostly as expected 13 The airport fixed effects are not only relevant in the flight-buffer regression, but also in the ground-buffer regression. Consider for example two flights that share the same turnaround airport and aircraft type, but that originate from different airports. These flights have the same minimum actual ground time, but they may have different load factors depending on the strength of demand and other forces. The potential difference in the load factor may affect the turnaround time and hence the ground buffer set by the airline. The inclusion of originating airport fixed effects (the origin airport of the previous flight) aims to control for this unobservable characteristic. 14 The connecting shares are computed using ticket data from the DOT's Airline Origin and Destination Survey (database DB1B), which comes from quarterly 10% sample of all tickets. For each route-airline-quarter combination in 2017, the share of connecting passengers on the route can be deduced from the ticket information. This share is then matched to the route-airline-month observations in our data. While some routes carry zero connecting passengers in actuality, other routes may show zero connections because the sampling process misses small numbers of passengers.
(for example, mean distance more than doubles to nearly 1700 miles moving to the two-flight sample).

Flight-buffer regressions with flight-time variability
The results of the flight-buffer regressions including flight-time variability are shown in Table 3. Many of the variables listed in Table 2 are omitted from these regressions (to be included later) on the grounds that they may be proxies for flight-time variability, thus being excludable given that variability is explicitly measured. In addition to other variables in Table   2, the excluded variables include the month dummies and the origin and destination fixed effects, all of which are likely determinants of flight-time variability. As mentioned above, the variability measure used in the regressions is the standard deviation of the actual flight times, computed by route, month and flight number for the previous year (2017).
In order to match the setup of our theoretical model, the regressions in Table 3 are based on a subsample of aircraft that operate only two flights a day (mainly flying coast-to-coast or to Alaska). We refer to this sample as 'two-flight sample', and it is expanded in later regressions to include aircraft operating more flights per day . In the table, column (1) shows the regression with the flight buffer of flight 1 set as dependent variable, column (2) has flight 2's flight buffer as dependent variable, and columns (3) and (4)  But column (3) shows that flight buffers are about half a minute longer for late afternoon and evening flights, which presumably correspond most often to flight 2 (the source of this discrepancy is unclear). 15 Table 4 presents the ground-buffer results with flight-time variability, again using the twoflight sample. Since the variability coefficients fo the ground buffer are somewhat sensitive to how variability is computed, we use two approaches. In columns (1) and (2), variability is the 2017 standard deviation of flight times computed by route, month and flight number, as before, whereas variability is computed only by route and flight number in columns (3) and (4), not controlling for month.

Ground-buffer regressions with flight-time variability
Greater time variability for flight 2 reduces the ground buffer in all four regressions, matching the results in Figure 3. The effect of flight 1's time variability is insignificant in the first two columns but significantly positive in columns (3) and (4), matching the results in Figure 2.
Thus, Table 4 shows that greater time variability for flight 2 (flight 1) reduces (raises or leaves constant) the ground buffer, in remarkably close correspondence to the simulation results.
Turning to the other covariates, the coefficients of the share of connecting passengers are uniformly positive and significant across the regressions, as expected. The point estimates indicate that a 0.1 increase in the connecting share on flight 2 raises the ground buffer by about 1 minute. The hub turnaround variable also has a positive and significant coefficient, and since the connecting share is held constant, this variable should capture the cost of ground time, with a negative coefficient expected. The positive estimate thus suggests that the hub turnaround variable may also help to capture the presence of connecting passengers. Note that a hub turnaround raises the ground buffer by about 13 minutes.
Among the airline dummies, notable results are the long ground buffers of Virgin America, Hawaiian, United and Alaska, and the short buffers of Allegiant. The Southwest coefficient, while significantly negative, shows a buffer only 2 minutes shorter than the reference carrier (American). But when all the low-cost carriers are represented by a common dummy, the variable's significant coefficient shows ground buffers about 8 minutes shorter than non-LCCs.
Columns (1) and (3) show that regional carriers have slightly shorter ground buffers than non-regional airlines.
The coefficients of the heavy-aircraft variable are insignificant, a finding that is reasonable given that ground buffers are already type-specific. The managerial-share coefficient is now significantly positive, indicating that ground buffers are longer in turnaround cities with a high managerial share, in an attempt to secure an on-time departure for flight 2.
The time-of-day coefficients show that the ground buffer monotonically increases as the turnaround between the two flights occurs later in the day. This effect is not captured in the model, but it could indicate that, when the two-flight sequence starts later in the day, the effects of time-varying congestion or other factors make a longer ground buffer optimal.

The determinants of flight time variability
The next step in the empirical analysis is to run regressions where flight-time variability is replaced by a host of variables that help to determine variability. The regressions thus include the previous variables along with proxies for variability but not variability itself, providing a different and perhaps more revealing picture of the factors affecting flight and ground buffers.
In this exercise, we present results for both the two-flight sample and a larger "unrestricted" sample described below.
Before showing these regressions, it is useful explore the relationship between flight-time variability and the set of proxy variables. These variables are the month dummies, which control partly for weather conditions, the origin and destination congestion variables (which presumably influence variability), the distance variable, the weekend dummy, and the origin and destination dummies, whose coefficients are not reported.
The results are shown in Table 5 for the two-flight sample, using a separate regression for each flight. With t statistics five to ten times those of other covariates, distance has a very precisely measured positive impact, showing that longer flights naturally have greater time variability. The congestion coefficients are also mostly positive and significant, as expected, while weekend flights show low time variability. The month dummies somewhat surprisingly indicate that the good-weather months of July and August have higher flight-time variability than the Winter months. This outcome may be due to high summer travel volumes, which can generate delays from various sources. 16

Flight-buffer regressions with variability proxies
The results of the flight-buffer regressions with variability proxies are presented in Table   6. The regressions in columns (1)-(4) are based on the two-flight sample and have the same structure as those in Table 3. The regressions in columns (5)-(7) remove the restriction to aircraft operating just two flights per day. The corresponding sample, which we refer to as the 'unrestricted sample', consists of aircraft that operate at most 8 flights in a day. 17 Other differences between the regression specifications are discussed below.
Where statistically significant, the monthly dummy coefficients have negative signs, indicating that compared to January, the reference month, flight buffers tend to be shorter in March. This result makes sense because January, February and March are winter months that bear the same bad-weather expectations and therefore the same delay concerns.
In the two-flight sample, the effect of weather on the flight buffer amounts (in columns (3) and (4)) to about 4 to 7 minutes of reduced buffer length during the Spring and summer months. In the unrestricted sample, this effect is still present, but with a slightly smaller impact, at about 5 minutes maximum. It should be noted that these month effects cannot be attributed to flight-time variability given the results of Table 6. If weather effects on flight buffers operated only through flight-time variability, then with a positive buffer-variability link, the month coefficients in Table 6 would follow the pattern seen in Table 5. The contrary results in Table 6 thus appear to show that the effect of weather may not operate entirely though flight-time variability. Instead, bad weather may slow all flights (requiring longer buffers) without making flight times more variable. Table 3, a hub origin has no effect on flight buffers. But the new specification generates a hub-destination effect that is significantly positive in all regressions instead of mostly insignificant. For hub-bound flights, airlines may lengthen flight buffers to avoid disrupting the travel of connecting passengers through late arrivals. Note that the coefficients of the connecting share, rather than being significantly (and inexplicably) negative as before, are now uniformly insignificant. The origin and destination congestion measures, which were not present in Table 3, have uniformly positive and significant coefficients, indicating that congestion at either endpoint raises flight buffers. Recalling from Table 5 that time variability is positively related to congestion, this positive congestion effect may thus operate through variability. 18

As in
With the exception of Southwest and Virgin America, the carrier coefficients are qualitatively similar to those in Table 3. The regional carrier dummy has the same negative coefficients as in that table, but the previous significant competition effect disappears in Table 6, with the coefficients uniformly insignificant. In addition the heavy-aircraft effect is now significantly negative. The new variable distance has a uniformly positive effect on flight buffers, and since distance and time variability are strongly related, this effect presumably operates through variability. The effect, however, is quantitatively small, on average about 12 seconds per hundred miles flown. As for weekend flights, the weekend dummy coefficients in the two-flight sample are insignificant, although the unrestricted sample shows a significantly positive effect. This effect cannot operate through variability, given that weekend flights have lower variability in Table 5, and its source is unclear. Note that the managerial-share variable does not appear in Table 6 given its collinearity with the airport dummies.
The time-of-day dummy coefficients in column (3) show the same pattern as in Table 3, and the flight-2 dummy coefficient is now significantly positive, showing longer flight buffers for the second flight in the two-flight sample instead of an insignificant difference. Turning to the unrestricted sample, the time-of -day coefficients in column (5) partly match those from the two-flight sample. Flights departing in the late afternoon have longer buffers than flights earlier in the day, although the buffer for evening flights is no different from the buffer for early-morning departures (the default period). Therefore, with more flights per day captured in the data, the time-of-day buffer pattern is somewhat more complex than in the two-flight sample.
To explore the time-of-day effect in a different way, we investigate the effect on its buffer of a flight's position in the day's flight sequence, using either flight-sequence dummies or the aircraft-rotation variable (see Table 2). 19 Column (6) of Table 6 shows results using flightsequence dummies in place of the time-of-day dummies for the unrestricted sample, and new implications emerge. With the exception of the negative flight-3 coefficient, the pattern is for flights 2 through 5 to have longer buffers than flight 1 (the default), while flights 7 and 8 have shorter buffers than flight 1. Because of its focus on just two flights, the model does not generate predictions for the cases covered by the unrestricted sample. But the empirical pattern appears to show a somewhat different logic than in the two-flight situation. In particular, delay propagation is apparently addressed through longer buffers for earlier, rather than later flights.
The shortest buffers are for last few flights of the day, given that the aircraft will soon terminate its daily operations, removing any concern about delay propagation. Although this pattern is the reverse of the one seen in the two-flight case, it could well be optimal in a more complex model with a longer flight sequence.
The regression in column (7) shows that this buffer pattern also emerges when the flight dummy variables are replaced by the continous aircraft-rotation variable, which appears in quadratic form. The positive sign of the aircraft rotation coefficient together with the negative coefficient on the quadratic term point towards an inverse U-shape relationship between the flight buffer and the rotation variable. In fact, the pattern of the parabola based on the estimated coefficients in column (7) is perfectly in line with the results of column (6), in which the set of flight dummies changes sign at flight 6, as seen in Figure 6. The non-linear effect of aircraft rotations on the flight buffer appears to reconcile two opposite forces: the first force pushes towards longer flight buffers to lessen the risk of delay propagation; the second force 19 The time-of-day dummies are not used in conjunction with either of the other two variables because they are almost functionally related. For example, while inclusion of both the time-of-day dummies and flight-sequence dummies would imply that independent variation is possible for these covariates, the fact that, say, the eighth flight of the day could never be a morning flight means that such independence is not present. For the same reason, we do not include the time-of-day dummies in the regressions in columns (1) and (2), as explained earlier.
pushes towards shorter flight buffers to maximize aircraft utilization. Table 7 shows ground-buffer regressions with variability proxies replacing the variability measure, doing so for both the two-flight and unrestricted samples. It is worth noting that the ground buffer is calculated at a flight's departing airport. In the two-flight sample, the observed airport corresponds to the departing airport of flight 2, whereas in the unrestricted sample, the observed airport is the departing airport of Flight i, with i = 2, ..., 8. Thus, the flight-specific variables in the regression, such as departure time, refer to the following flight.

Ground-buffer regressions with variability proxies
In this way, the first flight of the day is not included in the regressions for the unrestricted sample.
The effect of weather on ground buffers, as captured by the month dummies, is not as clear as in the flight-buffer regressions. The results show that in both samples, ground buffers are longest (relative to January) in the Fall months of September, October and November, a pattern that does not have a clear weather-based interpretation. However, in the unrestricted sample, the buffers over the March-August period are significantly shorter than in January, which is partly consistent with short ground buffers being scheduled in months with better weather.
The coefficients on the connecting-share and hub-turnaround variables are uniformly positive and significant across the regressions, as in Table 4. A congested turnaround airport has longer taxi times, which may prompt the airline to cut ground time. While this expectation is mostly confirmed by the negative coefficients on the congestion-turnaround variable, the effect is very small in magnitude: if an extra flight lands/departs when the observed flight lands, the airline reduces the ground buffer by about 5 seconds. 20 But adding the slot-control dummy reverses this effect in both samples, leading to a significantly positive congestion turnaround coefficient in column (6). The significantly negative slot-control effect in the two-flight subsample is large: ground buffers are almost 3 minutes shorter on average at such airports, which tend to be congested. 21 The pattern of the airline-dummy coefficients in the two-flight sample is somewhat different than in Table 4, with the coefficients of Delta and Frontier switching to significantly negative (Southwest's coefficient also becomes more negative). The coefficient pattern changes somewhat in the unrestricted sample, as seen in columns (4) and (6). The coefficients of the low-cost and regional-carrier dummy variables, however, follow the same pattern as in Table   4, being negative across regressions. The heavy-aircraft coefficient, previously insignificant, is now unstable in the two flight sample, but significantly negative in the unrestricted sample.
As in Table 4, the estimated coefficients of the time-of-day dummies show that airlines keep lengthening ground buffers across the day. This effect is strictly monotonic in each regression, rising from 2-7 minutes extra buffer in the morning (relative to early morning) to 4-19 minutes extra buffer in the evening. With the model portraying operation of only two flights, it cannot predict this pattern. But the pattern suggests that, when an aircraft operates many flights per day, ground buffers may play a more prominent role than flight buffers in mitigating delay propagation late in the day.
Airlines may operate with spare capacity during weekends, since most business travel is mid-week, thus explaining why they set slightly longer ground buffers during weekends, about two minutes longer on average. Our data show that flights scheduled during weekends are more punctual than non-weekend fights, and these longer ground buffers may be part of the reason. 22

Additional results
The empirical analysis above has analyzed the flight and ground buffers separately. That is, we have run separate reduced-form regressions for both types of buffers to indirectly test predictions of our theoretical model, with some success. However, the analysis so far has not investigated the model's prediction of a possible negative correlation between the flight and ground buffers. This prediction arose via possible constraints on the ground buffer, which would reduce it below the optimal value and lead to increases flight 1's buffer and possibly for flight 2's as well, creating an inverse association.
This prediction is difficult to test in a regression framework that puts one buffer (flight or ground) on the left-hand side and the other on the right-hand side, looking for a negative coefficient. The reason is that, since the choices of the flight and ground buffers are simultaneous, the right-hand-side buffer must be treated as endogenous. While this endogeneity rules out use of an ordinary least squares regression, an instrumental variable approach using two-stage least squares (2SLS) could be considered. But given that the buffers are generated within the same airline choice problem, thus being dependent on the same set of exogenous factors, such an instrumental variable (which affects one buffer but not the other) may not even exist.
The impossibility of 2SLS, however, does not preclude a simple correlation analysis, which is not intended to capture causal effects. Accordingly, Table 8  Another possibility that our empirical analysis has not considered so far is that past on-time performance may affect buffer decisions, since it is very natural that carriers would change their buffers to remedy past late arrivals. To capture this possibility, we included in the flight-buffer regressions a variable measuring the past on-time performance of the airline on the route.
The variable is constructed as follows: using the DOT's 2017 'Reporting Carrier On-Time Performance (1987-present)' dataset, we calculated by carrier-route-month the proportion of delayed flights. 24 The coefficient on this past-year delay variable is positive and statistically significant, confirming the presumption that carriers lengthen flight buffers if delays on the route in the past year were more frequent. The results are reported in the appendix Table   A1. Finally and importantly, the inclusion of the past-year delay does not alter qualitatively or quantitatively the estimated coefficients of the other covariates (except for the connecting passenger coefficient, which becomes significantly negative in columns (5)- (7)).

Conclusion
This paper has presented an extensive theoretical and empirical analysis of the choice of schedule buffers by airlines. With airline delays a continuing problem around the world, such an undertaking is valuable, and its lessons extend to other transport sectors such as rail and intercity bus service. One useful lesson from the theoretical analysis of a two-flight model is that the mitigation of delay propagation is done entirely by the ground buffer and the second flight's buffer. The first flight's buffer plays no role because the ground buffer is a perfect, while nondistorting, substitute. In addition, the apportionment of mitigation responsibility between the ground buffer and the flight buffer of flight 2 is shown to depend on the relationship between the costs of ground-and flight-buffer time.
The empirical results show the connection between buffer magnitudes and a host of variables, including the month of operation and distance of a flight, whether the flight operates early or late in the day, and congestion measures at the endpoints. In addition, the initial regressions relate buffer magnitudes to the variability of flight times, which simulations of the model identify as an important determining factor (the results show that high variability lengthens flight buffers).
Fruitful extensions to this work would most likely lie in the theoretical area. The model could be extended to include additional sequential flights, and the sketch of connecting traffic provided in the paper could be expanded into a full analysis. In addition, passenger scheduling preferences could be introduced, with a buffer-related extension of scheduled flight times possibly becoming less desirable if it leads to a divergence between a passenger's preferred and actual arrival times. The resulting models would be complex, but additional insights could be next, some 2018 observations are not matched with the 2017 data. generated, with relevance extending beyond the airline industry.
Another avenue of exploration could be in the area of market structure. For example, if a profit-maximizing airport sets the gate rental cost at an excessive level, the resulting decrease in ground buffers will impair airline mitigation of delay propagation, with negative effects on passengers. Alternatively, the entry barrier of gate shortages resulting from airport dominance by a large carrier (analyzed by Ciliberto and Williams (2010)) would also affect the ability of smaller carriers to address delay propagation via adequate ground time. Exploration of these issues could be illuminating.
Applying the same steps to (17), the b g -derivative of that expression equals Replacing the bracketed terms in (a4) and (a5) , respectively, the sum of (a4) and (a5) can be written as The derivative of the objective function with respect to b 1 builds on the previous results.
The b 1 -derivatives of (16) and (17) are identical to the b g derivatives, given by (a4) and (a5), with their sum equal to (a6). Since (2) also depends on b 1 , the b 1 -derivative of the objective function is then the expression in (3) (slightly rearranged) plus (a6): Using similar steps,

A2. Convexity of Ω
This appendix section first shows that the objective function Ω is strictly convex in b 1 and b 2 , with b g fixed at the optimal value, doing so for both cases of zero and positive buffer costs.
The first step is to compute the second derivatives of Ω with respect to b 1 and b 2 , assuming zero buffer costs. Using the shorthand Ω ij for ∂ 2 Ω/∂b i ∂b j , differentiation of (20) and (21) yields (after suppressing the multiplicative factor x + y) ) Observe that the same expressions apply when buffer costs are positive since they vanish in computing the second derivatives.
Ω 22 and Ω 12 are positive, while the sign of Ω 11 in (a9) is unclear. However, at the optimum in the zero-cost buffer case, the bracketed term in (a9) is zero, making Ω 11 positive and also ensuring satisfaction of the inequality in (11). The same conclusion holds in the positive-cost buffer case if c f > c g , in which case the bracketed term in (a9) is negative at the optimum.
With Ω 11 , Ω 22 > 0 and H ≡ Ω 11 Ω 22 − Ω 2 12 > 0 (a consequence of the inequalities in (a11)), Ω is thus strictly convex in b 1 and b 2 at the optimum. As a result, b 1 and b 2 values satisfying the first-order conditions yield a local minimum for Ω, holding b g fixed at its optimal value.
Convexity of Ω in all three buffers cannot be established analytically and must be assumed.    (a) The symbol denotes whether the variable is included in the flight buffer regressions (column b) or in the ground buffer regressions (column bg).
(c) All estimates include airport of origin and airport of destination fixed effects.