The untolled problems with airport slot constraints

Economics of Transportation ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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The untolled problems with airport slot constraints Joseph I Daniel n Department of Economics, University of Delaware, Newark, DE 19716, USA

art ic l e i nf o

a b s t r a c t

Article history: Received 13 June 2013 Received in revised form 24 January 2014 Accepted 25 January 2014

This paper examines the efficiency and practicality of airport slot constraints using a deterministic bottleneck model of landing and takeoff queues. It adapts this congestion pricing model to determine the optimal timing and quantity of slot permits for any number of slot windows. Aircraft choose their optimal operating times subject to the slot constraints, and airport queues adjust endogenously. The number and length of slot windows affects the congestion levels and efficiency gains. The atomistic bottleneck model is extended to include self-internalizing dominant traffic and atomistic fringe traffic. The model raises questions about the implementation of slot constraints that do not arise in standard congestion models. The theory explains (Daniel's, 2011) empirical findings that slot-constraints at Toronto are ineffective and suggests that recent proposals for slot constraints at US airports would be similarly ineffective. Effective slot constraints require many narrow slot windows, making slot auctions or markets difficult to implement. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Airport congestion Slot constraints Pricing Bottleneck Queuing

1. Introduction 1.1. The issue Slot constraints are restrictions on the number of landings or takeoffs that airports permit during specified time periods known as slot windows. For a typical example, an airport that can perform sixty landings per hour, issues sixty slot permits for landings during each hour of the day and requires aircraft to operate during their assigned hour. Most major commercial airports throughout the world impose slot constraints to control access to their runways, ostensibly to reduce congestion delays. In the US, however, most major airports that receive federal funding are available on a first-come, first-served basis without requiring slot permits.1 When airports become severely congested, airport authorities, airline officials, and policy makers generally seem to favor slot constraints over congestion tolls as a means of managing demand. They argue that slot constraints are simpler to implement because the airports need only limit the number of slot permits to the airport's capacity, then sell, auction, or give the permits away. Slot markets can price and allocate the slot permits efficiently. n

Tel.: 302 831 1913; fax: 302 831 6968. E-mail address: [email protected] 1 Airports assess landing fees based on aircraft weight, typically between one to five dollars per thousand pounds. These fees have nothing to do with the marginal social cost of serving the aircraft. The social cost consists primarily of the delays aircraft impose externally on other aircraft. Weight-based landing fees encourage too many operations by smaller aircraft compared to marginal-cost congestion tolls that promote efficient levels of operations by internalizing the external delays.

Congestion tolls, they argue, are too difficult (or politically inconvenient) for airports to assess correctly. Moreover, slot permits are supposed to avoid the problem of imposing different toll schedules on dominant airlines that already internalize their self-imposed delays than on fringe airlines that ignore the additional delays they impose on other aircraft. This case for slot constraints implicitly assumes that airports experience steady-state traffic during slot windows. Actual traffic patterns at major airports, however, exhibit rapid fluctuations that follow regular patterns due to airline scheduling practices. Airlines that operate hub-and-spoke networks schedule flights around passenger interchange periods. Consequently, many airports experience as many as ten substantial peaks during a day (see, Daniel and Harback, 2009). Actual traffic patterns call into question the effectiveness and practicality of slot-constraint systems that are based on steady-state traffic models that underlie most existing policy analysis. If traffic rates and queuing systems are in steady states, then restricting the number of hourly slot permits to the hourly airport capacity might reduce congestion. If congestion is actually caused by traffic rates fluctuating from slack to peak demand within hourly periods, then it is desirable to use a model that captures these features of the problem. In this paper, I develop a (dynamic) bottleneck model with multiple slot windows in which the airport authority chooses the timing of slot windows and the quantity of landings or takeoffs (operations) to permit during each window. Airlines choose when to operate their aircraft within the slot windows to minimize the costs of their queuing delays and of arriving before or after their most preferred time. A structural model of congestion has state-contingent queues that evolve endogenously in response to traffic adjustments. Congestion

http://dx.doi.org/10.1016/j.ecotra.2014.01.003 2212-0122/& 2014 Elsevier Ltd. All rights reserved.

Please cite this article as: Daniel, J.I, The untolled problems with airport slot constraints. Economics of Transportation (2014), http://dx. doi.org/10.1016/j.ecotra.2014.01.003i

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externalities cause traffic rates (i.e., the rates of arrivals at the landing or take off queue) to exceed service rates (i.e., the rates at which airports can perform landings or takeoffs) during a portion of each slot window, even when the number of slot permits per slot window is within the airport's capacity. While airports chose the number of slot permits per slot window, airlines chose aircraft operating times within the slot windows, resulting in equilibrium traffic patterns that exceed capacity during the portions of slot windows that are closest to the preferred operating times. In the bottleneck model, slot constraints cause the queue to empty periodically by preventing airlines from scheduling aircraft too early. This results in smaller peaks during each slot window rather than a single large queue that persists throughout the busy period. As the airport authority adds more slot windows, it constrains aircraft to narrower operating windows and the queue empties more frequently, thus limiting the accumulation of aircraft in the queue and reducing total queuing delay. The model also addresses the optimal timing of dominant and fringe airline operations, the efficient allocation of slots among dominant and fringe airlines, and the effect of slot constraints on the distribution of surpluses between dominant airlines and atomistic fringe aircraft. A policy section discusses the problems with current and proposed implementations of slot constraints, and the practical issues involved in designing and implementing efficient slot systems. The paper focuses on the more basic issues of timing and quantity of slot permits rather than how to design auctions or markets to distribute slot permits. The literature has largely overlooked these basic issues, but unless policymakers address them, the resulting slot-constraint system may have little effect on congestion at most airports, no matter how elaborately they design slot auctions or markets. A brief preview of the conclusions I derive from the model is as follows: (1) Effective slot-constraint systems require numerous narrow slot windows that force traffic to spread out over the peak period. Slot-constraint systems that hold the quantity of slot permits to the airport capacity over a single slot widow covering the entire peak period are completely ineffective. (2) In unconstrained equilibria, dominant airlines schedule some of their aircraft to operate at the service rate during the periods just before and after the atomistic traffic. These aircraft fully internalize their delays, while the remaining dominant aircraft join atomistic traffic and ignore the delays that they impose on other dominant aircraft. The fraction of internalizing aircraft varies from one to zero as fringe demand elasticity varies from zero to negative infinity. (3) If the airport authority has complete control over the allocation of slots, it will separate the dominant and fringe operations to enable the dominant airline to fully internalize all self-imposed delays. If the airport is unable to enforce this separation, then the dominant airline will schedule some aircraft atomistically, depending on the elasticity of fringe demand. The first-best optimum requires one slot permit and window for every service interval. 1.2. Background The International Air Transit Association's (IATA) Worldwide Slot Guidelines (WSG) provide for slot coordinators at highly congested airports who issue and distribute slot permits on a semi-annual basis. According to the guidelines, the fundamental considerations in allocating slots are preserving historical patterns of use, preventing “confiscation” of incumbent airlines' claims on slots, and allocating slots to new entrants only from new airport capacity. Twice a year, airline representatives submit slot requests that substantiate their past slot usage, and airport slot coordinators make preliminary slot distributions, then airline and airport representatives meet at three-day slot conferences to trade and finalize slot allocations. Slots may be traded in one-for-one

exchanges, but may not be sold. The airline industry appears to have designed this slot system as a means of restricting entry. Recent slot-constraint proposals for the US, however, seek to preserve free entry by providing for issuing of slot permits that would be valid for a period of ten years. Each year, one tenth of an airport's slots would expire and be re-issued. Airlines could resell slots, and presumably markets would develop for trading slots. There are various proposals concerning the type of initial auctions or markets to distribute newly issued slots. Since the values of slots depend on their combination with other slots (including those at other airports), these auctions or markets must be able to value an enormous number of potential slot combinations. Combinatorial auctions to distribute slots would potentially be the largest and most complex ever conducted. The fundamental justification for slot auctions or markets is that they are supposedly simpler than congestion tolling because an auction or market determines the price of slots rather than an administrative agency. To make combinatorial slot auctions or slot markets feasible would require issuing many undifferentiated slot permits with long slot windows to reduce the number of potential slot combinations. Since airlines are free to choose when to operate within the slot windows, however, longer windows with more undifferentiated permits reduce the airport's control over the timing of traffic and lengths of queues. There is a fundamental tradeoff between feasibility of slot auctions or markets and reduction of congestion. Proponents of slot-constraints largely ignore this issue because standard steadystate congestion models implicitly assume that traffic rates are constant within slot windows. Consequently, hourly slot windows appear to be adequate under steady state models. Any optimal system of congestion management must address two fundamental issues: setting the traffic quantity so that the marginal social cost of operations equals their marginal social benefit; and controlling the timing of traffic to minimize total social cost. The standard congestion models and most real-world slot constraint systems attempt to address the first issue but not the second. Slot-constraint systems proposed in the US, and the systems following the WSG all grant authority to operate during specific time windows – usually one-hour intervals. These wide time windows provide flexibility to accommodate random variation in operating times, facilitate exchange of slots, and reduce administrative and compliance costs, but they also make it impossible to achieve optimal traffic patterns and can render slot constraints largely ineffective. For example, Daniel (2011) shows that Toronto's Pearson International Airport experiences significant congestion in spite of its slot-constraint system that follows the WSG. The way airports collect and present traffic data reinforces the standard models' erroneous assumption of constant traffic rates within slot windows. Government and consulting reports typically aggregate airport traffic rates by hour. This practice hides the rapid fluctuations in traffic rates over shorter periods of time that are responsible for a significant amount of airport delays. Airports do not routinely collect data on the actual time spent in landing or takeoff queues. Instead, the primary measures of aircraft delays are on-time arrival and departure statistics. These report the number of aircraft that are more than 15 min late relative to the aircraft's scheduled operating time at the gate. Aside from the obvious issue of not recording delays of less than 15 min, on-time operating statistics do not reflect the additional time that airlines add to their scheduled travel times to allow for queuing delays on takeoff or landing, for traffic jams on the tarmac while trying to access the gates, or any of the other many regular causes of aircraft delay. Atlanta and Chicago O'Hare, for example, average approximately 20 min of queuing delay per aircraft, which is much higher than other US airports, while their on-time arrival statistics are no worse than most other major airports. The hourly data paints a

Please cite this article as: Daniel, J.I, The untolled problems with airport slot constraints. Economics of Transportation (2014), http://dx. doi.org/10.1016/j.ecotra.2014.01.003i

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misleading picture that is consistent with the standard model's steady-state traffic rates, while minute-by-minute data on traffic and delay patterns show rapid fluctuations that are inconsistent with the standard model, but are consistent with the bottleneck model.2 1.3. Review of the literature The standard congestion pricing model specifies an average (private) travel cost function for traversing a highway segment that is an increasing function of the current traffic volume during some fixed time period. The function is analogous to the average cost function of a firm that produces trips along the highway segment. The marginal (social) cost function lies above the average cost function and is also increasing in traffic volume. The vertical difference between average and marginal cost is the additional travel time that one more vehicle imposes externally on other vehicles. Travel demand is a decreasing function of the full price of travel, which consists of the travel time cost plus any toll for using the highway. The untolled equilibrium at the intersection of travel demand and the average travel-cost functions is socially inefficient because it does not account for the external cost of the trips. The optimal toll is the value of the vertical difference between the marginal and average travel times evaluated at the intersection of the travel demand and marginal social cost curves. It leads to the socially efficient traffic volume by internalizing the externality and equating marginal social costs and benefits of trips. Applying this model to airport slot constraints simply requires limiting the slot permits to the quantity where traffic demand crosses the marginal (social) cost curve. William Vickrey's (1969) bottleneck model provides a dynamic model of congestion with travelers adjusting their time of travel to minimize their costs of travel duration and early- or late-arrival time. Vickrey's model uses a deterministic queue that develops at a highway bottleneck and prevents travelers from all arriving at their destinations at their most preferred times. The queue length depends on the cumulative arrivals from when the queue was most recently empty, and each arrival affects future travel delays until the queue is empty again. In unconstrained equilibria, traffic rates and queuing patterns adjust endogenously over time to equate total costs of queuing and early or late arrival times of homogeneous travelers. In a tolled equilibrium, optimal tolls adjust continuously throughout the peak period to mimic the queuing costs of the no-toll equilibrium. The congestion toll completely eliminates queuing and converts all queuing costs into toll revenues. The model has several advantages over the standard model including: dynamic congestion, endogenous peaking of traffic and queues, modeling of traveler's choice of when to travel, and accounting for early- or late-time costs associated with scheduling decisions. The model applies to peak-period pricing, but it has not previously been applied to airport slot constraints or other forms of quantity constraints. The standard model is still the preferred framework among economists for modeling congestion. The economics literature largely ignored Vickrey's (1969) bottleneck model until Arnott, et al. (1990) reintroduced it by formalizing it, extending it to include a single-step toll, and modeling the optimal bottleneck capacity. Arnott et al. (1993) determines the optimal uniform, single-step, and continuous tolls with elastic demand. They argue 2 Daniel and Harback (2009a) document the rapid fluctuation of traffic patterns using minute-by-minute data on traffic and delays at the twenty seven largest US airports. Daniel and Pahwa (2000) compared the performance of the standard model, the deterministic bottleneck model, and the stochastic bottleneck model in replicating actual airport traffic patterns. They show that the standard model cannot produce these rapid fluctuations.

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that applying the standard steady-state model to subintervals of the peak period is conceptually unsound, but that the standard model represents a “semi-reduced form” of the entire peak period. They show that the efficiency gains from peak spreading are substantially greater than gains estimated with the standard model. They verify that the self-financing result that Mohring and Harwitz (1962) and Strotz (1965) proved for the standard model also applies to the bottleneck model, independently of the pricing regime.3 Daniel (1991) develops a deterministic bottleneck model with congestion that is non-linear in traffic rates and applies it to airport congestion. Daniel (1995) develops a bottleneck model with stochastic queuing and includes atomistic and non-atomistic traffic of Nash- and Stackelberg-dominant airlines. He empirically implements the stochastic bottleneck model using tower log data from Minneapolis-St. Paul airport and performs specification tests that indicate Northwest Airlines' scheduling of its flights is largely inconsistent with internalization of self-imposed delays. Daniel (2001) adds elastic demand, heterogeneous aircraft costs, and fringe aircraft with uniformly-distributed preferred operating times to the stochastic bottleneck model. Daniel (2009) develops a model of multiple-step tolling for a deterministic bottleneck with dominant and fringe traffic. It uses a similar framework to determine the efficient tolling rules when airlines choose their operating times optimally in response to tolls and queuing- early-, and late-time costs. A moderate sized literature examines the economics of airport slot constraints.4 Of this literature, only Verhoef (2008) and Brueckner (2009) explicitly model airport congestion and the optimal allocation of airport slots between dominant and fringe airlines. Verhoef (2008) demonstrates that in the presence of market power and uninternalized congestion, the first best pricing differentiates the tolls for dominant and non-dominant airline. Tradeable slots dominate undifferentiated pricing and approach the optimum when market power is unimportant, but slots may be less efficient than non-intervention when market power is significant. Brueckner (2009) includes congestion pricing, slot sales, and tradeable slots, but assumes away market power. He shows that direct slot sales by the airport fail to achieve the optimum allocation because they have a uniform price. Dominant airlines have too little traffic volume and fringe airlines have too much. When slot quantity is fixed under a slot auction or slot trading, congestion is also fixed and dominant carriers can bid up the price of slots so that the equilibrium is efficient. Verhoef and Brueckner use a standard congestion technology for which delay is a function of current period traffic volume, so the models do not capture changes in the timing of operations within the period. This paper contributes to the literature by developing a bottleneck model in which airports choose the timing and quantity of their slot-constraints, subject to dominant and fringe airlines choosing their operating times within the slot windows to minimize queuing-delay and early or late-time costs. The bottleneck model addresses important questions that cannot be formulated within the standard model. These include the effects of intertemporal traffic adjustments on the optimality of slot constraints, how distribution

3 Ralph Braid (1989) independently extends the bottleneck model to cover elastic demand. Arnott et al. (1990) extends the bottleneck model to heterogeneous travelers. 4 Much of the literature focuses on alternative methods of distributing slots, see e.g., Borenstein (1988), Starkie (1994), Mehndiratta and Kiefer (2003), Condorelli (2007), Verhoef (2008), and Brueckner (2009). A number of articles address the technical details of designing slot auctions; see Grether, Isaac, and Plott, C. (1989), Rassenti, Smith, and Bulfin (1991), de Vries and Vohra (2000) and Jehiel and Moldovanu (2003). The political economy of implementing slot constraints has also received attention from Rose (2003), Condorelli (2007) and Winston and de Rus (2008).

Please cite this article as: Daniel, J.I, The untolled problems with airport slot constraints. Economics of Transportation (2014), http://dx. doi.org/10.1016/j.ecotra.2014.01.003i

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of slots between airlines over time affect efficiency, how the duration of slot windows affects efficiency of equilibria, how slot constraints affect the ability of dominant carriers to internalize selfimposed delays, and the effect of slot constraints on strategic behavior by dominant airlines to preempt their most preferred periods.

2. The model 2.1. Airline decisions in the atomistic bottleneck model First consider the basic framework of Vickrey (1969) and Arnott et al. (1990, 1993), as applied to an airport with limited runway capacity facing demand for landings and takeoffs at rates that exceed the airport's service rate. In this section, assume that all aircraft operate atomistically, i.e., no airline operates multiple aircraft that impose delays on one another. Air traffic control alternates landing and takeoffs in such a manner that landing and takeoff queues operate independently of each other.5 The same model applies separately to both landings and takeoffs so the parameter names are the same, but the parameters values may be different. Airlines would like to schedule their aircraft during a particular peak period to operate at the most-preferred time, t*, which is a particularly desirable time for the airline passengers to travel.6 Runway capacity limits the landing and takeoff rates to s aircraft per minute. The rate at which aircraft join the queue at time t is r[t]. Deterministic queues develop at the runway bottleneck from the beginning of atomisitic arrivals at time tab, when the queue is empty, according to the equation: Z t q½t ¼ ðr½u  sÞdu: ð1Þ

entire peak period.7 Solving for r[t] separately when t is early, trt*  q[t]/s, or late, t 4t* q[t]/s, by substituting (1) into (2), and differentiating with respect to t while imposing the constant cost condition gives the aggregate traffic rates for periods in which atomistic aircraft operate.8 The solutions for atomistic traffic rates are

r½t ¼

8 0; > > > > α S; > < α β α > > α þ γ S; > > > : 1;

for C½t 4 C nw0 n for t r t n  q½t s and C½t ¼ C w0 n for t 4 t n  q½t s and C½t ¼ C w0 and

for C½t o C nw ði:e:; a finite number of instantaneous arrivalsÞ:

ð3Þ Eq. (3) fully characterizes the equilibrium traffic rates resulting from atomistic airlines choosing their aircraft operating times during a peak period. These are the same rates as in the original bottleneck models of Vickrey (1969) and Arnott, et al. (1990, 1993). In a slot-consrained bottleneck model, the airport authority takes traffic rates as given by Eq. (3) and chooses the number of slot permits to issue for various slot windows. Choosing the number of permits is equivalent to choosing the duration of the slot window, because it is never optimal for a deterministic queue to be idle during the peak period. The airport takes the number of slot windows parametrically so that any number is possible. Endogenous choice of the number of slot windows requires a model of the administration and compliance costs the slot-constraints system, which is beyond the scope of this paper. The focus here is on how any equilibrium configuration of slot-windows affects aircraft traffic patterns, operating costs, and airport delays. Steady state models implicitly assume that the airport can set traffic rates equal to (or below) capacity and thereby eliminate delays. Eq. (3) shows that this is impossible in a bottleneck model.

t ab

Because capacity is limited, aircraft must operate before or after their preferred times. An aircraft that joins the landing or takeoff queue at time t will spend q[t]/s min in the queue at a cost of $α per minute, and it will complete service at time tþq[t]/s. An early aircraft (completing service before t*) will have t* (tþq[t]/s) min of layover delay (early time) in excess of the optimal interchange time that starts at t*. The cost of additional layover time is $β per minute. A late aircraft (completing service after t*) will have (tþ q [t]/s)  t* min less than the optimal time to exchange passengers. The cost of reduced interchange time (late time) is $γ per minute. The sum of the queuing, early time, and late time costs as a function of the time when an (atomistic) aircraft joins the queue is:   q½t q½t q½t n þ β max 0; t n  t  þ γ max ½0; t þ  t : ð2Þ C½t ¼ α s s s

2.2. Airport decisions: multiple-window slot constraints with atomistic traffic

In equilibrium, atomistic traffic rates adjust to maintain constant costs C[t]¼ C*w among groups of identical aircraft across all times in which they operate. The subscript on C*w denotes the slot window. Costs are different across slot windows because slot constraints prevent traffic from moving between windows, whereas an unconstrained equilibrium allows aircraft to shift operating times freely, resulting in the same cost throughout the

To derive the optimal slot-constraint system for a given number of periods, the airport authority chooses the quantity of operations to permit during each slot window. The durations of the slot windows follow from the number of slot permits issued per window. The slot windows and permits designate the times at which aircraft are permitted to join the queue, not the time at which they complete service. Aircraft that arrive before their permitted window are not admitted to the queue and do not get any priority for arriving early. Aircraft must join the landing queue before the window expires, otherwise they must pay a fee sufficient to make late arrival undesirable. Early aircraft that arrive at the end of a window experience the longest queues and do not complete service until after the next window has begun. Similarly, departing aircraft may not push back from their gates until their window starts and must push back before it ends. Fig. 1 illustrates the qualitative properties of bottleneck equilibrium traffic patterns for the unconstrained, slot-constrained, and first-best cases using the framework of Eqs. (1)–(3). It is useful to understand the model graphically before developing the quantitative specification. The horizontal axis represents time, with the origin at the most preferred operating time t*. The vertical axis represents the aircraft index with the origin at the aircraft that

5 This is approximately true of actual airports operating under balanced traffic (landing and takeoff) conditions. The author's observations of traffic counts indicate that somewhat higher rates of takeoff are possible when there are no landings, but that no additional landings are possible when there are no takeoffs. 6 The model can have preferred operating times that are distributed over time, but this complicates the solution without fundamentally changing the traffic patterns or policy implications.

7 It follows that the optimal slot constraint equilibrium is different from and better than the optimal tolled solution, but for reasonable parameters the difference is quite small. 8 In the extended model, there are also periods with corner solutions when no traffic joins the queue because the cost is above the equilibrium level, or when a finite amount of traffic simultaneously joins the queue because the cost is lower than any other available operating time.

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20 Slot window 0

Slot window -2

Slot window -3

5

Slot window 1 Slot Window 2

Slot window -1

Slot Window 3

# of slot permits for window 2

Mass operations 10

Late aircraft 0

Slope Αs

Shaded areas are total queuing time (unconstrained)

Shaded area is total late time

Α Γ

Time from t zero

Early aircraft Time in Queue (unconstrained) Time in queue (slot constrained)

Aircraft in queue (unconstrained)

-10

# of slot permits for window 0

Shaded area is total queuing time (slot constrained)

Aircraft in queue (slot constrained)

Cummulative service completion

-20 Unconstrained arrivals for service

# of slot permits for window -2

Slot constrained arrivals for service

Slope Αs

Shaded area is total early time

Α Β

Cummulative operations

Slope s

-30

Minutes Early -60

-40

-20

Minutes Late 0

20

40

Fig. 1. Slot constraints with four early and three late slot windows.

completes service exactly at the most preferred time. Aircraft with negative indices experience early time and those with positive indices experience late time. The heavy diagonal line through the origin is the cumulative-service function that associates the index of the aircraft with the time it completes service (exits the queue). The cumulative-service function is the same for all three cases. It has slope equal to the service rate s. The first-best solution minimizes total aircraft operating-time costs, has arrival rates equal the service rate throughout the peak period so that there is no queuing. The two solid lines that connect on the horizontal axis above and to the left of the cumulative-service function represent the unconstrained cumulative-arrival function that associates the aircraft index with the time it joins the queue. This function has slopes given by Eq. (3). It is identical to original bottleneck equilibrium of Vickrey (1969), and Arnott et al., (1990, 1993), and it is also a special case of the model developed below when the airport has one early, and one late slot window and issues permits m aircraft. The remaining line segments above and to the left of the cumulative-service function represent the slotconstrained cumulative-arrival function, which are the described below. First consider the unconstrained equilibrium-cumulative arrival function. In bottleneck equilibria, homogeneous aircraft adjust their operating times over the peak period to assure that the sum of early-, late-, and queuing-time costs as given by Eq. (2) are constant whenever the arrival rate is positive (an interior solution). The first aircraft experiences only early time, the aircraft with index 0 completes service exactly at the most-preferred time t* and experiences only queuing time, and the last aircraft experiences only late time. During the early periods, the arrival rate (the slope of the cumulative arrival function) exceeds the service capacity (the slope of the cumulative service function) so

that the queue (the vertical distance between these functions) increases constantly to assure that additional queuing-time costs (α times the horizontal distance between these functions) exactly offset reductions in early-time costs (β times the horizontal distance between the service-completion function and the vertical axis). During the late period, the arrival rate drops below the service rate and the queue diminishes to assure that lower queuing costs just offset greater late-time costs (γ times the horizontal distance between the vertical axis and the servicecompletion function). Cumulative arrivals exceed the cumulative service capability of the airport throughout the entire peak period. Each aircraft adds its service time 1/s to the queuing time of every aircraft that joins the queue after it until the queue empties at the end of the peak period. Imposing slot constraints can limit this effect on subsequent aircrafts' queuing delays to only those aircraft that operate in the same slot window. Now consider a slot-constrained equilibrium with the number of early and late slot windows taken parametrically. The airport authority chooses the quantity of slot permits qi for each slot window i, subject to the airlines choosing arrival rates according to Eqs. (1)–(3). Choosing the number of slot permits for each window also determines the lengths of slot windows (qi/s) and their timing relative to t*, because it is never optimal to configure the widows in a way that would leave idle capacity during the peak period. Fig. 1 illustrates the case of four early and three late slot windows. The quantity of aircraft in each window is the vertical distance between the lightly shaded horizontal grid lines and the duration is the horizontal distance between the lightly shaded vertical grid lines. Starting at the bottom left of Fig. 1, the arrivals for the slotconstrained equilibrium will begin at the same instant as the unconstrained equilibrium and have the same arrival rate. The airport chooses the number of permits for this window and

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6

airport authority issues for early slot windows i¼  1,…,  y and late slot windows i¼1, …, z. As before, the axes are aircraft indices and time. The heavy diagonal line through the origin is the service-completion function, and the lighter connected line segments form the cumulative-arrival function. An aircraft's early time is the horizontal distance from the service completion function (prior to t*) to the vertical axis. It follows that the total early time during an slot window i is the area between the vertical axis and the service-completion function for aircraft operating in that window ð∑ij ¼11 q  j =s þ q  i =2 sÞðq  i Þ, where j indexes the subsequent early slot windows. Similarly, an aircraft's late time is the horizontal distance from the vertical axis to the servicecompletion function (following t*). It follows that the total late time during an slot window i is the area between the vertical axis and the service-completion function for aircraft operating in that window ð∑ij ¼11 qj =s þ qi =2 sÞðqi Þ , where j indexes the previous late slot windows. An aircraft's queuing time is the horizontal distance from the cumulative-arrival function to the service-completion function. It follows that the total queuing time during a slot window i is the area between the cumulative arrival function and the service completion function for all aircraft operating in that window. For early aircraft, this is a triangle with height qi and base equal to the queuing time at the end of the slot window. The slope of the service completion function is s so the run of the function is qi/s. Likewise, the slope (arrival rate) of the cumulative arrival function is αs⧸(α  β) so the run of the function is qi (α  β)⧸(αs). The queue at the end of the period is the time difference between the last aircraft's arrival time and its service completion time: qi/s qi (α  β)⧸(αs)¼qiβ⧸(αs). The area is q2i β⧸(2αs). For late slot windows, the total queuing time is the area of a triangle associated with the mass of aircraft that join the queue at

requires these aircraft to arrive at the queue before the time the slot window ends (at about t¼  57). These aircraft would prefer to operate later because the constraint system has lowered the queuing costs closer to t*, but the airport refuses to admit them to the queue after their slot window expires. The equilibrium cost C* 3 of the next slot window is lower than C* 4 so the arrival rate is 0 until the queue empties. At this point the early-time cost equals C* 3 and arrivals resume at the equilibrium rate. Note that the queue is significantly lower than it would be in an unconstrained equilibrium. This process repeats until the end of slot window i¼  1. The airport authority has the choice of not enforcing the constraint between slots  1 and 1. If it enforces the constraint, these periods are exactly like the others. If it does not enforce the constraint and allows aircraft with permits for slot windows 1 and 1 to operate in either period, then the equilibrium traffic will be just like unconstrained bottleneck equilibrium for the given number of permitted aircraft, q-1 þq1. This alternative may have significantly lower administrative costs or other advantages over enforcing the constraint. When the constraint is imposed, and for all other late windows, the airport prevents aircraft from joining the queue until the beginning of their slot windows. When these late windows begin, late-time costs are less than the equilibrium costs for the new slot window. Many aircraft arrive simultaneously and the airport randomizes their admission to the queue. The actual queuing cost jumps to twice the difference between the equilibrium cost and the late time cost. This assures that the expected costs at the beginning of a late slot are in equilibrium for that instant. No additional traffic will arrive until the queue diminishes to reestablish the equilibrium cost, then traffic resumes at the rate given by Eq. (3). Fig. 2 illustrates how to quantify the early-, late-, and queuingtime costs as functions of the number of slot permits qi that the

z

2 qj 2

Late Queuing Cost:

2

j 1

10

2s

qj2 2

s

2 2

Slope s

q[i] q[i]/s

Central Queuing Cost: Time

q 1 2

q1 2

2 s

2

z

l 1

q j

ql

l 1 j 1

s

2s

Late Time Cost :

s

ql

0

y

t1

t

k 1

k 1 q i Slope q k i 1

s

s

2s

q k

10

y

Early Queuing Cost: i 1

q i 2 2

s

q[-i]

20 q[-i]/s

yyy

y

t1t1 tt t1 t

kkk 11 k 1

Early Time Cost :

kkk 11 qq ii k 1 q i

qq kk q k qq kk q k

222ss iii 11 sss s 2s yi 1k 1 q i q k k 1 i 1

s

2s

q k

30 Cumulative Operations

60

40

20

0

20

Fig. 2. Geometric derivation of time costs.

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the instant the window starts plus the area of a triangle associated with aircraft operating in the tail of the window when C[t] ¼C*w and the slope (arrival rate) is αs⧸(α þ γ). At the beginning of a late slot window, the queue is empty of aircraft from the previous window and late time is increasing during the window, so there must be a group of arrivals sufficient to raise expected queuing and layover costs to the equilibrium level C*w. Let h be the number of aircraft that arrive at this instant. The airport randomizes their positions in the queue, so their queuing time will vary from 0 to h/ s, with expected queuing time h/(2s). Similarly the additional late time experienced by these aircraft will vary from 0 to h/(s), with expected value h/(2s). The additional late time experienced by an aircraft operating at the end of the window, when the queue empties is qi/s. Aircraft operating at the beginning and ending of the slot window, must experience the same equilibrium cost C*w. Therefore, it must be that (α þ γ)h/(2s)¼qi/(2s), which emplies h¼ 2γqi/(α þ γ). The total queuing time of all aircraft operating at the beginning of a late slot window is h2/(2s). The base of the triangle associated with the aircraft in the tail of the window is the queuing time when the first of them operates. There are qi–h ¼qi ((α  γ)/(α þ γ)) such aircraft. At arrival rate αs⧸(α þ γ), it takes them q (α  γ)/(αs) min to arrive, but they only require qi (α  γ)/ ((α þ γ)s) min to be served. The queue must have been busy with prior arrivals for qi ((α  γ)/(α þ γ))γ/(αs) min when the tail arrivals began. Multiplying this base by half the triangle's height (i.e., the number of aircraft) gives the total queuing time of the aircraft in the tail, γq2i ((α  γ)/(α þ γ))2/(2αs). The sum of these two triangles simplifies to γq2i /(2αs), which is the same as the queuing time of the late aircraft in an unconstrained bottleneck model. The airport authority chooses q  y ; …; q  1 ; q1 ; …; qz to minimize the total cost of aircraft operations in a multiple slotconstraint system with a given numbers of y early and z late slot windows. Its objective function is (

Minimize q  y ; …; q  1 ; q1 ; …; qz

y

∑

i¼2

!

α βðq  i Þ2 α βðq  1 Þ2 α gðq1 Þ2 þ þ 2αs 2αs 2αs

y

j

j¼1

s

∑

i¼1

 λ1

i1q

þ

y

z

i¼1

i¼1

! ! z qi ðq  i Þ þ ∑ 2s i¼1 !

∑ q  i þ ∑ qi  m  λ 2



s

γ q1 s

β qi

s

i1q j

∑

j¼1s

þ



s

γ qj s

 λ1 þ λ2

j¼1

s

γ qj s

β q  1 γ q1 

s

 λ1  λ2

γ s

β s

¼ 0;

ð5Þ

¼ 0;

ð6Þ

 λ1 ¼ 0; f or i ¼ 2; …; y;

ð7Þ

 λ1 ¼ 0; f or i ¼ 2; …; z;

ð8Þ

¼ 0;

ð9Þ

and ∑yi ¼ 1 q  i þ ∑zi ¼ 1 qi  m ¼ 0:

ð10Þ

To solve this system, sum Eq. (7) with i equal to 2 to y to obtain ∑yi ¼ 2

β qi s

þ ðy 1Þ∑yj ¼ 1

β qj s

 ðy  1Þλ1 ¼ 0:

ð11Þ

Now multiply Eq. (5) through by s/β, add it to (11), and isolate the summation on the LHS ∑yi ¼ 1 q  i ¼

ys 1 λ2 : λ1 þ ðyþ 1Þ ðyþ 1Þβ

ð12Þ

The same approach applied to (8) and (6) produces ∑zi ¼ 1 qi ¼

zs 1 λ2 : λ1  ðz þ 1Þ γ ðz þ 1Þ

ð13Þ

Substituting (12) and (13) into (10) gives ys 1 zs 1 λ1  λ2 þ λ2 ¼ m: λ1 þ ðy þ 1Þ ðz þ 1Þ γ ðz þ 1Þ ðy þ 1Þβ

ð14Þ



s

þ ∑yj ¼ 1

β qj s

∑zj ¼ 1

γ qj s

¼ λ2



β s

þ

γ s

 :

ð15Þ

The first two terms of (15) cancel out by Eq. (9). Substituting the summations with their values from (12) and (13) yields   y β z γ β γ þ λ1 þ λ2  λ1 þ λ2 ¼ λ2 : ð16Þ ðy þ 1Þ ðz þ 1Þ ðy þ1Þ s ðz þ1Þs s s

! ! qi ðqi Þ 2s

 :

s

β qj

þ ∑zj ¼ 1

s

β qj

y

þ ∑

s

γ qi

þ ∑yj ¼ 1

þ ∑zj ¼ 1

s

βðq  1 Þ γ ðq1 Þ s

β q1

β q  1 γ q1

α ð2 γ qi Þ2 þ þ ∑ 2 2 α s ðα þ γ Þ2 i ¼ 2 2 s ðα þ γ Þ þ ∑

The first order necessary conditions for problem (4) are

Subtracting (6) from (5) gives

!   α γ qi 2 ðα  γ Þ2

z

7

ð4Þ

Problem (4) is simply the sum of operating time costs as derived above, plus two constraints explained below. The first line of (4) is the cost of queuing time. The first two terms are the sum of all queuing costs from early slot windows, and the second two terms are the sum of all queuing costs from late slot windows. The second line is the sum of early- and late-time costs. The third line contains the constraints. The first constraint requires that the total number of aircraft is m. The second constraint requires that aircraft are indifferent between operating during slot-windows  1 and 1, so they may be treated as a common slot window. This constraint only binds when the numbers of early and late slot windows are not equal, and it only shifts aircraft between slot windows  1 and 1 – it has no effect on the optimal quantities of permits in the other periods.9 9 Eliminating the constraint results in the optimal number of permits in for these windows being given by Eq. (25) instead of (21). To verify that, set λ2 ¼ 0 in Eqs. (5, 6, and 16), solve (16) for λ1 , and use Eqs. (5)–(8) as before to solve for the optimal quantities.

Solving for λ2 in terms of λ1 gives  yðz þ 1Þ s  ðy þ 1Þ z s λ ¼ λ2 : β y ðz þ 1Þ þ γ ðy þ 1Þz 1



Substituting 17 into 14 and solving for parameters alone provides

λ1 ¼

ðβ y ðz þ 1Þ þ γ ðyþ 1ÞzÞβ γ m ðβ þ γ Þ2 y z s

:

ð17Þ

λ1 in terms of the ð18Þ

Solving (17) for λ2 .

λ2 ¼

ðy  zÞβ γ m

ðβ þ γ Þ2 y z

:

ð19Þ

Multiplying Eqs. (5) and (6) by s=β and s=γ respectively, adding them together to eliminate λ2, and using (10) to substitute m for the summations generates   s s þ λ1 ¼ 0: ð20Þ q  1 þq1 þ m 

β γ

Using (9), q  1 ¼ γ =β q1 , so (20) may be solved for q  1 and q1 q1 ¼

β ðβ y þ γ zÞ m γ ðβ yþ γ zÞ m and q  1 ¼ : ðβ þ γ Þ2 y z ðβ þ γ Þ2 y z

ð21Þ

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8

To find q  i for i¼2, …, y, subtract β q  1 =s from both sides of (7) to obtain

β qi s

þ ∑yj ¼ 1

β qj β q1 s



s

 λ1 ¼ 

β q1 s

; f or i ¼ 2; …; y:

ð22Þ

Adding (22) to itself for i¼ 2, …, y yields β qj

βq

þ ðy  1Þ∑yj ¼ 2 s  j ¼ ðy 1Þðλ1  β qs 1 Þ; or equivalently;   ðy  1Þ s λ1 ∑yj ¼ 2 q  i ¼ q1 ð23Þ y β ∑yj ¼ 2

s

Since (7) also implies qi ¼qj for i, j A ½2; y, (23) solves for qi by substituting the values of λ1 and q  1

γm qi ¼ f or i A ½2; y: ðβ þ γ Þy0

ð24Þ

A similar procedure applied to Eq. (8) produces qi ¼

βm for iA ½2; z: ðβ þ γ Þz0

ð25Þ

Substituting the q's into the objective function generates the optimal total, marginal, and average cost functions

β γ m2 β y ðz þ 1Þ þ γ z ðy þ 1Þ : Letting ðβ þ γ Þs 2 ðβ þ γ Þ y z βγ β y ðz þ1Þ þ γ z ðy þ 1Þ and Γ ¼ ; then δ¼ ðβ þ γ Þ 2 ðβ þ γ Þ y z δ Γ m2 2δΓm δΓm TC ¼

TC ¼

; MC ¼

s

s

; and AC ¼

s

:

ð26Þ

Eqs. (21), (24), and (25) specify how many permits the airport authority should issue per slot window qi and the duration of each window qi/s to minimize the sum of aircraft operating costs for given numbers of early and late slot windows. Eq. (26) gives the minimized value of operating costs (TC) and it gives the marginal (social) cost (MC) of an additional aircraft, and the average (private) cost (AC) that an additional aircraft experiences itself. As with the standard steadystate model, if there is no pricing of quantity constraints, additional aircraft will demand operations until their marginal willingness to pay equals the private cost, AC. This is inefficient because these aircraft impose delays on others in addition to those they experience. To obtain the efficient level of demand, the airport authority should limit the number of permits so that the marginal willingness to pay equals the marginal social cost (MC). The MC and AC functions are airport supply curves because they associate the quantity of operations with their cost. They are used as such throughout the remainder of this paper. In the atomistic case, for example, the airlines would choose m such that the marginal social benefit (demand) equals the average cost AC (which is the same across all time periods when there are no slot constraints), while the social planer should choose m such that the marginal social benefit equals the marginal social cost MC of adding an aircraft to the peak period (aircraft in different slot windows experience different operating costs, but slot constraints prevent them from equating individual operating cost with their marginal benefit). With linear demand, m¼ η  π p for example, the equilibrium and optimal number of aircraft are sη 2δm Γ ) m¼ p ¼ MC ¼ Γ s sþ δ Γ π U sη ) m¼ sþ 2 δ Γ π U

p ¼ AC ¼

δm s

ð27Þ

When the numbers of slot windows y and z both equal 1, the model produces the same total cost function, βγ⧸(β þ γ) m2/s, as the unconstrained equilibria of Arnott et al. (1993). If the airport imposes no quantity constraint (or ineffective constraints), the quantity demanded equals the unconstrained equilibrium quantity. The airport authority can halve congestion costs by half

Table 1 Values of Γ: percent of minimum cost achieved by number of slot windows. Early\late

1

2

3

4

5

10

50

100

1 2 3 4 5 10 50 100

200 167 156 150 147 140 135 134

183 150 139 133 130 123 118 117

178 144 133 128 124 118 112 112

175 142 131 125 122 115 110 109

173 140 129 123 120 113 108 107

170 137 126 120 117 110 105 104

167 134 123 117 114 107 102 101

167 134 123 117 114 107 102 101

Note: assumes β ¼ 7.5 and γ ¼ 15.

setting both y and z equal 2 and issuing the optimal number of slot constraints. This equilibrium is the same as Arnott's uniformtolled equilibrium, except that the slot owners keep the value of the un-assessed toll. In the limit as y and z go to infinity, the model produces the same total cost function, βγ⧸(2(β þ γ))  m2/s as the fine toll of Arnott et al. This multiple-slot window model provides a general unified solution covering the entire range of precision in slot constraint systems. Table 1 shows how the efficiency of the slot-constraint system varies with the number of early and late slot windows.10 The table gives the values of Γ using cost parameters, α, β, and γ, that are typical of those Daniel and Harback (2008) estimate for major hub airports in the US, but the overall efficiency results in the table are not particularly sensitive to variations in these parameters. The cost parameters do affect the relative advantage of early versus late slot windows. Four slot windows with the optimal quantities of permits recover half of the efficiency loss from congestion, but eight slot windows are necessary to recover eighty percent. Such slot windows would be about ten minutes long for a typical peak. Slot constraints with time windows that cover the entire peak period cannot spread atomistic traffic within the peak periods, so they cannot improve the efficiency of aircraft scheduling. Moreover, the total traffic volume over the complete peak period is equal to the service capacity, so this type of quantity constraint does not reduce the amount traffic that may participate in the peak. It follows that such slot constraints are completely ineffective for atomistic traffic. Marketability of these slot permits cannot improve their efficiency because they confer identical operating rights, so market prices cannot differentiate between operating times. Real-world slot-constraint systems that are designed for steadystate traffic typically have one-hour windows. Such slot-constraint systems do little to reduce congestion, but they may serve as barriers to entry by potential competitors (Dresner et al., 2002). Studies of traffic and congestion patterns at US and Canadian airports by Daniel and Harback (2009) and Daniel (2011) demonstrate that slot constraints did not eliminate congestion at Toronto's Pearson International, Washington National, LaGuardia, and J.F. Kennedy Airports. Conclusion 1. Optimal slot-constraint systems spread the flow of traffic during peak periods to reduce the accumulation of aircraft in the queues. Airports cannot directly control the traffic rates within slot windows, but they can effectively turn traffic flows on and off by controlling admission to the queue. They can configure the slot windows and quantities of permits to periodically turn the flow on and allow the queue to fill, then turn the flow off and allow the queues to empty. By reducing the accumulation of aircraft in the queue, slot constraints reduce the total amount of queuing delay. Slot constraint systems increase in effectiveness as 10 Table 1 also applies in the dominant-fringe equilibria, supra. It includes the efficiencies resulting from rescheduling aircraft, not from optimizing the number of aircraft.

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the number of slot windows increases and the duration of the slot windows narrows. 2.3. Airline decisions: dominant and fringe aircraft Now suppose the airport authority is passive and allows dominant and fringe airlines to schedule their aircraft in unconstrained bottleneck equilibria. A dominant airline coordinates its traffic to internalize delays that its aircraft impose on each other, while fringe airlines operate their aircraft independently of one another (atomistically). The dominant airline schedules d ¼d[pd] aircraft per peak period, where pd is the private price of a dominant aircraft operation, to be determined below. The fringe airlines operate f¼ f[pf] aircraft per peak period where pf is the private price (AC) experienced by an atomistic aircraft. The dominant airline acts as a Stackelberg leader by scheduling its aircraft subject to the atomistic traffic following it and scheduling aircraft to satisfy the equilibrium conditions given by Eqs. (1)–(3). Therefore, the aggregate traffic rates must satisfy Eq. (3) during the period in which the fringe operates, even if some dominant aircraft also operate at the same time. The “atomistic peak” refers to the period in which the atomistic fringe and possibly dominant aircraft operate, as opposed to the “peak period” which includes the internalizing dominant aircraft in addition to the atomistic peak. Let rd[t] and rf[t] and be the dominant and fringe arrival rates. The best (scheduling) responses of the atomistic fringe as functions of the dominant airline's arrival rates are 8 0; > > h i > > > > < Max 0; αsαβ r d ½t h i r f ½t; r d ½t ¼ > > Max 0; αsþα γ r d ½t > > > > : 1; ðfinite simultaneous arrivalsÞ

for C½t4 C n

for late periods; and for C½to C n :

ð28Þ The fringe best response function in Eq. (28) shows that the dominant airline cannot change the aggregate traffic rates to internalize its aircrafts' costs by rescheduling them within the atomistic peak. To internalize self-imposed delays, the dominant airline must shift some aircraft out of the atomistic peak. The least costly rate at which to schedule these internalizing aircraft is the service rate s, because they cannot obtain service more rapidly than rate s and there is no advantage in scheduling aircraft at rates below s. Let tdb and tde be the times on opposite sides of the atomistic peak at which dominant aircraft begin and end operations. These times must be such that their costs are identical, otherwise the airline could reduce costs by moving aircraft from the higher- to the lower-cost period. So β(t* tdb)¼ γ(t* þtde) and tde  tdb ¼(dþ f)/s. Let tab and tae denote the beginning and ending of the atomistic peak. It follows that the dominant airline's traffic pattern for aircraft not scheduled during the atomistic peak is r d ½t ¼ s; for t ϵ½t db ; t ab  and for t ϵ½t ae ; t de ;

ð29Þ

where t db ¼ t n 

there are no slot constraints so there is one unconstrained period (i.e., y¼z ¼1), in which case Γ reduces to one, and after substituting (fþ x) for m, the private price (AC) that fringe aircraft experience is δ(f þx)/s. Substituting the private price of fringe operations in the fringe demand function and solving for equilibrium quantity yields s

δ f^ ½x þ x s δðf þ xÞ e ; and f^ ½x ¼ η  π pf ½f ; x ¼ s s s ηs  π δ x ) f^ ½x ¼ : ð30Þ sþπ δ Adding the total cost of the dominant aircraft in the atomistic peak and substituting the fringe demand gives the dominant airline's objective function. The dominant airline's problem is to choose the number of aircraft to schedule during the atomistic peak subject to the fringe demand depending on the dominant airline's choice ( ) e e δ ðd  xÞððd  xÞ þ 2ðf^ þ xÞÞ δ ðf^ þ xÞx Minimize þ ; x 2s s e η sδ π x : s:t: 0 r x r d; and f^ ¼ sþδ π

γ dþf β dþf and t de ¼ t n þ : βþγ s βþγ s

Let x be the number of aircraft that the dominant airline schedules during the atomistic peak. There are γ/(β þ γ)  (d x) early aircraft that experience early time of (d þx þ2f)γ/((β þ γ)2s) and β/(β þ γ)  (d-x) late aircraft that experience late time of (d þx þ2f)β/((β þ γ)2s). Multiplying the numbers of aircraft by their time values and average delay times gives the total cost of the internalizing dominant aircraft, δ(d-x)(d þx þ2f)/(2s). Recall from Eq. (26) that the optimized total, marginal, and average cost functions are of the form Γδm2/s, 2Γδm/s, and Γδm/s where Γ is the parameter specified in Eq. (26). In this subsection,

ð31Þ

The first term of (31) is the total cost for the internalizing dominant aircraft and the second term is the total cost of the noninternalizing dominant aircraft. It is easy to see that the solution to this problem is to choose the number of aircraft to schedule atomistically as follows: e x^ ½d ¼

for early periods;

9

δπd : sþδ π

ð32Þ

Let ϕ ¼ δπ⧸(sþ δπ) denote the fraction of dominant aircraft scheduled during the atomistic peak. Substituting ϕd for x[d] in the dominant airline's average cost (AC) yields the airport supply function, which determines the private price for dominant operations in unconstrained equilibria. Let the supply and demand functions for dominant airline operations be    2 ped ½f ; d ¼

δ

2 f þ d 1þϕ 2

and

s

d ½ped  ¼ μ  ρ ped ½f ; d: e

ð33Þ

The number of aircraft in the unconstrained equilibrium simultaneously satisfies the supply and demand functions as given in Eqs. (30) e e and (33). Let f^ and d^ be the number of aircraft in the unconstrained e

e

equilibrium. Let p^ f and p^ d be the equilibrium full prices of fringe and dominant aircraft. The simultaneous solutions of (30) and (33) for unconstrained equilibrium prices and quantities are

μ δ ^e e 2 e 2 ^e ^η f^ ¼ f φ δ ρ sðϕ þ 1Þ þ 2 s  φð2 δ π ϕ sÞ; p^ f ¼ s ðf þ ϕ d Þ;

e μ η δ e 2 e e d^ ¼ ð2 δ π s þ 2 s2 Þ  ð2 δ ρ sÞ; and p^ d ¼ ð2f^ þ ð1 þ ϕ Þd^ Þ; 2s φ φ ð34Þ

where

φ ¼ 2 s2 þ δ2 π ρðϕ  1Þ2 þ δ sðρðϕ2 þ 1Þ þ 2 π Þ: Note that ϕ is the fraction of aircraft that the dominant airline schedules atomistically. This fraction equals zero when the slope of the fringe demand curve is zero and approaches one as the slope goes to minus infinity. Eq. (29) describes the behavior of the (1  ϕ)d internalizing dominant aircraft. Conclusion 2. In the unconstrained dominant-fringe equilibrium, the dominant airline fully internalizes when fringe demand is perfectly inelastic; partially internalizes when demand is imperfectly elastic; and approaches fully atomistic behavior as the fringe

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approaches perfectly elastic demand. The dominant airline internalizes delays by scheduling some aircraft to operate at the service rate just before and after the atomistic peak. 2.4. Airport decisions with dominant and fringe aircraft In this section, the airport authority takes the numbers of early (y) and late (z) slot windows parametrically (again, it may choose these, but the choice is not endogenous to this model) and chooses the number of permits and hence the timing of the slot windows. As before, airlines choose the timing of their operations within the slot windows in which they are allowed to operate. Several equilibria are possible, depending on whether the airport can control the distribution of slots between dominant and fringe airlines. When an airport controls the distribution of permits, then it can chose the socially optimal number of aircraft of each type. It can also facilitate internalization by the dominant airline by segregating it from the fringe. Both dominant and fringe airlines will prefer to operate during the more central slot windows closer to t*, because these slot windows have lower equilibrium costs. For this reason, and perhaps others, the airport authority may not be permitted to discriminate between airlines when assigning slot permits. When the airport does not control the distribution of slots between dominant and fringe airlines, then it cannot choose the optimal number of each type of operation or facilitate internalization by segregating operations. 2.4.1. Deriving the segregating equilibria Suppose the airport authority can determine the number of aircraft and assign slot windows for each airline. As before, Eqs. (21), (24), and (25) determine the timing of the slot windows during the atomistic peak and the optimal amount of traffic to assign to each slot window as functions of the number of peak operations (f þx). Eq. (26) specifies the social cost of these atomistic aircraft operations. Adding the social cost of the internalizing dominant aircraft gives the social cost of all aircraft operating during the busy period: !   ðd xÞ ðf þ xÞ δ ðf þ xÞ2 C s ¼ δ ðd  xÞ þ þΓ ð35Þ 2s s s Differentiating (35) with respect to f, determines the marginal social cost (MC) of fringe operations; ∂C s δ ðd  xÞ 2 Γ δðf þ xÞ þ : ¼ s s ∂f

ð36Þ

The airport authority equates the marginal social cost (MC) of fringe operations with the marginal social benefit given by its inverse demand function s η  f δ ðd  xÞ 2 Γ δðf þ xÞ η s  δ π d  ð2 Γ  1 Þδπ x þ ) f^ ½x; d ¼ ¼ : s s π s þ 2Γδ π

ð37Þ

Note that this is the socially optimal number of fringe aircraft in a slot-constrained bottleneck equilibrium, which accounts for the external delays the fringe aircraft impose on others – it is not the privately optimal number from the unconstrained equilibrium. The airport authority also chooses the number of dominant aircraft that behave atomistically, x, to minimize social costs by setting the derivative of Eq. (35) with respect to x equal to zero,11 11 The social benefits of dominant operations are unaffected by whether they internalize or not.

s substituting f^ ½x; d from Eq. (37) into the result, and solving for x^ ½d

∂C s ð2Γ  1Þδðf þx½dÞ δ π dη s ¼ 0 ) x^ ½d ¼ ¼ : ð38Þ s ∂x sþδ π s Note that f^ ½x; d and x^ ½d cannot simultaneously both be positive, so non-negativity constraints require that x^ ½d be zero whenever there is fringe traffic in the atomistic peak (and vice versa). The airport authority would not schedule dominant aircraft among the fringe aircraft when the alternative is to have them fully internalize their delays. It remains for the authority to choose the number of (internalizing) dominant operations by setting their marginal social cost (MC) equal to their marginal social benefit and s ^ Finally, substitutes substituting f^ ½x; d from Eq. (37) to solve for d. ^d back into (37) to obtain the reduced-form solutions for the total number of slot constraints to issue: ∂C s δðf þd þ ð2Γ  1Þðf þ x½dÞx' ½dÞ μ  d ¼ ¼ ) s ∂d ρ d^ ¼ f^ ¼

sðsμ  δηρ þ 2Γδμπ Þ s2 þ ð2Γ  1Þδ ρπ þsδðρ þ 2Γπ Þ 2

sðsη þ δηρ  δμπ Þ s2 þ ð2Γ  1Þδ ρπ þ sδðρ þ 2Γπ Þ 2

:

and ð39Þ

There are two ways of implementing this slot constraint system. The first approach is for the airport authority to make a total of f^ slot permits available to the fringe to operate during the atomistic peak. It divides them among the slot windows as specified in Eqs. (21), (24), and (25). The airport authority could ^ β þ γ) and βd/( ^ β þ γ) slot also issue the dominant airline γd/( permits to operate immediately before and after the atomistic peak. These permits are only necessary if additional demand for off-peak periods requires that the airport reserve them for the ^ β þ γ) dominant airline. Second, the airport authority makes γd/( ^ β þ γ) slots available for the dominant aircraft to operate and βd/( just before and after t*. The dominant airline will schedule them at the service rate. The airport authority then issues f^ slot permits to the fringe and divides them among the slot windows as specified in Eqs. (21), (24), and (25). With homogeneous aircraft, these implementations have the same social cost, but the airlines with the central most slot windows experience lower operating costs. Fig. 3 illustrates the timing of dominant and fringe traffic, under the first implementation of slot constraint system. The thick dark line represents the unconstrained traffic pattern of the atomistic peak that is similar to the atomistic bottleneck equilibrium. Congestion externalities lead atomistic traffic to exceed the service rate during the early periods, because it ignores the effect of queuing delays it imposes on other traffic. The traffic rate falls below the service rate exactly when new operations complete service late. This allows the queue to empty at exactly the end of the atomistic peak. When a dominant airline internalizes selfimposed delays, it moves operations out of the atomistic peak to the adjacent regions on either side, as indicated by the light gray regions. This traffic operates at the service rate and creates no queuing. The solution in Eq. (32) determines the fraction of dominant traffic remaining in the atomistic peak. The darker shaded regions illustrate the slot-constrained traffic rates. The early atomistic traffic rate still exceeds the service rate because it must still satisfy the bottleneck equilibrium condition in Eq. (2). The airport cannot effectively set the traffic rates, but it can use the slot constraints to halt the traffic flow periodically to permit the queue to empty. Slot constraints spread the traffic more evenly over the peak – not by reducing traffic rates but by creating gaps and shifting traffic to low traffic periods. Slot constraints may also reduce the overall number of aircraft operations and affect the

Please cite this article as: Daniel, J.I, The untolled problems with airport slot constraints. Economics of Transportation (2014), http://dx. doi.org/10.1016/j.ecotra.2014.01.003i

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11

1.0

Traffic Rate

Unconstrained Atomistic Traffic Internalizing Dominant

0.8

Slot Constrained Atomistic Traffic

0.6

Service Rate

0.4

Internalizing Dominant

Internalizing Dominant

Atomistic Fringe and Dominant Traffic with Slot Constraints

0.2

Time 0.0

100

50

0

50

Fig. 3. Comparison of traffic patterns.

distribution of dominant aircraft between the atomistic peak and the dominant internalizing periods. In the equilibrium of Eq. (39), the airport authority would shrink the atomistic peak by moving all dominant operations to the gray regions. In the next scenario, however, the dominant airline may leave some traffic in the atomistic peak to preempt fringe operations. 2.4.2. Non-segregating equilibria Now suppose that the airport authority cannot issue slot permits that temporally segregate the dominant and fringe operations, or set the number of dominant operations to the socially optimal level. The dominant airline chooses the number of its aircraft to schedule to minimize its private costs (AC) subject to fringe demand under the slot constraint system. Eqs. (35)–(37) describe the social cost and the fringe behavior as before. The dominant objective function differs from the social cost in Eq. (35) in that the (fþx)2 in the last term is replaced by (fþx)x because the dominant airline is only concerned with its own private costs. The dominant airline accounts for the fringe's best response to its schedule as in Eq. (28), but the fringe now faces the full price given by Eq. (26). The dominant airline's problem and its optimal choice of x are

Minimize x



δ ðd xÞ

   ðd  xÞ ðf þ xÞ δ ðf þ xÞx þ þΓ ; 2s s s

η s  δ π d  ð2 Γ  1Þδπ x ; s þ 2Γδ π s ð3Γ  2Þπδ d  ðΓ  1Þs η s ) f^ d ) x^ d ¼ ð2 Γ 1Þðs þ 2 π δÞ 2 s2 η þsδð d þ η þ ΓηÞπ 3dΓδ π 2 : ¼ ðs þ 2δπ Þðs þ 2Γδπ Þ s

s:t: f^ ½x; d ¼

ð40Þ

The full price and the number of dominant operations are atcsd ¼

δðs2 ð  1 þ 2Γ Þðd þ ηÞ þ sΓδðdð  2 þ 4Γ Þ þ ð  1 þ 3Γ ÞηÞπ  dΓ 2 δ2 π 2 Þ sð  1 þ 2Γ Þðs þ 2δπ Þðs þ 2Γδπ Þ

s sð1  3Γ ÞΓδ ηρπ þ ð  1 þ 2Γ Þð2s2 ð1 þ Γ Þδμπ þ sðs2 μ  sδηρ þ 4Γδ μπ 2 ÞÞ : d^ d ¼ 2  Γδ ðsð4  8Γ Þ þ ΓδρÞπ 2 þ ð  1 þ 2Γ Þðs2 ðs þ δρÞ þ 2sδðs þ sΓ þ ΓδρÞπ Þ 2

2

ð41Þ The equilibrium in (40) and (41) has mixed dominant and fringe traffic during the atomistic peak and spans the whole range of possibilities, from fully internalizing behavior to fully atomistic behavior (in the limit) by the dominant airline as it perceives the fringe demand responsiveness varying from perfectly elastic to perfectly inelastic. Conclusion 3. If the airport authority can fully determine when the dominant and fringe aircraft operate, then it will separate their operations to enable the dominant airline to fully internalize. The airport either reserves all the peak-period slot permits for the fringe and the dominant airline fully internalizes by scheduling all aircraft off-peak, or it allocates all central peak slots to the dominant airline and the fringe schedules its aircraft on either side of them. These solutions have the same total social cost when dominant and fringe aircraft have the same time costs, but the aircraft operating in the central window have lower private costs than those outside the central window. If the airport is unable to enforce this separation, then the dominant airline will schedule some of its aircraft atomistically during the atomistic peak, as in the unconstrained equilibrium. Depending on the fringe airlines' demand elasticity, the dominant firm may full internalize, partially internalize, or behave atomistically.

Please cite this article as: Daniel, J.I, The untolled problems with airport slot constraints. Economics of Transportation (2014), http://dx. doi.org/10.1016/j.ecotra.2014.01.003i

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12

The (first-best) optimal slot-constraint system requires one slot permit and slot window per service interval. As the number of slot windows in Eqs. (40) and (41) approaches infinity (or more realistically one per service interval), the atomistic aircraft fully internalize the social cost of their operations, eliminating the distinction between internalizing and atomistic behavior. The airport authority can treat the “internalizing” dominant aircraft the same as all the “atomistic” aircraft because it no longer matters when the dominant aircraft operate during the busy period.

3. Policy discussion and conclusions Proponents of slot-constraint systems argue that congestion tolling requires assessing different congestion prices for dominant and fringe aircraft, but that price differentiation is politically difficult and imposes onerous information requirements. In principle, marketable slot permits can solve the optimal pricing problem without needing the airport authority to set tolls. Auctions or markets for slot permits improve on undifferentiated congestion pricing, because dominant airlines out bid the fringe and capture more slots, without the airport authority needing to set differentiated prices. These arguments are based on the standard congestion model for which reducing steady-state traffic rates is the only means of internalizing delays. The deterministic bottleneck model, with its dynamic congestion and endogenous traffic rates, is more applicable to airport traffic because it allows airlines to internalize delays by rescheduling aircraft. In a dynamic model, slot permits must have different slot windows to support different prices. Airports cannot prevent traffic from peaking within slot windows. Slot-constraint systems face an inherent conflict between efficient aircraft scheduling and feasibility of distribution: the former requires many narrow slot windows while the latter requires few slot windows of long deration. More slot windows means vastly more permutations of slot permits for the system to allocate.12 Aircraft also deviate randomly from their intended arrival times, making narrow slot windows problematic. The standard model ignores these dynamic issues while focusing on optimizing steady-state traffic rates. This approach would be reasonable if internalizing congestion were primarily a matter of reducing the total traffic volume rather than adjusting the timing of operations. Under the current weight-based pricing system, however, the market for airport operations clears because queuing time fluctuates rapidly to equate supply and demand. Reducing delays involves replacing the fluctuation in queuing costs with congestion fees, slot prices, or quantity restrictions, not simply reducing the traffic level. Calculating the marginal social costs of airport operations to determine the appropriate toll schedule is not really very difficult. In principle, the correct toll structure is simply the value of queuing-delay time under the untolled traffic pattern. The Federal Aviation Administration routinely calculates aircraft operating costs and passenger and crew time costs for use in cost-benefit analyses of airport improvement projects. It also collects information on airline fleet characteristics at all the major airports. These values and a correct understanding of the congestion technology are sufficient to determine the external delays. Daniel (1995) develops a stochastic bottleneck model and Daniel and Harback 12

The value of a slot permit depends on combining it with other slots. Within a single airport, slot portfolios must coordinate aircraft arrival and departure times with connecting aircraft and gate availability. With multiple airports, slot portfolios must coordinate arrival and departure times throughout the route network. Airlines must also coordinate slot combinations across time, by days, months, and seasons to provide coherent schedule frequency.

(2008) estimate time dependent airport delays from flight-level data at major airports. They show that the stochastic bottleneck model matches the delay patterns very well, and they determine toll schedules for all major US airports. Tolling solves or avoids a number of additional problems posed by slot constraints. The cost of administering time-dependent tolls should be comparable to the costs of administering the current weight-based pricing system. Control towers already gather the information needed to assess time varying tolls, based on when aircraft join arrival and departure queues. Aircraft would arrive at the airport and pay the toll, just as they currently arrive and “pay” the queuing time. Slot permits, however, must be distributed before airlines schedule their aircraft. They require some provision be made for random shocks to aircraft operating times, while assuring that airlines actually schedule aircraft (and operate them on average) at the permitted times. With tolling, airlines could schedule aircraft by checking the airport tolling schedule and traffic patterns to determine the full price of operations, without having to obtain slot permits. If the toll levels were correct, there would be minimal queuing. If the toll levels were incorrect, queuing delay would equate supply and demand as it currently does. Persistent queuing would provide a signal to revise the toll schedule. New airlines would be free to enter an airport market as long as they valued an operation more than its social cost, without obtaining prior permission. Slot-constrained airports may experience increasing concentration over time because of entry barriers. Tolled airports do not necessarily need to differentiate tolls between types of aircraft to optimize traffic patterns.13 In the absence of tolling, slots do require administrative rationing with differential treatment of airlines. Tolling generates sufficient revenues to pay for optimal airport capacity, and revenues provide signals to indicate the need for additional capacity. Depending on how slots are distributed, slots permits may confer unpriced benefits on airlines, depriving the airport of revenue. These benefits also give rise to costly rent-seeking behavior. Finally, tolling allows more flexibility to adjust to changing conditions over time than slot constraints, which often depend on historical usage patterns and/ or confer long-lasting property rights that are difficult to change. If slot constraints were the only policy available to mitigate airport delays, then the policy's difficulty in attaining the first-best optimum would be of little relevance. Congestion tolling, however, avoids these problems and requires only marginal changes in airport and airline practices. Daniel (2009) develops a multi-step tolling model and discusses its implementation. This paper demonstrates that effective slot constraint systems require issuance of slot permits that have many narrow slot windows and differentiate between dominant and fringe aircraft. Given the high administrative costs of the IATA's WSG system14 it is worth considering how the system will cope with the administrative costs of distributing highly differentiated slot permits, or alternatively; whether the minimal efficiency gains obtainable with hourly slot window are worth the high administrative and distribution costs.

Statement of contribution This paper develops a deterministic bottleneck model of airlines scheduling aircraft operations at congested airports. It demonstrates that slot constraints limiting the number of operations by hour would have little effect on congestion. The operating windows associated 13 A referee points out that it may be unfair to compare a slot-constraint system with a first best pricing system. Tolls would likely be by steps, which would have a similar effect on efficiency as the slot windows. I argue that adding steps to the toll schedule would impose much lower administrative and compliance costs than additional slot windows. 14 The WSG system involves biennial slot conferences attended by about one thousand people for three days to redistribute slot permits.

Please cite this article as: Daniel, J.I, The untolled problems with airport slot constraints. Economics of Transportation (2014), http://dx. doi.org/10.1016/j.ecotra.2014.01.003i

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with slots would have to be very short to approach the efficiency of first-best pricing. Distributing highly differentiated slots leads to high administrative costs for the airlines and airport. Acknowledgement The author would like to thank two anonymous referees for their many helpful comments and suggestions. References Arnott, R., de Palma, A., Lindsey, R., 1990. Economics of a Bottleneck. J. Urban. Econ. 24 (1), 111–126. Arnott, R., de Palma, A., Lindsey, R., 1993. A structural model of peak-period congestion: a traffic bottleneck with elastic demand. Am. Econ. Rev. 83 (1), 161–179. Borenstein, S., 1988. On the efficiency of competitive markets for operating licenses. Quart. J. Econ. 103 (2), 357–385. Brueckner, J.K., 2009. Price vs. quantity-based approaches to airport congestion management. J. Public Econ. 93 (5–6), 681–690. Braid, R., 1989. Uniform versus peak-load pricing of a bottleneck with elastic demand. J. Urban. Econ. 26 (3), 320–327. Condorelli, D., 2007. Efficient and equitable airport slot allocation. Riv. Politic. Econ. 97 (1–2), 81–104. Daniel, J., 1991. Peak-load congestion pricing of hub airport operations with endogenous scheduling and traffic-flow adjustments at Minneapolis-St. Paul airport. Transp. Res. Rec. 1298 (1991), 1–13. Daniel, J., 1995. Congestion pricing and capacity of large hub airports: a bottleneck model with stochastic queues. Econometrica 63 (2), 327–370. Daniel, J., 2001. Distributional consequences of congestion pricing at large hub airports. J. Urban. Econ. 50 (2), 230–258. Daniel, J., 2011. Congestion pricing of Canadian airports. Can. J. Econ. 44, 290–324.

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Please cite this article as: Daniel, J.I, The untolled problems with airport slot constraints. Economics of Transportation (2014), http://dx. doi.org/10.1016/j.ecotra.2014.01.003i