Tmnspn. Rex-8. Vol. Ml. No. 4. pp. 287-297. 1990 0191.261S/‘X S3.OOt.

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Printed in Great Bnram. 0 1990 Rrgamon Press plc

AIRPORT GATE POSITION ESTIMATION FOR

MINIMUM TOTAL COSTS -APPROXIMATE CLOSED
FORM SOLUTION

S. C. WIRASINGHE
and S. BANDARA
The University of Calgary, Department of Civil Engineering, 2500 University Drive N.W..
Calgary, Alberta, Canada, T2N lN4

(Received 14 November 1988; in revisedform 22 September 1989)

Abstract-A method to determine the optimum number of gate positions at an airport terminal,
which minimizes the sum of the cost of gates and the cost of delays to aircraft is presented. It is
based on an approximate procedure to determine the total deterministic delay to aircraft caused by
a limited number of gates, given information regarding the peaking of the aircraft arrival rate, and
the number of peaks per day. Closed-form solutions are obtained for the cases of one peak and
several identical nonoverlapping peaks, respectively. The optimum number of gates required for the
Calgary International Airport, based on a common gate use policy, is reported.

1. INTRODUCTION

The number of aircraft gate positions is a significant element to be considered in

planning new terminal buildings. The cost of delaying aircraft due to the lack of gates has
to be balanced against the cost of constructing and maintaining gate positions. Further,
the number of gates is a critical input for determining the airport terminal configuration
(Wirasinghe, Bandara, and Vandebona, 1987).
Most studies related to aircraft gate positions can be divided into planning and
operational studies. The planning studies deal with the estimation of the gate position
requirement for a given demand (Horonjeff, 1975; Steuart, 1974). The operational studies
deal with the assignment of aircraft to existing gate positions. Restrictions and preferences
at the gate positions and the optimization of aircraft delay and passenger-walking can be
handled by the latter type of models which require an extensive amount of information
(Hamzawi and Mangano, 1986; Babic et al., 1984).
In this paper, an analysis to obtain the number of gate positions that will minimize the
sum of the cost of gates and cost of total deterministic delay to aircraft is presented in the
context of long term planning. It is based on minimal information assumed to be available
regarding the peaking characteristics of the aircraft arrival rate.

2. OPTIMUM (MINIMUM COST) GATE POSITION REQUIREMENT

The amount of delay to aircraft is basically dependent on the aircraft arrival rate and
the aircraft service rate. The aircraft service rate is a function of the number of gates
available, G, the gate occupancy time, T, and the aircraft separation time (buffer time), ts,
at a gate. The aircraft separation time is defined as the time between a departure from a
gate position and the next arrival, which consists of the push out or power out time, the
time required by the departing aircraft to clear the apron area, the time required to prepare
the gate for the next arrival, and the time required by the arriving aircraft to move in from
the apron entrance to the gate position. If G gates are provided and the mean gate
occupancy time and the mean aircraft separation time at a gate are 7 and ?$, respectively,
then these gates can serve aircraft at an approximate mean rate

F=G/(T+FJ . (1)

The time period over which the delays are calculated is a critical factor in the analysis.
Since queues usually vanish at least once during a 24 hour period and the variation of the
287
288 S. C. WIRASINGHE
and S. BMDARA

hourly aircraft arrivals is large compared to the daily and monthly variation of aircraft
arrivals, a 24 hour time interval is used for the analysis.
Let ~1 = aircraft service rate in aircraftjhr,
K = average cost of delay to airlines and passengers/aircraft/hr, and
W = total deterministic delay to aircraft/day, due to lack of gates.

The total deterministic delay per day, W, is a function of a number of parameters li in

addition to p. Let

W=f(cL,% (2)

We assume that the marginal capital, maintenance, and operating cost of a gate position
per day, C, is a constant.
Then the sum of the delay cost and gate cost (total cost)

Z=p( T + t,)C+fl&X)K (3)

where, from Eqn 1, p( 7 + ?J is the number of gates.

To obtain the optimum service rate, Z has to be minimized with respect to p. The
related optimum number of gates is then obtained from eqn (1).
The model can also be used to determine the optimum gate requirement for a terminal
with a preferential gate use policy, if data on aircraft or airline groupings are available.
Then the gate numbers for each group are simply estimated separately.

3. APPROXIMATE DETERMINATION OF DELAY TO AIRCRAFT

If future schedules of aircraft arrivals are known and if all aircraft are on time, the
delays based on a given number of gates can be assessed exactly for that schedule and the
optimum number of gates could be determined simply by calculating the total cost for
various numbers of gates. However, it is unlikely that such schedules will be known several
years in advance. Nor are all aircraft likely to be on time. The following analysis assumes
that the best information available will be in the form of rates of aircraft arrivals.
An airport or a specified portion of an airport, such as a group of gates designated
for wide body aircraft only, where any arriving aircraft can use any gate is considered. This
portion can be represented by a multiple server system where there are many gates to serve
common arrivals (Newell, 1982). If the gate capacity is not reached, the service rate, as a
function of time, 1, will be equal to the arrival rate of aircraft at the gates, A(t), and hence
there will be some idling gates. When the arrival rate exceeds the service rate a queue will
form. Because of the availability of idle gates, a queue will not form as soon as the arrival
rate exceeds the mean service rate. The time lag to form the queue will depend on the
relative magnitude of the arrival rate before it reaches the service rate, and it will be fairly
small unless there are large numbers of gates and a very small initial arrival rate. Hence for
the following calculations it is assumed that the queue will form as soon as the arrival rate
exceeds the mean service rate.
Consider an arrival rate curve as shown in Fig. 1. If the arrival rate equals the service
rate at times t, and t,, respectively, a queue will start to build up at t, and it will reach a
maximum at t,. When the aircraft arrival rate, A(r), is less than p, the queue starts to
dissipate and vanishes at time t,, as described by Newell (1982).
The queue at any time t is given by

Q(1)= ll,[_4(1)-p]dt in t, <t<r,. (4)

As shown in Fig. 1, the queue at f is the area between the arrival rate curve and the service
rate curve between 1, and t. Since the queue vanishes at f,, the (positive) area between the
two curves during the interval (tz, t,) should be equal to the (negative) area between the
same two curves during the interval (f2, f3). The total delay is given by
 Airport gate position estimation 289

t, 12 t t3

TIME
Fig. 1. Arrival rate curve.

w= jf3Q(r)dr (5)
11

if the function A(1) is continuous over the period t, to t3.

Data on hourly aircraft arrivals were collected for Calgary International Airport.
Similar data for Denver and Atlanta Airports were reported by Shirazi (1980). Cumulative
hourly step functions for aircraft arrivals were drawn from the above data. These were
smoothed to obtain a continuous cumulative aircraft arrivals curve as a function of time.
The aircraft arrival rate at a given time (Figs. 2a,b) is the slope of the cumulative arrivals
curve at that time. A peak is defined as a period during which the arrival rate exceeds the
average arrival rate. Several of the peaks can be approximated by a concave parabolic or
triangular shape (Figs. 2a,b). If a triangular or concave parabolic shape can be assumed
for the peak, the only information needed to calculate the total deterministic aircraft delay
are (i) the maximum arrival rate, A,, (ii) the mean arrival rate, z, and (iii) the time, To,
during which the mean arrival rate is exceeded.
When the shape of a peak is unclear a suitable approximation can be obtained by

120

r
,*,

:-
\_ ,.L
..a.’

- ATLANTA .. DENVER /
I
J,l,lll,l,,ll,,,l,1,1,11,,1,,,i,’i~,~ “,.,,iiW

13:oo moo 17:oo 19:oo

TIME

Fig. 2.(a) Arrival rate curves for Denver and Atlanta airport
290 S. C. W~USINGHEand S. BANDARA

500 7:oo 9:oo ll:oo moo 15:oo 17:oo 19:oo 21:oo

TIME
Fig. 2.(b) Arrival rate curve for Calgary airport.

measuring the area between the arrival rate curve and the horizontal line representing the
mean arrival rate.

Let E=(A,-x)T,,/(Area Measured). (6)

For the peak to be a triangle or a parabola E should be equal to 2, 1.5, respectively.

However, the peaks can be approximated by a triangle and a parabola with an error of
about lo%, if

2.25>E> 1.75 (7)

and

1.75>E>1.30, (8)

respectively.
If E>2.25 or < 1.30 neither the triangular nor parabolic approximations will be suitable.
Then delays to aircraft have to be estimated using graphical techniques.

3. I Calculation of Delays

Parabolic arrival rate (flat peak). Newell (1982) has derived an expression for total
deterministic delay for an arrival rate function at least twice differentiable near the maxi-
mum value. If the arrival rate curve can be represented by a function

A(t)=A(tJ-P(t-t,)Z (9)
where A(t) = aircraft arrival rate at time t,
t, = time at which the arrival rate is a maximum, and
P = a constant coefficient,

the total deterministic delay, for a service rate, cc, is given by

W=9[A(r,)-/~]~/4/3. (10)

If the arrival rate pattern can be approximated by a parabola, (Fig. 4), it can be represented
by
 Airport gate position estimation 291

Fig. 3. Triangular shaped peak.

A(f)=& -a(t- To/2)2. (11)

At t=O and t= To,

A(t) =?i. (12)

Thus, x=A,,,-aTo214, (13)

and a=4(A,-x)/T*. (14)

The eqn (11) is similar to eqn (9) given above. Therefore the total deterministic delay can
be written by observing eqn (10):

Triangular arrival rate (sharp peak). If the arrival rate pattern is to be approximated
by a triangle, as shown in Fig. 3,

X+a in OrtsTd2
A(t)= x+at (To-t) in Td2ctsT,. (16)
t
It can be shown (Bandara, 1989), that the total deterministic delay for a triangular peak is
given by

w= To2(Am
- PY
(17)
2J2(A, -x)2.

4. OPTIMIZATION

The number of gates provided should be such that the mean service rate falls between
the maximum and the mean arrival rates. If the service rate, p, is less than the mean arrival
rate, 2, delays will increase continuously. On the other hand, if the service rate is greater
than the maximum arrival rate, A,,,, there will be no delays, but some gates would be
always idle. It is assumed in the following analysis that the service rate should be suffi-
ciently higher than the mean arrival rate to prevent the overlapping of queues caused by
separate peaks.
292 S. C. WIRMINGHEand S. BANDARA

If the maximum and mean aircraft arrival rates are A, and 2 per hour, respectively,
and the time during which the arrival rate exceeds its mean value, TO, are known, the
function A(t) is defined depending on the expected shape of the peak. The total delay
during the peak is obtained as a function of p from eqn (15) or (17) as appropriate. Then
the total cost can be represented as in eqn (3). By differentiating the total cost function
with respect to p and equating to zero, the optimal service rate, p*, is obtained. The related
gate requirement G is obtained from eqn (1).

One parabolic shaped peak per day

Using eqns (3) and (15), the total cost due to delay is

Setting the derivative of eqn (18) with respect to p to zero and simplifying, the following
expression for the optimal service rate is obtained:

/A*=&--0.89(A,,,-~)[(~+t,)C]/[T,2K]. (19)

The second derivative of eqn (18) with respect to p is positive and hence the solution is a
minimum.
Because of the assumption that the queue vanishes before A(t) reaches 1 again, t,
should always be less than TO. Referring to Fig. 4, it can be shown that t, < TOif

/~>0.75A,+0.25x. (20)

One triangular shaped peak per day

Similarly, the optimum service rate for a triangular shaped peak is

/J*=A,/-0.97(&-x)[(T+ i;)C’j’~2/[T02K]“2. (21)

In order to satisfy the condition t, < TO, it can be shown from eqn (21) that it is necessary
to have a service rate

/L>o.59&+0.41 X. (22)

It can be seen that for both the arrival patterns: parabolic and triangular, the optimum
service rate p* is sensitive to TO, the time during which the arrival rate exceeds its mean
value. The smaller the TO, the smaller the p* for a given A,. Sensitivity of p* to the

t
a

TIME
Fig. 4. Parabolic shaped peak.
 Airport gate position estimation 293

independent parameters is comparatively higher for the case of a parabolic arrival rate
pattern, which has a higher effect on delays due to the flat peak. If the capital and
operating costs represented by C are neglected, p* tends to A,,, for both the cases.

Modified formulae
When the shape of the expected arrival rate curve is available, it is not essential to
consider the mean arrival rate, 2, and the corresponding To value, to calculate the
expected delay. Instead, a convenient arrival rate A’, which is closer to A,,, and gives a
better symmetry to the arrival rate curve about the time of maximum arrival rate, and the
corresponding time, r,,, the time during which the arrival rate exceeds the value A’ could
be used as shown in Fig. 5. In this case 2 and T, in eqns (19) and (21) have to be
substituted by A ’ and T,‘, respectively. The modified equations for the optimal service
rates are given by

Cr*=Am-0.89(A,-A?[(T+i,)q/[(To’)*~ (23)

for one parabolic shaped peak per day, and

/.4*=Am-0.97(A,-Afl)[(T+ i;)q”*/[(T,~)*Ay* (24)

for one triangular shaped peak per day, respectively.

In order to satisfy the assumption regarding t, < to’ the above optimum results should
be subjected to the conditions

/.~>0.75 A,+0.25 A’ (25)

and

p>O.59 A,+0.41 A’ (26)

for the parabolic and triangular shapes, respectively.

c 1

I- TO 1

TIME

Fig. 5. Asymmetrical peaks.

294 S. C. WIRASLYGHE
and S. BAVDAM

Several peaks per day

If more than one peak exists during a day, there are additional steps to the above
optimization procedure. To arrive at the optimum service rate in the minimum number of
steps possible, the optimization should be first carried out considering the delay due to the
highest peak only. If the service rate obtained falls above all the other peaks, it will be the
optimum service rate. If the value obtained falls below any of the other peaks, the optimi-
zation has to be repeated considering the delays due to all the peaks which are greater than
the service rate obtained in the previous step. If the service rate obtained from the second
optimization falls above any of the peaks considered in the total cost function, the optimi-
zation procedure has to be continued neglecting the effect due to the peaks which are less
than the new service rate obtained, until the service rate does not fall above any of the
peaks which have been considered.
Let p, be the service rate obtained in the ith step. Then the total cost function for the
i+ 1th step is given by

where ni is the number of peaks that have an arrival rate greater than ,L+ If n, > 1, the
optimization procedure has to be continued until n,=~;+~.
If there are n identical nonoverlapping peaks, the eqns (19) (21) (23), and (24) are
applicable with the gate cost C replaced by C/n.

5. EXAMPLES

5. I Calculation of delays
The aircraft arrival rate curve can be obtained if the expected hourly aircraft arrivals
are known. When the hourly aircraft arrivals are available, a cumulative aircraft arrival
curve is drawn and the slopes of the smoothed cumulative arrival curve are used to
determine the aircraft arrival rates at different times.
Operational data obtained from Calgary International Airport for December 21, 1984
are used in the following example. The arrival rate for the day (Fig. 6) is obtained from the
cumulative arrivals curve. The arrival rate curve shows four major peaks around 07.30,
12.30, 14.30, and 17.30 hours, respectively.
For the calculation of delays, peaks 1, 2, and 4 are considered as parabolic shaped

ll-

600 7:oo 9:oo no0 moo 15:oo 17:oo 19:oo 21:oo

TIME

Fig. 6. Summary of peak periods-Calgary airport.

 Airport gate position estimation 295

peaks and the peak 3 is treated as a triangular peak. A constant aircraft separation time of
10 minutes is considered for the calculations. Further, a service rate of 7 aircraft/hr is
selected instead of the mean service rate as the first gives a better symmetry for the peaks
(Fig. 6). A,, A’, and To’ for the four peaks are given in Table 1.
Expected delays for different service rates are calculated to the nearest minute based
on the appropriate formulation selected from eqns (15) and (17). For purposes of compari-
son, the gate assignment simulation model of Hamzawi and Mangano (1986), which
estimates delays to the nearest 5 minutes, is used to calculate the delays for actual aircraft
arrivals and departures and a 10 minute aircraft separation time. Results from the analyti-
cal and simulation models are given in Table 2.
It can be seen that the delays calculated from the proposed model are less than the
simulated values. This difference is small when the number of gates provided is close to the
number of gates actually required to ensure no delays on the apron. The difference
between calculated and simulated delays increases as the difference between the number of
gates provided and the number that is actually required increases. One of the reasons for
the above difference is the overlapping of queues. For example, the queue build up for the
peak 4 may not vanish by 1900 hr. and can continue further if the number of gates
provided is very small. But the analytical model does not account for these delays since we
assume for planning purposes that the overlapping of queues is not acceptable. The other
reason is the rounding off of the simulation model results to the nearest 5 minutes.

5.2 Optimum number of gates

As discussed by Shirazi (1980) the direct operating cost of an aircraft will depend on
the expenditures for fuel and oil, flight crew time and maintenance. The cost of delay to a
passenger will depend on the travellers trip purpose and may be proportional to the wage
rate. Since the evaluation of the above cost values is beyond the scope of this paper, gate
costs given by the Federal Aviation Administration (1977) and operational cost of aircraft
and cost of delay to passengers used by Shirazi (1980), approximately adjusted to 1984
Canadian dollars, are used here:

Cost of a gate position (C) = 33500 $/year.

Average cost of delay/aircraft (K) = 2370 $/hr.
Calgary International Airport: 7 = 0.69 hr., t,=O.17 hr.

The expected delay in each peak period, for a service rate p, is calculated using an
appropriate formulation for delay from eqns (15) or (17) modified for A ’ and T,‘.
Considering the delay due to the highest peak only and using eqn (19), we get p, =7.8
aircraft/hr. The value p, is less than the maximum arrival rate of all the peaks (i.e., n, =4
and > 1). Hence consider the delays due to all the peaks. Then, from eqn (27), the total
cost

335000
Z,=O.86x- /L+415(9.5-/++717.7(9-#
365

+45.6(10-/~)~+2444.3(8.8-~)~. (28)

By differentiating eqn (28) with respect to p and equating to zero we get p2=8.84 aircraft/
hr. The value pZ is greater than the maximum arrival rate for peak 4. (i.e., n, =4, n,=3).
Therefore we neglect the delay due to peak 4. By differentiating the new total cost function
with respect to g and equating to zero, p3=8.9 aircraft/hr. (i.e., n,=n,=3).
It can be shown that s>O and p3 satisfies the assumption that t3< T,’ for all
BP,
the peaks. Hence, p3 can be considered as the optimum service rate. From eqn (l), G,,=
7.7. Hence 7 or 8 gates, whichever gives the minimum total cost, should be provided. For 7
gates the total cost is $9298.50 and for 8 gates the total cost is $7389.30. Therefore 8 gates
are selected.
296 S. C. WIRASWGHE
and S. BANDARA

Table 1. Peak characteristics

Peak Number

1 2 3 4

A,,, (aircraftjhr) 9.5 9.0 10.2 8.8

A’ (aircraftihr) 7.0 1.0 7.0 1.0
7, (min) 41 62 42 109

A 4.16 aircraft/hr
T+T* 0.69 hr. +0.17 hr. =0.86 hr.

5.3 Long term planning

For long term planning purposes the following approximation may be used. We
assume that the following information is available at the planning stage: The expected
peak hour aircraft arrivals, Ap, and the maximum (instantaneous) rate at which the
aircraft will arrive at any particular time, A,. Since the maximum rate of arrival will
occur during the peak hour period, the shape of the arrival rate curve during the peak can
be approximated by a parabola or an isosceles triangle (Bandara, 1989). As the number of
aircraft arrivals during a one hour time period is equal to xp, the above peaks can be
treated as a parabola or a triangle with

Am=Amp, A’= ; &-+ A,,,, TO’=1;

or

Am=Ampr A’= :A_ iA mp, TO’= 1. (30)

2 p 2

respectively, and eqns 23, 24 can be used to obtain the optimal service rate.
Consider the following example of an airport with one peak per day:

Peak hour aircraft arrivals (A,) = 30 aircraft/hr.

Maximum rate of arrivals (A_mp) = 45 aircraft/hr.
Mean gate occupancy time (T) = 45 min.
Mean aircraft separation time (t,) = 10 min.
Cost of a gate position (c) = 365000 $/year
Average cost of delay (K> = 3500 $/year.

Using eqn sets (29) and (30), the peak described above can be approximated by a parabolic
shape with

Table 2. Results of analytical and simulation models

Delay (Minutes)

Number of Service Total

Gates Rate Peak 1 Peak 2 Peak 3 Peak 4
G c1 (Analytical) Analytical Simulation

9 10.5 0 0 0 0 0 0
8 9.3 1 0 1 0 2 10.0
7 8.1 13 15 11 30 69 70.0
6 7.0 40 72 38 201 351 485.0
 Airport gate position estimation 297

A,=45 aircraftjhr., A ‘=20 aircraft/hr. and To’= 1 hr.

or by a triangular shape with

A,,,=45 aircraft/m, A’=22.5 aircraft/m and To’= 1 hr.

respectively.
If the peak is assumed to be a parabolic one, from eqns (23) and (l),

p*=39.1 and G*=36 gates.

If the peak is assumed to be a triangular one, from eqns (24) and (l),

~*=33.8 and G*=31 gates.

As this approximation deals with an isolated peak, the conditions for t, < To’ need not be
checked.
If there are two independent identical peaks per day, the optimal service rate is
obtained by using half the value of the gate cost, C, used above. Then, the values of G* are
39 and 34, respectively.

Acknowledgement-This research was supported in part by the Natural Sciences and Engineering Research
Council of Canada under Grant No. A471 1. We wish to thank the referees for several valuable comments.

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