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Developments in airport capacity analysis
DEVELOPMENTS
IN AIRPORT
STEPHEN L. M. HOCKADAY
CAPACITY
ANALYSIS
and ADIB K. KANAFANI
Peat, Marwick, Mitchell & Co. Burlingame, California and University of California Berkeley, California, U.S.A. (Received 6 October 1973) Abstract-Runway capacity is defined as the maximum number of aircraft operations that can be handled during a specific period of time, under given operating conditions. The most important determinants of capacity are the aircraft mix, the length of the common approach path, and the operating strategy. Aircraft are postulated to deviate from intended paths while approaching a runway to land. These deviations are assumed to be normally distributed random variables with zero means. In order to maintain the probability of violations of aircraft separation rules, controllers are assumed to introduce buffers between aircraft in order to absorb the randomness in their separations. A capacity model is constructed with these postulates. The model yields runway capacity for various operating strategies and permits the choice of the optimal strategy for a given and intended arrival-departure mix. The application of the model is demonstrated with data from New York’s La Guardia Airport.
INTRODUCTION D@initions
Airport capacity analysis serves two purposes: (1) to estimate the ability of the various components of an airport system to handle passenger and aircraft flows; and (2) to estimate the delays experienced in the system at different traffic flow levels. Thus, in airport planning, capacity analysis permits both the determination of facility size requirements and evaluation of the effectiveness of alternative system component designs. The concept of total airport system capacity is a recent one. Previous capacity analyses have concentrated on the runway component of the airport system. In addition, there has not been a clear distino tion between airport capacity and airport level of service, commonly measured by delay. Most past capacity analyses were aimed at the estimation of delays at specific volume levels or the estimation of volumes that would lead to specific levels of delay. With the increasing awareness of strict and enduring environmental, energy, and physical constraints on airport growth it is becoming increasingly important to assess the true ability of an airport system to handle air traflic. In other words, “level of service” is becoming a term too vague to be used as a system performance measure. In this paper, the development of runway capacity analysis is reviewed, and a model of runway capacity is presented. The model is aimed at estimating the true capacity of a runway, or a system of runways, under varying operating conditions and strategies.
qf capacity
Although it might appear that there should only be a single definition for the capacity of a service facility, a number of different definitions have been used in airport capacity analysis. These definitions derive from two capacity concepts: “ultimate capacity” and “practical capacity”. “Ultimate capacity” is the maximum number of aircraft that can be handled by a facility during a specified time period under conditions of continuous demand. It is calculated as the reciprocal of the weighted average service time of the aircraft using the facility. “Practical capacity”, on the other hand, is defined as the number of aircraft operations that can be handled by a facility during a specified period of time such that the average delay to all processed aircraft equals a certain specified amount. The concept of practical capacity links delay with capacity, and by so doing, directly implies that the capacity of a facility is influenced by the arrival pattern of customers seeking service; i.e. by the user demand characteristics. In principle, this influence is not appropriate as it is possible under this definition to change the capacity of a runway (by rescheduling the arrival of aircraft and reducing the average delay) without changing the physical or operating characteristics of the runway. Nevertheless, the concept of practical capacity has been developed and has acquired wide use in runway capacity analysis during the last decade. 171
112 Runway
STFPHEN
capacity
L. M. HOCKADAY and Aom K. KANAFANI
analyses
Interest in runway capacity analysis dates back more than two decades. Most of the earlier work on the subject was concerned with the estimation of aircraft delays and employed models of queuing theory. Bowen and Pearcey (1948) using a Poisson arrivals model with constant service time, derived steady state average landing delays. Pearcey (1948) extended the work and derived the steady state delay distributions. Using a similar queuing model, Galliher and Wheeler (1958) derived nonstationary delay distributions to landing aircraft. The effect of approach path separations on landing aircraft delays was investigated by Oliver (1962). Mixed landing and takeoff runway operations were studied by the Airborne Instruments Laboratory (AIL) in 1963. AIL derived steady state and nonstationary delay distributions for both landing and takeoffs, assuming Poisson arrivals to the takeoff queue and a displaced exponential gap distribution for the landing arrivals into the landing queue. The effect of runway use priority rules in mixed operations on aircraft delays was investigated by Pestalozzi (1964). Based on the work of AIL, a series of airport capacity manuals was developed for the U.S. Federal Aviation Administration (FAA). These manuals, based on the practical capacity concept, are widely used in airport planning. One of the earliest models of runway capacity using the ultimate capacity concept was due to Blumstein (1960) who identified the factors that affect this capacity as the length of the common approach path, the aircraft speeds, and separation and runway occupancy times. Baron (1968) presented what seemed like an alternative capacity manual using a deterministic model of ultimate capacity for various types of operations. The next step was the incorporation of errors in navigation and air traffic control. This was done by treating the service times as stochastic variables. The U.S. National Bureau of Standards (1969) defined capacity as the limit (as T + co) of E{N(T)/T}, where N(T) is the number of aircraft served during the time interval ‘T; and analyzed capacity for various random distributions of the service time. Harris (1969) postulated normally distributed errors in aircraft interarrival times at the entry gate and at the runway threshold. He derived ultimate capacity by allowing time separation buffers to account for these errors. Harris applied his model to a single runway case, for t Extensions of the model to multiple runway configurations are not presented in this paper. The principles are the same as those presented here, but the algebra involved becomes quite tedious.
both exclusive arrival and departure a mixed operations runway. A model
of runway
runways,
and to
capacity
In this section, a model of runway capacity using the ultimate capacity concept is presented. The model is based on the work of Harris (1969) and incorporates a number of additional features. These include accounting for the effect of wake turbulence created by large jet aircraft separations, and the derivation of optimal operating strategies for specific proportions of arrivals and departures in the mix. The model also extends runway capacity analysis to the cases of multiple runways of various configurati0ns.f The basic postulate of the model is that aircrafi attempt to arrive at points in space (the entry gate 01 the runway threshold) at particular times in order to maintain intended separations. It is further postulated that the deviations from intended flight paths are normally-distributed with zero means, and that the deviations ofdifferent aircraft are independent. Because of these deviations from intended flight paths, air traffic controllers increase the separation between aircraft by buffers designed to reduce the probability of violation of aircraft separation rules. The runway capacity model consists of three consecutive steps as follows: A. Aircraft separations. Computation of time intervals between aircraft performing landing or takeoff operations. B. Capacity computations. Manipulation of these intervals to produce capacity estimates for variou: operating strategies. C. Operating strategy selection. Selection of combina. tion of operating strategies that produces the highesl capacity for any arrival-departure mix. In presenting the model, the case of a single runway is considered. A. Aircrqft
separations
Air traffic rules specify minimum separations betweer aircraft using the same system of runways. To repre. sent these rules the following variables are defined: AROR(i) Arrival runway occupancy requirement. The time separation that needs to be maintained between two consecutive arriving aircraft in order to ensure that the first aircraft clears the runway before the second aircraft touches down. The index refers to the class of the leading aircraft. AASR(ij) Arrival-arrival separation requirement. The time separation between two arriving aircraft (a lead aircraft of class i followed by an aircraft of class j: that needs to be maintained to ensure that ail separation rules are not violated.
Developments in airport capacity analysis DROR(i) Departure runway occupancy requirement. The time separation between a departure and a following operation that needs to be maintained to ensure that the departing aircraft of class i clears the runway before the next operation can ‘use it. DDSR(ij) Departure-departure separation requirement. The time separation that needs to be maintained between two consecutive departures class i followed by class j to ensure conformity to air separation rules. DASR(j) Departure-arrival separation requirement. The time separation that needs to be maintained between an oncoming arrival of class j and the release of a departure to ensure conformity to air separation rules.
173
rule would have to be respected at the threshold, and a larger separation would be required at the entry gate. In the case where ui 2 uj, the minimum separation would have to be respected at the entry gate and a larger separation would evolve at the threshold. These two cases are shown on the time-space diagrams of Figs. 1 and 2. If Tj is the separation at the threshold, then the probability of violation of the rules, pv, is given by for v(i) s v(j) pv = Prob.
(2)
Arrival runway occupancy requirement, AROR (i) Let Tj be the actual time separation between two consecutive arrivals, i followed by j. Under the basic assumption of the model: T$ _ N(Ti. afi) where u 4is the standard deviation of interarrival time.t The same assumptions apply to the runway occupancy time, AR(i), of the lead aircraft i: AR(i)
-
A buffer time separation B(ij) is introduced the above equation, where
B(ij) = u/D- ‘( 1 - p,). The arrival-arrival separation requirement maintain Tj 2 AASR(ij), where
and where Q-’ is the cumulative distribution function.
i
standard
normal
(3) is then to
wd __ + B(ij)
N(AR(i).q&J
where 0 ARis the standard deviation of arrival runway occupancy time. The runway occupancy requirement AROR(i) is set such that the rule IT;i> AR(i) is violated with a small probability pU(e.g. 0.05). Then AROR(i) = AR(i) + Jo’, + oL@- ‘(1 - p,) (1)
to ensure
4.4
for v(i) 5 v(j) AASR(ij) =
(4)
I
for u(i) > u(j)
or AASR(ij) = $;
+ r~,@- ‘( 1 - p,)
Arrival-arrival separation requirements, AASR(ij) The operating rules for the case of two consecutive arrivals, i followed by j, depend on the visibility conditions. Under visual flight rules (VFR) only separation requirements due to wake turbulence are applied, while under instrument flight rules (IFR) additional separation requirements are applied throughout the length of the common approach path. In deriving the requirements for AASR(ij), only the general case of IFR is considered here. Let aij be the minimum separation as specified by the rule, and y be the length of the approach path, and vi and vj be the respective ground speeds of aircraft i and j. In the case where vi 5 vi (the overtaking case), the minimum separation t If each aircraft’s deviation from its intended flight path is go, then ^. the time separation I_ _I error between the adjacent aircraft 1s gwen by cA = ,,/Zo&
Departure runway occupancy requirement, DROR(i) Let DR(i) be the runway occupancy time of departing aircraft i. The model postulates that DR(i) N N(DR(i), &)
(6)
where aDR is the standard deviation of departure runway occupancy time. The value taken for the runway occupancy requirement depends on the type of operation following the departure i. For a departure-departure sequence the controller has positive control over the release of each departure. Therefore, no buffer is added, and the random variable itself is used in the model.
STEPHENL. M. HWKA~AY
Distance approach Exit
J
and ADIR K. KANAFANI
along path
--IARORb+
Runway I threshold A
Time
t Entrygate
t
Distance approach
7k Exit\
Runway threshold A
”
along path
4
7y, (AA)-
L
ARORW,
111
TI me
Entry_ gate
Fig. 2.
Developments
in airport capacity analysis
For a departure-arrival sequence, since there is no positive control over the arrival operation, a buffer is added to account for variations in runway occupancy times. The runway occupancy requirement is then given by DROR(i)
= DR(i) + rrn,$‘- ‘( 1 - p,).
(7)
Note that variations in the arrival time of the trailing aircraft over the threshold are then so small that they may be disregarded.
the probability of an aircraft pair with an aircraft class i followed by an aircraft class j is ~(0) = p(i)&).
DDSR(ij)
= t,
(8)
where td takes on different numerical values depending on weather and on the presence of wake turbulence.
Departure-arrival separation requirement DASRCj) A dcparturc is not released if an arriving aircraft ,j will be within &,,, of the departure. In order to allow for the variations in the times between departure clearance and the start of departure roll, a buffer is added to this separation. The departure-arrival separation requirement is given by + o,W ’ (1 - p,).
DASR(j) = +
(9)
I B. Computation of capacity Runway capacity is calculated as the reciprocal of the weighted average separation times for different operations. In this section three types of capacity are considered: arrival capacity (landings only); departure capacity (takeoffs only); and mixed operations capacity (landings and takeoffs). Arrival capacity. Let TAA(ij) be the expected interarrival time between two consecutive landings, i followed by j. Then TAA(ij) = max[AASR(ij),
AROR(i)]
(10)
which means that the interarrival separation is governed by the larger of the air separation rule requirement and the runway occupancy requirement. If the proportions of aircraft of classes i and j in the mix are p(i) and p(j), respectively (and in the absence of aircraft sequencing), then it can be assumed that
(11)
Note that sequencing can be included in this model by directly specifying the values of p(ij). Using equation (IO), it is possible to average over all aircraft types and to obtain the expected interarrival time TAA as TAA = c p(ij)TAA(ij).
(12)
ij
Departure-departure separation requirement, DDSR(ij) In practice, traffic controllers estimate interdeparture time separations, t,,, that need to be applied in order to maintain distance separations specified by air traffic rules. These time separations usually include whatever buffers the controllers add to allow for deviations in times needed to execute departures. DDSR(ij) is then simply given by
175
The arrival capacity CAP(AA) CAP(AA)
is then given by
= &A.
(13)
Departure capacity. Let TDD(ij) be the expected separation between two consecutive departures, i followed by j. Then TDD(ij) = max[DR(i),
DDSR(ij)]
(14)
that is, the departure separation is governed by the larger of the runway occupancy rule and the air separation rule. Averaging over aircraft types as before gives the expected interdeparture service time TDD as TDD = c p(ij)TDD(ij)
(15)
ij
and the departure
capacity CAP(DD) CAP(DD)
as
= &
Mixed operations capacity. Computing the mixed operations capacity consists of inspecting interarrival times to determine whether any departures can be released within them. Based on the preemptive priority given to arrivals by air trafhc rules, departures can be released only if a sufficient interarrival gap is available so that the separation requirements are satisfied. In developing the model it is assumed that there is no need to consider more than three departures interleaved within any single interarrival gap. This simplifies the computations and is quite a realistic assumption for near-capacity situations in mixed operations. Let d(ij, kern) be the actual number of departures of an ordered triplet of potential departures of aircraft classes k, 8, and m, consecutively, that can leave within an interarrival gap TAA(ij). Since TAA(ij), TDD(kG), TDD(Gm), and AR(i) are random variables, d(ij, kCm) is also a random variable. It should be noted that the variance of the interdeparture time interval is zero when air separation requirements govern, because then
STIPHFY
I76
L. M.
HOCKADAY
controller has positive control on the departure. recalling equation (15): &kf)
=
I&,
Thus,
DR(k) 2 DDSR(kl)\
, otherwise
\O
(17)
I
then d(ij, kfm) 2 0 with probability
1
d(ij, k/m) 2 1 if TAA(ij) - AR(i) 2 Max[DASR(j),
DROR(k)].
Pl(ij. Urn) = 1 ~ @ DROR(k)} ~ y_TAA(ij)
X ,;o:
+
1
+
AW
AIIIB
K.
KANAPANI
N 1(ij, k/m) = 0 if no departure can leave, which act with probability 1 - Pl(ij, k/m); Nl(ij, kfm) = 1 i least the first departure (class k) leaves, which oc( with probability Pl(& kfm); N2(ij, kfm) = 0 if at IT one departure leaves, which occurs with probab 1 - P2([j. k/m); N2(ij, k/m) = 1 if at least 2 depart\ leave, which occurs with probability P2([j, ki N3(ij, k/m) = 0 if at most 2 departures leave, wf occurs with probability 1 - P3(ij, kfm); N3(ij, k/-m) if 3 departures leave, which occurs with probab P3(ij, kfm). Then the expected number of departures of class I first of a triplet of potential aircraft departures classes hkf. in an interarrival gap ij is
This occurs with probability
Max(DASR(,j),
and
m(ii, hkf)
c18l
o&q
d(ij, k/m) 2 2
Similarly, the expected number as second of a triplet kh/ is
if
m(ij,
TAA(ij) - AR(i) - TDD(W)
= Pl(ij, hkf).
of departures
of cla
k/z/) = P2([j, khf)
and as third of a triplet k/h, is
2 Max[DASR(j),
DROR(/)]
m(ij,
k/k)
= P3([j, k/h).
This occurs with probability The expected number D(ij, k) of departures in an interarrival gap (ij) is then
P2([j, k/m) = 1 - @
X
Max (DASR(J). DROR(/)) -zGVJ+ARo+~ Tcrm+rrX,+-;;(k/
(19) )
I
X [N(ij,
d(ij, k/m) = 3
DROR(m)].
P3(ij, k/m) = 1 - @ DROR(m)} - TAA(ij)
X v/c;
+
ai,
_t_=Wkf) +
oi(kf)
+?%?([j, kkf)
+m(ij,
+ TWfm) + ai
d({j, k/m) = 3 with probability
k/k)]
to
x [Pl(ij, kkf)
+ P2([j, kkf)
+ P3(ij, k/k)].
The expected number of departures average interarrival gap is then
This occurs with probability
+ A!(‘,
kkf)
D(ij, k) = 1 p(k)p(f b(h) k
- TDD(m)
2 Max[DASR(j),
Max(DASR(j),
b(h)
which is equivalent
if TAA([j) - AR(i) - TDD(kf)
D(& k) = 1 dk)df k
of cla
1
(20)
0.
From these formulae the expected number of departures of a definite class k is calculated as the sum of three quantities corresponding to the expected number of departures of class k as the first, second, or third departure in the triplet. Let Nl(ij, kfm), N2(ij, k/m), and N3([j, k/m) be (0, 1) random variables defined as follows:
of class k in
D(k) = 1 p(ij)D(ij, 12)
i
ij
and the flow of departures
of class k is
F(k) = D(k)/TAA However, summing these flows will produce an 01 estimate of the departure capacity of the airfield, to the mix distortion resulting from the select process of the departures (aircraft requiring a la separation from preceding departures are being , favored). A conservative estimate of the depart
Developments
capacity, allowing for a total departure correct mix of aircraft, is obtained as?
in airport
capacity
177
analysis
flow with the
Note that the arrival capacity is identical to that of an arrival only runway, i.e. CAP,(DA) and the corresponding
= CAP,(AA)
(29)
per cent arrivals PA(DA) is CAP,(DA)
PA(DA) = [cAP,(DA)
+
CAP,(m)]
(30)
100
0
%
Fig. 3.
C. Operating strategy selection The capacity of a runway system depends on the operating strategy used. Typically, if the runway is operated exclusively for departures, it would have a larger capacity than if it were operated exclusively for arrivals. The mixed operations capacity is normally somewhere between the two capacities depending on the relative proportions of arrivals and departures in the traffic stream. The departure mix is represented by the per cent of the total traffic consisting of arrivals. It is possible to serve any given per cent of arrivals with a number of different operating strategies. The purpose of this step of the model is to illustrate how these strategies can be combined to achieve the highest capacity or any given percentage of arrivals. A single runway can be operated under any one or a combination of the following strategies: Arrivals only. Departures only. Mixed operations. These strategies and the corresponding capacities are shown on Fig. 3. Under the “arrivals only” strategy the per cent of arrivals in 100, and the flow corresponds to arrival capacity. Likewise, under the “departures only” strategy, the per cent of arrivals is zero and the flow corresponds to departure capacity. If these two strategies are combined in such a way that each is used a certain proportion of the time over a period such as one hour, then it is possible to obtain flows as represented by line AB for the varying proportions for each strategy. However, as it is often possible to insert departures in a gap between two arriving aircraft, it is possible to obtain flows higher than those shown by line AB if a mixed operations strategy is adopted. t Capacity in fact lies between the values before and after the correction for mix distortion. However, the differences involved are usually too small to be of any significance. $ For a descrlptlon of the data sources, see Douglas Aircraft Co. <“Itrl. (1973).
arrivals
This is shown on Fig. 3 by line ACD. This line should theoretically connect A and B, but does not because the model is formulated with a maximum of three departures interleaved during any interarrival gap. However, it can be seen that the envelope ACB represents the combinations of strategies that yield the highest flow rate and thus is the capacity envelope. This envelope defines the runway capacity for any percentage arrivals and shows the strategy required to achieve this capacity. The calculations required to derive such an envelope are extensive, although quite simple in nature. They are efficiently performed with the aid of a digital computer. Model application. The application of the single runway capacity model is demonstrated in this section. For this demonstration, data were obtained from observations at La Guardia Airport in New York.f Departure capacity, arrival capacity, and mixed operation capacity are calculated for the single runway. This is followed by the derivation of a highest capacity envelope showing capacity for any per cent of arrivals. The following data are used in the calculations: aircraft mix: i = 1, 903% jets (DC8, B707, B727, B737, DC9) i = 2, 9.2% light aircraft (DH6, FA27, BE99). Mean arrival runway occupancy
time
AR(i) = (38,47) sec. Standard
deviation
of arrival runway occupancy (TAR= 6
Standard
deviation
XC.
of interarrival gA = 10 sec.
times
time
178
STI.PHEN
Probability
L. M. HOCKADAY and
CAPn(DA)
pr = 0.05.
CAP,,,(DA) = CAPA
Length of common
The following table summarized the capacities as calculated by themodel and compares them with actual flow rates observed at La Guardia Airport, New York, during capacity operations.
approach
paths
;’ = 8.4 miles. Mean departure
runway occupancy time DR(i) = (39, 34).
deviation
Hourly mixed operation Arrival flow Departure flow Per cent arrivals
of DR(i) crDR= 3 sec.
Time separation
between
departures 39
39
41
34 set
I I
t&j) = Distance
of arrival from departure
release
6du = 0 miles. Standard
= 35 aircraft!hr.
speeds ri = (130, 120) knots.
Standard
= 30 aircraftihr.
and
rules &j = 3 miles, for all i, j.
Approach
KANAFAI\;I
Mixed operations capacit)
of rule violation
Separation
AIIIH K.
deviation
of time to start a departure
roll
cc = 0 sec. Using thedata shown above it is possible to compute the separation requirements using equations (I), (5). (7). (8). AROR(i) = (66.2, 57.2) sec. AASR([j) =
99.53
125.84
99.53
106.45
DROR(i)
= (43.9, 38.9) sec.
DDSR(ij)
=
I
f;
;;
I
DASR( j) = 0 sec. With these calculated :
separations
the various
capacities
Arrival cupucit)~ CAP,(AA)
= ‘If6tt = 35 aircraft/hr.
Departure capacit) CAP,(DD)
= y9y
= 92 aircraft/hr.
are
Cuhlated 65 35 30 54 “<,
Ohserwd 65 33 32 5 I iII)
Using the calculated capacities it is possible to derive the envelope showing the highest flow rate possible for any given per cent of arrivals in the opxitions mix. This envelope is used in the selection of runway operating strategy for various types of demand. The envelope in Fig. 4 shows the figures obtained in this application. Point A represents the arrival capacity, 35, and point B the departure capacity, 92. Line AB gives the flow rates possible with combinations of these two strategies with varying proportions during I hr. Point C represents the mixed operations capacity. It lies on a line AD which gives the flow rates obtainable under mixed operations with varying amounts of departure releases between arrivals. Along the line AD, departures are released during available gaps. Different per cent arrivals are obtained by stretching the interarrival separation. The line CR shows the fows possible with the combination of the
Developments in airport capacity analysis mixed operations strategy with the departures only strategy. The envelope ACB then gives the highest flow rates possible and allows the user to select the operating strategy to meet any arrival-departure mix.
SUMMARY The capacity of a system of runways is defined as the maximum flow rate of operations that can be accommodated under specified operating conditions. Capacity is usually calculated for specific periods of time, most commonly for 1 hr. A model for calculating runway capacity is presented in this paper. The model is based on the postulate that aircraft deviate from intended paths and that these deviations are normally distributed random variables. For any given per cent of arrivals, capacity can be maximized by selecting from among three basic strategies and their combinations: arrivals only, departures only, and mixed operations. The proposed model is applied to calculate the single runway capacity using data observed at La Guardia Airport in New York. The results of the model compare sufficiently well with observed volumes at the airport. This favorable comparison may be attributed to the model’s detailed incorporation of operating rules and strategies, and of the stochastic nature of aircraft operations. The main advantage of the proposed approach compared with existing approaches is thought to be the fact that it separates capacity analysis from delay analysis. Although the two are closely related, they are completely different performance measures of runway system operations.
179
Acknowledgement-The models described in this paper were developed by the authors in cooperation with the project staff as part of a study entitled “Procedures for Determination of Airport Capacity” performed for the U.S. Federal Aviation Administration, by Peat, Marwick, Mitchell & Co. Numerous staff members have contributed directly or indirectly to the paper. Their assistance was greatly appreciated.
REFERENCES
Airborne Instruments Laboratory (1963). Operational Evaluation of Airport Runway Design and Capacity. AIL Report 7601-6. Deer Park, Long Island. Baran G. (1968) Airport Capacity Analysis. Report D6-23415. The Boeing Company, Seattle, Washington. Blumstein A. (1960). An Analytical Investigation of Airport Capacity. Report TA-1358-8-L Cornell Aeronautical Laboratory. Bowen E. G. and Pearcey T. (1948). Delays in the flow of air traflic. J. H. Azrontrf,t. Sot. 52, 251 2%. Douglas Aircraft Company, Peat, Marwick, Mitchell & Co., American Airlines (1973), Procedures for Determination of Airport Capacit;. Phase I Interim Report. Galliher H. P. and Wheeler R. C. 11958). Nonstationarv queuing probabilities for landing Long&tion of aircraff. O/X RLJS.6, 264~275. Harris R. M. (1969). Models for Runway Capacity Analysis. The Mitre Corporation Technical Report MTR-4102. Washington, D.C. Oliver R. M. (1962). Delays in Terminal Air Traffic Control. Symposium on Airport Capacity, Institute of Transportation and Traffic Engineering, Berkeley, California. Pearcey. T. (1948). Delays in the landing of air traffic. ./. K. AoYJ/lol,t. Sr,c. 52, 799 8 12. Pestalozzi G. (1964). Priority rules for runway use. Op. Res. 12, 941-950. Shapiro 0. (1965). An Airport Capacity Handbook: Results of a Federal Aviation Administration Research Program in Mathematical Models and Techniques for Determining Airport Capacity. Operations Research Society of America, Twenty-seventh Annual Meeting, Boston, Massachusetts.
R&umP La capacite des pistes d’atterrissage est dbfinie comme &tant le nombre maximal de manoeuvres d’avions pouvant etre acceptC au tours d’une pkriode de temps spkcifique dans les conditions opkrationnelles donnL:cs. Pkirmi les facteurs dkterminants de capacitt les plus importants, figurent le mklange de types d’appareils. la longueur de la trajectoire d’approche commune, et la strattgie opkrationnelle. On postule que les appareils dkvient de la trajectoire pr&uc lorsqu’ils approchent de la piste d’atterrissage. On suppose que ces dkviations sont des variables aleatoires h distribution normale dont la moyenne est nulle. On suppose que, pour maintenir la probabi1itC des violations de rtgles de siparation des appareils. les contrbleurs introduisent des kltments tampons entre les appareils pour compenser la nature al&atoire de leur stparation. A partir de ces postulats, on construit un modele de capacitC. Le modtle fournit la capacitk de pistes d’atterrissage pour diverses strattgies optrationnelles et permet de choisir la strattgie optimale pour un mklange donnt- d’arrivtes et de departs prCvus. On dCmontre l’application du modele ti l’aide de don&es provenant de I’ACroport de La Guardia g New York.
1x0
STEPHEN L. M. HOCKADAY and Arnn KANAFANI
Zusammenfassung-Die Leistungsfihigkeit von Start- und Landebahnen wird als die unter vorherrschenden Betriebsbedingungen gr6Btmogliche Anzahl von Flugbewegungen wLhrend eines bestimmten Zeitraumes definiert. Die wichtigsten EinfluBgrijDen fur die Leistungsfahigkeit sind Art und Mischung der Fluggerlte, die Llnge des gemeinsamen Anflugweges und die Betriebsregelung. Es wird davon ausgegangen, dal3 die Flugzeuge beim Landeanflug vom vorgegebenen Kurs abweichen. Diese Abweichungen sollen einer Normalverteilung mit dem Mittelwert Null entsprechen. Urn die Wahrscheinlichkeit von Unterschreitungen des Mindestanflugabstandes nicht zu grog werden zu lassen, wird angenommen, dalj die Fluglotsen Abstandspuffer zwischen aufeinanderfolgenden Flugzeugen vorhalten, urn die Zufalligkeiten in der Anflugfolge auszugleichen. Mit diesen Vorgaben wird ein Leistungsfahigkeitsmodell entwickelt, das Leistungswerte bei unterschiedlichen Betriebsregelungen liefert und die Auswahl einer optimalen Strategie bei gebener oder vorgegebener Abfolge von Starts und Landungen gestattet. Die Verwendungsmiiglichkeiten des Modells wird mit Hilfe von Betriebsdaten des Flughafens La Guardia in New York nachgewiesen.
IN AIRPORT
STEPHEN L. M. HOCKADAY
CAPACITY
ANALYSIS
and ADIB K. KANAFANI
Peat, Marwick, Mitchell & Co. Burlingame, California and University of California Berkeley, California, U.S.A. (Received 6 October 1973) Abstract-Runway capacity is defined as the maximum number of aircraft operations that can be handled during a specific period of time, under given operating conditions. The most important determinants of capacity are the aircraft mix, the length of the common approach path, and the operating strategy. Aircraft are postulated to deviate from intended paths while approaching a runway to land. These deviations are assumed to be normally distributed random variables with zero means. In order to maintain the probability of violations of aircraft separation rules, controllers are assumed to introduce buffers between aircraft in order to absorb the randomness in their separations. A capacity model is constructed with these postulates. The model yields runway capacity for various operating strategies and permits the choice of the optimal strategy for a given and intended arrival-departure mix. The application of the model is demonstrated with data from New York’s La Guardia Airport.
INTRODUCTION D@initions
Airport capacity analysis serves two purposes: (1) to estimate the ability of the various components of an airport system to handle passenger and aircraft flows; and (2) to estimate the delays experienced in the system at different traffic flow levels. Thus, in airport planning, capacity analysis permits both the determination of facility size requirements and evaluation of the effectiveness of alternative system component designs. The concept of total airport system capacity is a recent one. Previous capacity analyses have concentrated on the runway component of the airport system. In addition, there has not been a clear distino tion between airport capacity and airport level of service, commonly measured by delay. Most past capacity analyses were aimed at the estimation of delays at specific volume levels or the estimation of volumes that would lead to specific levels of delay. With the increasing awareness of strict and enduring environmental, energy, and physical constraints on airport growth it is becoming increasingly important to assess the true ability of an airport system to handle air traflic. In other words, “level of service” is becoming a term too vague to be used as a system performance measure. In this paper, the development of runway capacity analysis is reviewed, and a model of runway capacity is presented. The model is aimed at estimating the true capacity of a runway, or a system of runways, under varying operating conditions and strategies.
qf capacity
Although it might appear that there should only be a single definition for the capacity of a service facility, a number of different definitions have been used in airport capacity analysis. These definitions derive from two capacity concepts: “ultimate capacity” and “practical capacity”. “Ultimate capacity” is the maximum number of aircraft that can be handled by a facility during a specified time period under conditions of continuous demand. It is calculated as the reciprocal of the weighted average service time of the aircraft using the facility. “Practical capacity”, on the other hand, is defined as the number of aircraft operations that can be handled by a facility during a specified period of time such that the average delay to all processed aircraft equals a certain specified amount. The concept of practical capacity links delay with capacity, and by so doing, directly implies that the capacity of a facility is influenced by the arrival pattern of customers seeking service; i.e. by the user demand characteristics. In principle, this influence is not appropriate as it is possible under this definition to change the capacity of a runway (by rescheduling the arrival of aircraft and reducing the average delay) without changing the physical or operating characteristics of the runway. Nevertheless, the concept of practical capacity has been developed and has acquired wide use in runway capacity analysis during the last decade. 171
112 Runway
STFPHEN
capacity
L. M. HOCKADAY and Aom K. KANAFANI
analyses
Interest in runway capacity analysis dates back more than two decades. Most of the earlier work on the subject was concerned with the estimation of aircraft delays and employed models of queuing theory. Bowen and Pearcey (1948) using a Poisson arrivals model with constant service time, derived steady state average landing delays. Pearcey (1948) extended the work and derived the steady state delay distributions. Using a similar queuing model, Galliher and Wheeler (1958) derived nonstationary delay distributions to landing aircraft. The effect of approach path separations on landing aircraft delays was investigated by Oliver (1962). Mixed landing and takeoff runway operations were studied by the Airborne Instruments Laboratory (AIL) in 1963. AIL derived steady state and nonstationary delay distributions for both landing and takeoffs, assuming Poisson arrivals to the takeoff queue and a displaced exponential gap distribution for the landing arrivals into the landing queue. The effect of runway use priority rules in mixed operations on aircraft delays was investigated by Pestalozzi (1964). Based on the work of AIL, a series of airport capacity manuals was developed for the U.S. Federal Aviation Administration (FAA). These manuals, based on the practical capacity concept, are widely used in airport planning. One of the earliest models of runway capacity using the ultimate capacity concept was due to Blumstein (1960) who identified the factors that affect this capacity as the length of the common approach path, the aircraft speeds, and separation and runway occupancy times. Baron (1968) presented what seemed like an alternative capacity manual using a deterministic model of ultimate capacity for various types of operations. The next step was the incorporation of errors in navigation and air traffic control. This was done by treating the service times as stochastic variables. The U.S. National Bureau of Standards (1969) defined capacity as the limit (as T + co) of E{N(T)/T}, where N(T) is the number of aircraft served during the time interval ‘T; and analyzed capacity for various random distributions of the service time. Harris (1969) postulated normally distributed errors in aircraft interarrival times at the entry gate and at the runway threshold. He derived ultimate capacity by allowing time separation buffers to account for these errors. Harris applied his model to a single runway case, for t Extensions of the model to multiple runway configurations are not presented in this paper. The principles are the same as those presented here, but the algebra involved becomes quite tedious.
both exclusive arrival and departure a mixed operations runway. A model
of runway
runways,
and to
capacity
In this section, a model of runway capacity using the ultimate capacity concept is presented. The model is based on the work of Harris (1969) and incorporates a number of additional features. These include accounting for the effect of wake turbulence created by large jet aircraft separations, and the derivation of optimal operating strategies for specific proportions of arrivals and departures in the mix. The model also extends runway capacity analysis to the cases of multiple runways of various configurati0ns.f The basic postulate of the model is that aircrafi attempt to arrive at points in space (the entry gate 01 the runway threshold) at particular times in order to maintain intended separations. It is further postulated that the deviations from intended flight paths are normally-distributed with zero means, and that the deviations ofdifferent aircraft are independent. Because of these deviations from intended flight paths, air traffic controllers increase the separation between aircraft by buffers designed to reduce the probability of violation of aircraft separation rules. The runway capacity model consists of three consecutive steps as follows: A. Aircraft separations. Computation of time intervals between aircraft performing landing or takeoff operations. B. Capacity computations. Manipulation of these intervals to produce capacity estimates for variou: operating strategies. C. Operating strategy selection. Selection of combina. tion of operating strategies that produces the highesl capacity for any arrival-departure mix. In presenting the model, the case of a single runway is considered. A. Aircrqft
separations
Air traffic rules specify minimum separations betweer aircraft using the same system of runways. To repre. sent these rules the following variables are defined: AROR(i) Arrival runway occupancy requirement. The time separation that needs to be maintained between two consecutive arriving aircraft in order to ensure that the first aircraft clears the runway before the second aircraft touches down. The index refers to the class of the leading aircraft. AASR(ij) Arrival-arrival separation requirement. The time separation between two arriving aircraft (a lead aircraft of class i followed by an aircraft of class j: that needs to be maintained to ensure that ail separation rules are not violated.
Developments in airport capacity analysis DROR(i) Departure runway occupancy requirement. The time separation between a departure and a following operation that needs to be maintained to ensure that the departing aircraft of class i clears the runway before the next operation can ‘use it. DDSR(ij) Departure-departure separation requirement. The time separation that needs to be maintained between two consecutive departures class i followed by class j to ensure conformity to air separation rules. DASR(j) Departure-arrival separation requirement. The time separation that needs to be maintained between an oncoming arrival of class j and the release of a departure to ensure conformity to air separation rules.
173
rule would have to be respected at the threshold, and a larger separation would be required at the entry gate. In the case where ui 2 uj, the minimum separation would have to be respected at the entry gate and a larger separation would evolve at the threshold. These two cases are shown on the time-space diagrams of Figs. 1 and 2. If Tj is the separation at the threshold, then the probability of violation of the rules, pv, is given by for v(i) s v(j) pv = Prob.
(2)
Arrival runway occupancy requirement, AROR (i) Let Tj be the actual time separation between two consecutive arrivals, i followed by j. Under the basic assumption of the model: T$ _ N(Ti. afi) where u 4is the standard deviation of interarrival time.t The same assumptions apply to the runway occupancy time, AR(i), of the lead aircraft i: AR(i)
-
A buffer time separation B(ij) is introduced the above equation, where
B(ij) = u/D- ‘( 1 - p,). The arrival-arrival separation requirement maintain Tj 2 AASR(ij), where
and where Q-’ is the cumulative distribution function.
i
standard
normal
(3) is then to
wd __ + B(ij)
N(AR(i).q&J
where 0 ARis the standard deviation of arrival runway occupancy time. The runway occupancy requirement AROR(i) is set such that the rule IT;i> AR(i) is violated with a small probability pU(e.g. 0.05). Then AROR(i) = AR(i) + Jo’, + oL@- ‘(1 - p,) (1)
to ensure
4.4
for v(i) 5 v(j) AASR(ij) =
(4)
I
for u(i) > u(j)
or AASR(ij) = $;
+ r~,@- ‘( 1 - p,)
Arrival-arrival separation requirements, AASR(ij) The operating rules for the case of two consecutive arrivals, i followed by j, depend on the visibility conditions. Under visual flight rules (VFR) only separation requirements due to wake turbulence are applied, while under instrument flight rules (IFR) additional separation requirements are applied throughout the length of the common approach path. In deriving the requirements for AASR(ij), only the general case of IFR is considered here. Let aij be the minimum separation as specified by the rule, and y be the length of the approach path, and vi and vj be the respective ground speeds of aircraft i and j. In the case where vi 5 vi (the overtaking case), the minimum separation t If each aircraft’s deviation from its intended flight path is go, then ^. the time separation I_ _I error between the adjacent aircraft 1s gwen by cA = ,,/Zo&
Departure runway occupancy requirement, DROR(i) Let DR(i) be the runway occupancy time of departing aircraft i. The model postulates that DR(i) N N(DR(i), &)
(6)
where aDR is the standard deviation of departure runway occupancy time. The value taken for the runway occupancy requirement depends on the type of operation following the departure i. For a departure-departure sequence the controller has positive control over the release of each departure. Therefore, no buffer is added, and the random variable itself is used in the model.
STEPHENL. M. HWKA~AY
Distance approach Exit
J
and ADIR K. KANAFANI
along path
--IARORb+
Runway I threshold A
Time
t Entrygate
t
Distance approach
7k Exit\
Runway threshold A
”
along path
4
7y, (AA)-
L
ARORW,
111
TI me
Entry_ gate
Fig. 2.
Developments
in airport capacity analysis
For a departure-arrival sequence, since there is no positive control over the arrival operation, a buffer is added to account for variations in runway occupancy times. The runway occupancy requirement is then given by DROR(i)
= DR(i) + rrn,$‘- ‘( 1 - p,).
(7)
Note that variations in the arrival time of the trailing aircraft over the threshold are then so small that they may be disregarded.
the probability of an aircraft pair with an aircraft class i followed by an aircraft class j is ~(0) = p(i)&).
DDSR(ij)
= t,
(8)
where td takes on different numerical values depending on weather and on the presence of wake turbulence.
Departure-arrival separation requirement DASRCj) A dcparturc is not released if an arriving aircraft ,j will be within &,,, of the departure. In order to allow for the variations in the times between departure clearance and the start of departure roll, a buffer is added to this separation. The departure-arrival separation requirement is given by + o,W ’ (1 - p,).
DASR(j) = +
(9)
I B. Computation of capacity Runway capacity is calculated as the reciprocal of the weighted average separation times for different operations. In this section three types of capacity are considered: arrival capacity (landings only); departure capacity (takeoffs only); and mixed operations capacity (landings and takeoffs). Arrival capacity. Let TAA(ij) be the expected interarrival time between two consecutive landings, i followed by j. Then TAA(ij) = max[AASR(ij),
AROR(i)]
(10)
which means that the interarrival separation is governed by the larger of the air separation rule requirement and the runway occupancy requirement. If the proportions of aircraft of classes i and j in the mix are p(i) and p(j), respectively (and in the absence of aircraft sequencing), then it can be assumed that
(11)
Note that sequencing can be included in this model by directly specifying the values of p(ij). Using equation (IO), it is possible to average over all aircraft types and to obtain the expected interarrival time TAA as TAA = c p(ij)TAA(ij).
(12)
ij
Departure-departure separation requirement, DDSR(ij) In practice, traffic controllers estimate interdeparture time separations, t,,, that need to be applied in order to maintain distance separations specified by air traffic rules. These time separations usually include whatever buffers the controllers add to allow for deviations in times needed to execute departures. DDSR(ij) is then simply given by
175
The arrival capacity CAP(AA) CAP(AA)
is then given by
= &A.
(13)
Departure capacity. Let TDD(ij) be the expected separation between two consecutive departures, i followed by j. Then TDD(ij) = max[DR(i),
DDSR(ij)]
(14)
that is, the departure separation is governed by the larger of the runway occupancy rule and the air separation rule. Averaging over aircraft types as before gives the expected interdeparture service time TDD as TDD = c p(ij)TDD(ij)
(15)
ij
and the departure
capacity CAP(DD) CAP(DD)
as
= &
Mixed operations capacity. Computing the mixed operations capacity consists of inspecting interarrival times to determine whether any departures can be released within them. Based on the preemptive priority given to arrivals by air trafhc rules, departures can be released only if a sufficient interarrival gap is available so that the separation requirements are satisfied. In developing the model it is assumed that there is no need to consider more than three departures interleaved within any single interarrival gap. This simplifies the computations and is quite a realistic assumption for near-capacity situations in mixed operations. Let d(ij, kern) be the actual number of departures of an ordered triplet of potential departures of aircraft classes k, 8, and m, consecutively, that can leave within an interarrival gap TAA(ij). Since TAA(ij), TDD(kG), TDD(Gm), and AR(i) are random variables, d(ij, kCm) is also a random variable. It should be noted that the variance of the interdeparture time interval is zero when air separation requirements govern, because then
STIPHFY
I76
L. M.
HOCKADAY
controller has positive control on the departure. recalling equation (15): &kf)
=
I&,
Thus,
DR(k) 2 DDSR(kl)\
, otherwise
\O
(17)
I
then d(ij, kfm) 2 0 with probability
1
d(ij, k/m) 2 1 if TAA(ij) - AR(i) 2 Max[DASR(j),
DROR(k)].
Pl(ij. Urn) = 1 ~ @ DROR(k)} ~ y_TAA(ij)
X ,;o:
+
1
+
AW
AIIIB
K.
KANAPANI
N 1(ij, k/m) = 0 if no departure can leave, which act with probability 1 - Pl(ij, k/m); Nl(ij, kfm) = 1 i least the first departure (class k) leaves, which oc( with probability Pl(& kfm); N2(ij, kfm) = 0 if at IT one departure leaves, which occurs with probab 1 - P2([j. k/m); N2(ij, k/m) = 1 if at least 2 depart\ leave, which occurs with probability P2([j, ki N3(ij, k/m) = 0 if at most 2 departures leave, wf occurs with probability 1 - P3(ij, kfm); N3(ij, k/-m) if 3 departures leave, which occurs with probab P3(ij, kfm). Then the expected number of departures of class I first of a triplet of potential aircraft departures classes hkf. in an interarrival gap ij is
This occurs with probability
Max(DASR(,j),
and
m(ii, hkf)
c18l
o&q
d(ij, k/m) 2 2
Similarly, the expected number as second of a triplet kh/ is
if
m(ij,
TAA(ij) - AR(i) - TDD(W)
= Pl(ij, hkf).
of departures
of cla
k/z/) = P2([j, khf)
and as third of a triplet k/h, is
2 Max[DASR(j),
DROR(/)]
m(ij,
k/k)
= P3([j, k/h).
This occurs with probability The expected number D(ij, k) of departures in an interarrival gap (ij) is then
P2([j, k/m) = 1 - @
X
Max (DASR(J). DROR(/)) -zGVJ+ARo+~ Tcrm+rrX,+-;;(k/
(19) )
I
X [N(ij,
d(ij, k/m) = 3
DROR(m)].
P3(ij, k/m) = 1 - @ DROR(m)} - TAA(ij)
X v/c;
+
ai,
_t_=Wkf) +
oi(kf)
+?%?([j, kkf)
+m(ij,
+ TWfm) + ai
d({j, k/m) = 3 with probability
k/k)]
to
x [Pl(ij, kkf)
+ P2([j, kkf)
+ P3(ij, k/k)].
The expected number of departures average interarrival gap is then
This occurs with probability
+ A!(‘,
kkf)
D(ij, k) = 1 p(k)p(f b(h) k
- TDD(m)
2 Max[DASR(j),
Max(DASR(j),
b(h)
which is equivalent
if TAA([j) - AR(i) - TDD(kf)
D(& k) = 1 dk)df k
of cla
1
(20)
0.
From these formulae the expected number of departures of a definite class k is calculated as the sum of three quantities corresponding to the expected number of departures of class k as the first, second, or third departure in the triplet. Let Nl(ij, kfm), N2(ij, k/m), and N3([j, k/m) be (0, 1) random variables defined as follows:
of class k in
D(k) = 1 p(ij)D(ij, 12)
i
ij
and the flow of departures
of class k is
F(k) = D(k)/TAA However, summing these flows will produce an 01 estimate of the departure capacity of the airfield, to the mix distortion resulting from the select process of the departures (aircraft requiring a la separation from preceding departures are being , favored). A conservative estimate of the depart
Developments
capacity, allowing for a total departure correct mix of aircraft, is obtained as?
in airport
capacity
177
analysis
flow with the
Note that the arrival capacity is identical to that of an arrival only runway, i.e. CAP,(DA) and the corresponding
= CAP,(AA)
(29)
per cent arrivals PA(DA) is CAP,(DA)
PA(DA) = [cAP,(DA)
+
CAP,(m)]
(30)
100
0
%
Fig. 3.
C. Operating strategy selection The capacity of a runway system depends on the operating strategy used. Typically, if the runway is operated exclusively for departures, it would have a larger capacity than if it were operated exclusively for arrivals. The mixed operations capacity is normally somewhere between the two capacities depending on the relative proportions of arrivals and departures in the traffic stream. The departure mix is represented by the per cent of the total traffic consisting of arrivals. It is possible to serve any given per cent of arrivals with a number of different operating strategies. The purpose of this step of the model is to illustrate how these strategies can be combined to achieve the highest capacity or any given percentage of arrivals. A single runway can be operated under any one or a combination of the following strategies: Arrivals only. Departures only. Mixed operations. These strategies and the corresponding capacities are shown on Fig. 3. Under the “arrivals only” strategy the per cent of arrivals in 100, and the flow corresponds to arrival capacity. Likewise, under the “departures only” strategy, the per cent of arrivals is zero and the flow corresponds to departure capacity. If these two strategies are combined in such a way that each is used a certain proportion of the time over a period such as one hour, then it is possible to obtain flows as represented by line AB for the varying proportions for each strategy. However, as it is often possible to insert departures in a gap between two arriving aircraft, it is possible to obtain flows higher than those shown by line AB if a mixed operations strategy is adopted. t Capacity in fact lies between the values before and after the correction for mix distortion. However, the differences involved are usually too small to be of any significance. $ For a descrlptlon of the data sources, see Douglas Aircraft Co. <“Itrl. (1973).
arrivals
This is shown on Fig. 3 by line ACD. This line should theoretically connect A and B, but does not because the model is formulated with a maximum of three departures interleaved during any interarrival gap. However, it can be seen that the envelope ACB represents the combinations of strategies that yield the highest flow rate and thus is the capacity envelope. This envelope defines the runway capacity for any percentage arrivals and shows the strategy required to achieve this capacity. The calculations required to derive such an envelope are extensive, although quite simple in nature. They are efficiently performed with the aid of a digital computer. Model application. The application of the single runway capacity model is demonstrated in this section. For this demonstration, data were obtained from observations at La Guardia Airport in New York.f Departure capacity, arrival capacity, and mixed operation capacity are calculated for the single runway. This is followed by the derivation of a highest capacity envelope showing capacity for any per cent of arrivals. The following data are used in the calculations: aircraft mix: i = 1, 903% jets (DC8, B707, B727, B737, DC9) i = 2, 9.2% light aircraft (DH6, FA27, BE99). Mean arrival runway occupancy
time
AR(i) = (38,47) sec. Standard
deviation
of arrival runway occupancy (TAR= 6
Standard
deviation
XC.
of interarrival gA = 10 sec.
times
time
178
STI.PHEN
Probability
L. M. HOCKADAY and
CAPn(DA)
pr = 0.05.
CAP,,,(DA) = CAPA
Length of common
The following table summarized the capacities as calculated by themodel and compares them with actual flow rates observed at La Guardia Airport, New York, during capacity operations.
approach
paths
;’ = 8.4 miles. Mean departure
runway occupancy time DR(i) = (39, 34).
deviation
Hourly mixed operation Arrival flow Departure flow Per cent arrivals
of DR(i) crDR= 3 sec.
Time separation
between
departures 39
39
41
34 set
I I
t&j) = Distance
of arrival from departure
release
6du = 0 miles. Standard
= 35 aircraft!hr.
speeds ri = (130, 120) knots.
Standard
= 30 aircraftihr.
and
rules &j = 3 miles, for all i, j.
Approach
KANAFAI\;I
Mixed operations capacit)
of rule violation
Separation
AIIIH K.
deviation
of time to start a departure
roll
cc = 0 sec. Using thedata shown above it is possible to compute the separation requirements using equations (I), (5). (7). (8). AROR(i) = (66.2, 57.2) sec. AASR([j) =
99.53
125.84
99.53
106.45
DROR(i)
= (43.9, 38.9) sec.
DDSR(ij)
=
I
f;
;;
I
DASR( j) = 0 sec. With these calculated :
separations
the various
capacities
Arrival cupucit)~ CAP,(AA)
= ‘If6tt = 35 aircraft/hr.
Departure capacit) CAP,(DD)
= y9y
= 92 aircraft/hr.
are
Cuhlated 65 35 30 54 “<,
Ohserwd 65 33 32 5 I iII)
Using the calculated capacities it is possible to derive the envelope showing the highest flow rate possible for any given per cent of arrivals in the opxitions mix. This envelope is used in the selection of runway operating strategy for various types of demand. The envelope in Fig. 4 shows the figures obtained in this application. Point A represents the arrival capacity, 35, and point B the departure capacity, 92. Line AB gives the flow rates possible with combinations of these two strategies with varying proportions during I hr. Point C represents the mixed operations capacity. It lies on a line AD which gives the flow rates obtainable under mixed operations with varying amounts of departure releases between arrivals. Along the line AD, departures are released during available gaps. Different per cent arrivals are obtained by stretching the interarrival separation. The line CR shows the fows possible with the combination of the
Developments in airport capacity analysis mixed operations strategy with the departures only strategy. The envelope ACB then gives the highest flow rates possible and allows the user to select the operating strategy to meet any arrival-departure mix.
SUMMARY The capacity of a system of runways is defined as the maximum flow rate of operations that can be accommodated under specified operating conditions. Capacity is usually calculated for specific periods of time, most commonly for 1 hr. A model for calculating runway capacity is presented in this paper. The model is based on the postulate that aircraft deviate from intended paths and that these deviations are normally distributed random variables. For any given per cent of arrivals, capacity can be maximized by selecting from among three basic strategies and their combinations: arrivals only, departures only, and mixed operations. The proposed model is applied to calculate the single runway capacity using data observed at La Guardia Airport in New York. The results of the model compare sufficiently well with observed volumes at the airport. This favorable comparison may be attributed to the model’s detailed incorporation of operating rules and strategies, and of the stochastic nature of aircraft operations. The main advantage of the proposed approach compared with existing approaches is thought to be the fact that it separates capacity analysis from delay analysis. Although the two are closely related, they are completely different performance measures of runway system operations.
179
Acknowledgement-The models described in this paper were developed by the authors in cooperation with the project staff as part of a study entitled “Procedures for Determination of Airport Capacity” performed for the U.S. Federal Aviation Administration, by Peat, Marwick, Mitchell & Co. Numerous staff members have contributed directly or indirectly to the paper. Their assistance was greatly appreciated.
REFERENCES
Airborne Instruments Laboratory (1963). Operational Evaluation of Airport Runway Design and Capacity. AIL Report 7601-6. Deer Park, Long Island. Baran G. (1968) Airport Capacity Analysis. Report D6-23415. The Boeing Company, Seattle, Washington. Blumstein A. (1960). An Analytical Investigation of Airport Capacity. Report TA-1358-8-L Cornell Aeronautical Laboratory. Bowen E. G. and Pearcey T. (1948). Delays in the flow of air traflic. J. H. Azrontrf,t. Sot. 52, 251 2%. Douglas Aircraft Company, Peat, Marwick, Mitchell & Co., American Airlines (1973), Procedures for Determination of Airport Capacit;. Phase I Interim Report. Galliher H. P. and Wheeler R. C. 11958). Nonstationarv queuing probabilities for landing Long&tion of aircraff. O/X RLJS.6, 264~275. Harris R. M. (1969). Models for Runway Capacity Analysis. The Mitre Corporation Technical Report MTR-4102. Washington, D.C. Oliver R. M. (1962). Delays in Terminal Air Traffic Control. Symposium on Airport Capacity, Institute of Transportation and Traffic Engineering, Berkeley, California. Pearcey. T. (1948). Delays in the landing of air traffic. ./. K. AoYJ/lol,t. Sr,c. 52, 799 8 12. Pestalozzi G. (1964). Priority rules for runway use. Op. Res. 12, 941-950. Shapiro 0. (1965). An Airport Capacity Handbook: Results of a Federal Aviation Administration Research Program in Mathematical Models and Techniques for Determining Airport Capacity. Operations Research Society of America, Twenty-seventh Annual Meeting, Boston, Massachusetts.
R&umP La capacite des pistes d’atterrissage est dbfinie comme &tant le nombre maximal de manoeuvres d’avions pouvant etre acceptC au tours d’une pkriode de temps spkcifique dans les conditions opkrationnelles donnL:cs. Pkirmi les facteurs dkterminants de capacitt les plus importants, figurent le mklange de types d’appareils. la longueur de la trajectoire d’approche commune, et la strattgie opkrationnelle. On postule que les appareils dkvient de la trajectoire pr&uc lorsqu’ils approchent de la piste d’atterrissage. On suppose que ces dkviations sont des variables aleatoires h distribution normale dont la moyenne est nulle. On suppose que, pour maintenir la probabi1itC des violations de rtgles de siparation des appareils. les contrbleurs introduisent des kltments tampons entre les appareils pour compenser la nature al&atoire de leur stparation. A partir de ces postulats, on construit un modele de capacitC. Le modtle fournit la capacitk de pistes d’atterrissage pour diverses strattgies optrationnelles et permet de choisir la strattgie optimale pour un mklange donnt- d’arrivtes et de departs prCvus. On dCmontre l’application du modele ti l’aide de don&es provenant de I’ACroport de La Guardia g New York.
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STEPHEN L. M. HOCKADAY and Arnn KANAFANI
Zusammenfassung-Die Leistungsfihigkeit von Start- und Landebahnen wird als die unter vorherrschenden Betriebsbedingungen gr6Btmogliche Anzahl von Flugbewegungen wLhrend eines bestimmten Zeitraumes definiert. Die wichtigsten EinfluBgrijDen fur die Leistungsfahigkeit sind Art und Mischung der Fluggerlte, die Llnge des gemeinsamen Anflugweges und die Betriebsregelung. Es wird davon ausgegangen, dal3 die Flugzeuge beim Landeanflug vom vorgegebenen Kurs abweichen. Diese Abweichungen sollen einer Normalverteilung mit dem Mittelwert Null entsprechen. Urn die Wahrscheinlichkeit von Unterschreitungen des Mindestanflugabstandes nicht zu grog werden zu lassen, wird angenommen, dalj die Fluglotsen Abstandspuffer zwischen aufeinanderfolgenden Flugzeugen vorhalten, urn die Zufalligkeiten in der Anflugfolge auszugleichen. Mit diesen Vorgaben wird ein Leistungsfahigkeitsmodell entwickelt, das Leistungswerte bei unterschiedlichen Betriebsregelungen liefert und die Auswahl einer optimalen Strategie bei gebener oder vorgegebener Abfolge von Starts und Landungen gestattet. Die Verwendungsmiiglichkeiten des Modells wird mit Hilfe von Betriebsdaten des Flughafens La Guardia in New York nachgewiesen.