output analysis in distributed simulation

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OPTIMAL ALLOCATION OF ARRIVALS AND DEPARTURES AT
AN AIRPORT
Corresponding Author:
Asmita Rai, Master’s Student, Aerospace Engineering Department, Indian Institute
of Technology Bombay, Powai, Mumbai – 400076, [email protected] , Tel:
+91-22-25767127, Fax: +91-22-25722602
Supplementary Author:
Rajkumar S. Pant, Associate Professor, Aerospace Engineering Department, Indian
Institute of Technology Bombay, Powai, Mumbai – 400076
[email protected]
Tel: +91-22-25767127, Fax: +91-22-25722602
1
OPTIMAL ALLOCATION OF ARRIVALS AND DEPARTURES AT
AN AIRPORT
Asmita Rai
Master’s Student
[email protected]
Rajkumar S. Pant
Associate Professor
[email protected]
Aerospace Engineering Department, Indian Institute of Technology Bombay,
Powai, Mumbai – 400076, Tel: +91-22-25767127, Fax: +91-22-25722602
Abstract
This paper discusses a model for determination of the air traffic flow at the arrival
and departure fixes that result in optimal allocation of the total capacity of an airport.
An existing model was enhanced by incorporating additional variables and control
parameters, using which the effect of reduction in airport system capacity can be
investigated. Further, the effect of allocating differing priorities to the aircraft of
competing airlines arriving and departing through these fixes on the optimal flows
and queues generated can also be studied. The results of a case study demonstrating
the effect of these enhancements are also presented.
Index Terms: Air Traffic Management, Optimization, Arrival & Departure
Capacity
Nomenclature
a ij
Total number of aircraft arrival demand through jth arrival fix in ith time slot
bik, t
Ct
Total departure demand of aircraft of tth airline through kth departure fix in ith
time slot
Total number of aircraft of tth airline in the arrival queue of jth arrival fix in ith
time slot
Total terminal capacity
d ik
Total demand for departure through kth fix in ith time slot
Faf
Fdf
Capacity of an arrival fix
cij, t
g
i
j, t
h ik, t
L
M
N
P
Capacity of a departure fix
Total number of arriving aircraft of tth airline passing through jth arrival fix in
ith time slot
Total number of aircraft of tth airline passing through kth departure fix in ith
time slot
Total number of arrival fixes
Total number of airlines operating at the airport
Total number of time slots
Total number of departure fixes
2
q ik, t
ui
vi
Total number of aircraft of tth airline in the departure queue of kth departure
fix in ith time slot
Total arrival demand of aircraft of tth airline through jth arrival fix in ith time
slot
Arrival capacity of the runway system in ith time slot
Departure capacity of runway system in ith time slot
w ij
Total number of aircraft arriving through jth arrival fix in ith time slot
Xij
Total number of aircraft in the arrival queue of jth arrival fix in ith time slot
Yki
Total number of aircraft in the departure queue of kth departure fix in ith time
slot
Total number of aircraft departing through kth arrival fix in ith time slot
Priority assigned to the tth airline
Fraction of terminal capacity available for use
Arrival priority (used to control the flow of arrival and departure in an
interrelated manner)
rj,i t
z ik
αt
β
λ
1. Introduction
Air traffic congestion during arrival and departure at major airports is one of the
most severe and fastest growing problems in the air transportation industry. In the
USA alone, nation-wide annual losses due to air traffic delays are estimated in
billions of dollars (Gilbo, 2003). Traffic congestion can be handled either by
building new infrastructure to accommodate more traffic, or by ensuring optimal
utilization of the resources available. There is a limit to the amount of new airports
or runways that can be built; hence the latter approach is more cost effective and
practical. FAA is trying to implement many methods to improve the present
condition of airport congestion, but still there is a need for developing ideas to
mitigate congestion in arrival and departure phases. Dear and Sherif (1991) have
proposed a methodology, which they call “Constrained Position Shifting” (hither to
referred as CPS), for sequencing and scheduling of aircraft in high density terminal
areas. Horonjeff (1994) and Virendra (1999) calculated arrival delay per aircraft
when only arriving sequence of aircraft was considered on the runway. Runway
configuration can also be modeled and simulated using TAAM (Total Airspace and
Airport Modeler) (2002) which is a four dimensional flight path simulator, to ensure
maximum capacity utilization. Hesselink and Basjes (1998) have shown that airport
capacity can be increased by good planning of the sequences. They have given a
tool, which helps the controllers in making optimal departure sequences. The study
was concentrated on the runway departure planning, Constraint satisfaction
technique was used to specify the problem and with the help of the optimization
function is used to select the best sequence. Anagnostakes and Idris (2000), gave the
concept of a departure planner (DP) that can be utilized to assist air traffic
controllers in improving performance of departure operations and optimize runway
time allocation. Idris and Bhuva (2002) have presented a model for predicting the
3
taxi-out time, i.e., the time taken by the flight from the gate to the takeoff. This time
varies, and its estimation can be helpful in congestion problem. The taxi out
prediction helps to better estimate the takeoff time and the delays as well. Carr
(2002) has studied how the closure of departure fixes plays an important role in
congestion. This model was developed for departing traffic with restriction of
closure. The validity of this model was checked with Monte Carlo Simulation of 40
hours. Kaltenhauser (2003) has provided details Tower and Apron simulator
developed at DLR, which can be used to study the interaction of machines with the
system, Such a simulator also helps in optimization of capacity. This kind of real
time simulation can be helpful in training new air traffic controllers, and in testing
new or modified airports, and air traffic controller tools. So the previous literature
available mainly focuses on individual contribution of each airport component in
traffic, this paper looks at methodologies for optimal utilization of arrival and
departure capacity of an airport as a whole that can lead to increase in traffic flows,
and hence reduction in the overall queue over a given time interval.
2. Background
Gilbo (1993) has suggested a method for estimation of capacity of runway system of
an airport under various operational conditions, and presented a method to optimize
the capacity to best satisfy the expected traffic demand. In this model, the arrival and
departure capacities are assumed as interdependent variables whose values depend
on arrival/departure ratio in the total airport operations. An arrival–departure
capacity curve, as shown in Figure 1 was proposed, which consists of some pairs of
number of arrivals and corresponding numbers of departures that can be handled at
an airport within a specified time slot.
Figure1 to be inserted here
This study was further extended by Gilbo (1997) to consider the runway (i.e., the
airport) and arrival and departure fix capacities as a single system resource. This is
achieved by accounting for the interaction between runway capacity and capacity of
the fixes to optimize the traffic flow through the airport system, as shown in Figure
2.
Figure 2 to be inserted here
This collaborative optimization model was modified later to accommodate user
priorities and converted into a decision support tool for solving traffic flow
management problems at airports (Gilbo & Howard, 2000). This tool was
successfully evaluated in real life decision making situations at St. Louis Lambert
International airport (Gilbo, 2003). Recently, some researchers (Idrissi and Li, 2006)
have solved the same capacity allocation problem using a CSP algorithm and
replicated results presented in (Gilbo, 1997).
4
To gain an understanding of Gilbo’s model, it was coded and coupled to OPL, which
is a standard optimization tool developed by Van Hentenryck (2002), for the demand
profile used in the case study by Gilbo (1997) reproduced in Table 1.
Table 1 to be inserted here
Several optimal solutions were obtained for the case in which equal priority was
given to arrival and departure flows. The resulting flows and queues obtained in one
such case are compared with those listed in (Gilbo, 1997), as shown in Table 2 and
3, respectively.
Table 2 to be inserted here
Table 3 to be inserted here
It can be seen that while the summation of the total arrival and departure flows (and
queues) in all the fixes in each time slot is the same, there are differences in the
individual values of the flows (and queues in each slot). Further, the summation of
total flows (and queues) through each (arrival and departure) fix is also the same.
Since the objective function in (Gilbo, 1997) depends only on the total flows (or
queues) in each time slot and each fix, its value is the same in both these solutions.
This study indicated that the solution obtained by Gilbo (1997) is non-unique, and
one of the ways to make it unique is to assign priorities to flow through each
individual fix. The next section explains the modifications made to this model to
enhance its capabilities.
3. Modifications to the model by Gilbo (1997)
Gilbo’s model (1997) allocates capacities with a view to minimize the delays or
queues at arrival and departure fixes. The current study aims at enhancing this model
by incorporating certain additional features. The first of these modifications allows
one to estimate the effect of reduction in the capacity of the terminal on the optimal
traffic flows and queues that result. This is done by introducing a control parameter
( β ) in the model, which puts a restriction on the total available airport system
capacity. These could be due to features such as inclement weather conditions,
equipment outages, and excessive levels of workload on the air traffic controllers.
The terminal capacity Ct is obtained as the maximum value of the summation of
arrival and departure capacities at various points on the Capacity curve. Hence, the
summation of arrival and departure capacities ( u i and v i , respectively) for a
particular time slot cannot exceed the available terminal capacity C t .
(u i +vi )  (Ct .β)
… (1)
For instance, the peak total capacity of the arrival departure capacity were shown in
Figure 1 is 48 (24 arrivals and 24 departures), which reduces to 36, for β =0.75. The
representation of this on the Capacity curve is shown in Figure 3. Equation (1)
5
modifies the Capacity curve excluding data points which are operating beyond
specified percentage of terminal capacity represented by β , as shown in Figure 3.
Figure 3 to be inserted here
The second modification enables one to assign priorities to the various aircraft that
arrive via the same fixes. While it is always possible to assign equal priority to the
aircraft of all airlines that arrive (or depart) via the same fix, this feature allows one
to simulate the real life scenarios in which the aircraft of a particular airline might be
assigned higher priority over the others. This modification also results in unique
solutions being obtained for the optimization problem, since the objective function is
now affected by the exact number of aircraft of a particular airline arriving (or
departing) via a particular fix in a given time slot.
4. Formulation of Objective Function and Constraints
The optimization problem is formulated as maximization of the linear objective
function consisting of the cumulative arrival and departure flows at the airport over a
given time period, as shown in Equation (2). It can be seen that the objective
function includes the priority α t assigned to each airline. The term (N-i+1) assigns a
lower priority to the time slots which are in further away in time.
max
N
L
M
P
M
i=1
j=1
t=1
k=1
t=1
 (N-i+1)(( (λ.( αt .gij, t )))+( ((1-λ).( α t .h ik, t ))))
…. (2)
Apart from the Capacity curve constraints (as shown in Figure 3), the following
additional constraints have to be kept in mind while obtaining the optimum flow:
1) All the variables should be non negative, and the queues, flows, demands of
aircraft and capacities assigned should be integer.
2) Total arrival queue at the end of (i+1)th time slot is equal to sum of total
arrival demand in ith time slot at jth fix and total arrival queue in ith time slot
at the jth fix, minus flow through the jth arrival fix in ith time slot, i.e.,
i
i
i
Xi+1
…. (3)
j =X j +a j -z j
th
th
3) Arrival queue of t airline at the end of (i+1) time slot is equal to the sum
of total arrival demand at ith time slot and total arrival queue at ith time slot,
minus the flow through the arrival fix in ith time slot, i.e.,
i
i
i
ci+1
…. (4)
j, t =c j, t +rj, t -g j, t
th
4) Total departure queue at the end of (i+1) time slot is equal to the sum of
total departure demand and departure queue at the ith time slot, minus flow
through the departure fix at the ith time slot, i.e.,
Yki+1 =Yki +d ik -zik
…. (5)
th
th
5) Departure queue of t airline at the end of (i+1) time slot is equal to sum of
total arrival demand in ith time slot at kth fix and total departure queue in ith
time slot at kth fix minus flow through the departure fix, i.e.,
6
i
i
i
qi+1
k, t =q k, t +bk, t -h k, t
6)
7)
8)
9)
…. (6)
Total arrival flow through arrival fixes in i time slot cannot exceed the
arrival fix capacity in that time slot, i.e.,
…. (7)
w ij  Faf
th
Total departure flow through departure fixes in i time slot cannot exceed the
departure fix capacity in that time slot, i.e.,
z ik  Fdf
…. (8)
th
Total arrival and departure capacity in i time slot cannot exceed the
available Terminal capacity, i.e.,
(u i +vi )  (Ct .β)
…. (9)
th
Total aircraft arrival flow in i time slot through all arrival fixes cannot
exceed the summation of total arrival demand up to the ith time slot, i.e.,
th
L
i
L
 w  a
i
j
j=1
ie
j
…. (10)
ie=1 j=1
10) Total aircraft departure flow in ith time slot through all departure fixes cannot
exceed summation of total departure demand up to the ith time slot, i.e.,
P
i
P
 zik   diek
k=1
…. (11)
ie=1 k=1
11) Total actual arrivals in ith time slot through jth arrival fix should be equal to
the summation of aircraft arrivals of all M airlines in ith time slot, i.e.,
M
w ij =  g ij,t
…. (12)
t=1
12) Total actual departures in ith time slot through kth departure fix should be
equal to the summation of aircraft departures of all M airlines in ith time slot,
i.e.,
M
z =  h ik,t
i
k
…. (13)
t=1
13) Total arrival queue in ith time slot through jth arrival fix should be equal to the
summation of arrival queues of all aircraft of M airlines in ith time slot, i.e.,
M
X ij =  cij, t
…. (14)
t=1
14) Total departure queue in ith time slot through kth departure fix should be
equal to the summation of departure queues of all aircraft of M airlines for ith
time slot, i.e.,
M
Yki =  q ik, t
…. (15)
t=1
15) Total arrival demand in ith time slot through jth arrival fix should be equal to
the summation of arrival demand of all M airlines in ith time slot, i.e.,
M
a ij =  rj,i t
t=1
7
…. (16)
16) Total departure demand in ith time slot through kth departure fix should be
equal to the summation of departure demand of all M airlines in ith time slot ,
i.e.,
M
d ik =  b ik,t
…. (17)
t=1
17) Total arrival flow should not exceed the runway arrival capacity, i.e.,
L
w
i
j
 ui
…. (18)
j=1
18) Total departure flow should not exceed the runway departure capacity, i.e.,
P
z
i
j
 vi
…. (19)
k=1
5. Hypothetical example to illustrate the enhanced model
To illustrate the usefulness of the model, the total demand through each fix in each
time slot listed in Table 1 was hypothetically split into arrival and departure demand
for four airlines. The optimized flows (and the resulting queues) thus obtained are
presented in this section.
Table 4 and 5 list the assumed demand data for the each arrival and departure fix
respectively, for four airlines ( M =4), whose aircraft arrive and depart via 4 arrival
( L =4) and 4 departure fixes ( P =4) during 12 time slots (N=12). The maximum
capacity of each arrival and departure fix in any time slot is assumed to be 10
( Fdf = Faf =10). The terminal capacity Ct for the capacity curve shown in Figure 2 is
(24+24) =48, which is the point in the capacity curve where the maximum arrival
departure operations are possible.
Table 4 to be inserted here
Table 5 to be inserted here
5.1 Optimized flows and resulting queues for various airlines
The model was run for the above data, by assigning priorities of 0.7, 0.10, 0.15 and
0.05 to airlines from A1 to A4, respectively, while assigning equal priority to arrival
and departure (λ=0.5) operations and assuming that full terminal space is available
( β =1). The arrival departure flows obtained for airlines A1 to A4, respectively, are
shown in Table 6 and 7.
Table 6 to be inserted here
Table 7 to be inserted here
It can be seen that for the airline A1, which was assigned the highest priority (α1=
0.70), the arrival and departure flows in almost all the fixes match the demand. The
only exception to this is the flow through the third arrival fix in the ninth time slot. A
queue of 4 is generated here, since the demand (14) in this time slot is more than the
8
maximum capacity of this fix (10). On the other hand, flows are assigned to airline
A4 (which has the lowest priority since α4=0.05) only in those slots, where capacity
is available after fulfilling the demands and outstanding queues of the other three
airlines which are assigned higher values of αt.
Figure 4 shows the reduction in total queue of all airlines as their priority factor αt is
increased from 0 to 1, while assigning equal priority to the other three airlines.
Figure 4 to be inserted here
It can be seen that all airlines have substantial queues, till their own αt < 0.25; since
in such situations, all the other airlines have a higher αt assigned to them. Queues for
all airlines are seen to sharply reduce for cases with αt  0.25. The queues for all
airlines are reduced to zero when αt for them exceed 0.5, which shows the
incremental effect of αt on reduction of queues. Figure 5 shows the effect of
assigning different priorities to each airline.
Figure 5 to be inserted here
It can be seen that the queues for all airlines reach high value during time slots 3, 4,
9 and 10, during which the demand is high. However, the queues for all airlines are
reduced to zero in time slot 6, since there is low demand in this and the previous slot.
Similarly the queues reduce to zero in the last time slot. However, drastic changes in
queues are seen in proportion to the assigned priorities. Figure 6 reinforces this
observation and shows how the cumulative queues vary for all airlines.
.
Figure 6 to be inserted here
It can be seen that when airline A1 is assigned a priority of 0.7, its queue is reduced
to zero in almost all time slots in which it had non-zero queue earlier, when all
airlines had equal priority (of 0.25 each). Cumulative queues over all the time slots
for airline A1 and A3 decreased from 55 to 4 and 33 to 3 respectively. However, the
cumulative queues for other airlines increase from 51 to 55 for airline A2 and 81 to
158 for airline A4.
5.2 Effect of control parameter β on arrival and departure flows
Equation (1) introduced the control parameter ( β ) which can be varied to
investigate the effect of changes in overall capacity of the terminal airspace. For
instance, the shaded region in Figure 3 indicates the allowable capacity with
β =0.75. Running the model for demand data listed in Table 1, for any specific value
of β , (say β =0.75, i.e., system operating at 75 % of its peak capacity) we can
obtain the respective arrival and departure flows which now optimizes the flow in
this scenario, as shown in Tables 8 and 9.
Table 8 to be inserted here
9
Table 9 to be inserted here
It is clear that in this hypothetical case, the total arrival and departure flow in each
time slot does not exceed 36 (0.75Ct). When the available terminal capacity is less
than full capacity, then the total flow is reduced, and the model optimally allocates
the reduced flow among the fixes and to respective airlines in the various time slots
to maximize the total flow.
6. Conclusions
The model presented in this paper enables determination of optimal air traffic flows
through the arrival and departure fixes of an airport, meeting a specified demand
profile over some time slots for the aircraft of competing airlines. The introduction
of a control parameter further enables modeling the effect of changes in total airport
capacity on the traffic that can be handled through these fixes. The results for a
hypothetical case study establish the efficacy of this model in allocating the flows
through various fixes over the time slots that result in minimum overall queues. It is
felt that this enhanced model can be utilized to create a decision support tool for air
traffic managers for better management of the traffic in real life scenarios.
References
Gilbo, E. P. (2003), ‘Arrival/Departure Capacity Tradeoff Optimization: a
Case Study at the St. Louis Lambert International Airport (STL)’, 5th USA/Europe
Air Traffic Management R&D Seminar, Budapest.
Dear, R.G. and Sherif, Y. S. (1991), “An algorithm for computer assisted
sequencing and scheduling of terminal area operations”, Transpn, Res.-A, Vol 25 A.
Horonjeff R., and McKelvey F.X. (1994), “Planning and Design of
Airports”, Chapter 8, pp 304-305, 4th ed, McGraw-Hill.
Bazargan .M, flemming K.and, Subramanian P. (2002), “A Simulation study
to investigate runway capacity using TAAM”, Proceedings of the 2002 Winter
Simulation Conference.
Hesselink and Basjes (1998), “Mantea Departure Sequencer: Increasing
Airport capacity by planning optimal sequences”, 2nd USA/EUROPE air traffic
management R&D seminar.
Anagnostakes.I. Idris.H., Clarke, Feron E., Hansman R.J. (2000), Odoni and
Hall W.D,”A Conceptual Design of a Departure Planner Decision Aid”, 3rd
USA/Europe Air Traffic Management R&D Seminar.
10
Carr.F, Evans. A., Clarke J. and Feron E. (2002), ”Modeling and Control of
Airport Queuing Dynamics under Severe Flow Restrictions”. Proceedings of the
American Control conference.
Sven Kaltenhauser (2003),”Tower and airport simulation flexibility”,
Simulation Modeling Practice and Theory 11,187–196.
Gilbo, E. P. (1993), ‘Airport Capacity: Representation, Estimation,
Optimization’, IEEE Trans. on Control Systems Technology, Vol. 1(3), pp.144-154.
Gilbo, E. P. (1997), ‘Optimizing Airport Capacity Utilization in Air Traffic
Flow Management Subject to Constraints at Arrival and Departure Fixes”, IEEE
Trans. on Control Systems Technology, Vol. 5, pp. 490-503.
Gilbo, E. P. and Howard, K. (2000), ‘Collaborative Optimization of Airport
Arrival and Departure Traffic Flow Management Strategies for CDM’, 3rd USA /
Europe Air Traffic Management R&D Seminar, Italy.
Idrissi, A and Li, C. M. (2006), ‘Modeling and optimization of the capacity
allocation problem with constraints’, IEEE International Conference on Research,
Innovation and Vision for the Future, pp.107-116.
Van Hentenryck, P., Michel, L. (2002), ‘The modeling language OPL, A
short overview’, in Optimization Software Class Libraries, Voss. S., Woodruff, D.
L., eds., OR/CSI Series, Kluver Academic Publishers, The Netherlands.
Idris H., Clarke J., Bhuva R. and Kang L. (2002), ‘Queuing Model for TaxiOut Time Estimation’, Technical report, Massachusetts Institute of Technology.
Virendra Kumar and Satish Chandra (1999), “Air transportation planning and
design”, Chapter 5, pp 119-121, 1st ed, Galgotia Publication.
11
Time
Slot
#
1
2
3
4
5
6
7
8
9
10
11
12
∑
1
10
13
15
9
2
1
4
2
5
2
3
2
68
Arrival Demand
Fixes
2
3
4
11
1
4
14
5
6
12
7
8
15
2
3
2
2
0
1
5
6
0
4
6
3
7
8
2
14
19
2
12
9
6
2
2
6
0
4
74
61
75
∑
26
38
42
29
6
13
14
20
40
25
13
12
278
1
9
8
2
4
2
1
4
9
8
5
3
1
56
Departure Demand
Fixes
2
3
4
9
9
9
8
8
8
2
2
3
4
4
3
2
2
1
3
3
3
4
4
5
8
8
8
8
10
8
7
5
5
3
3
4
0
0
0
58
58
57
∑
36
32
9
15
7
10
17
33
34
22
13
1
229
Table 1: Arrival and departure demand for each fix in each time slot (Gilbo, 1997)
12
Time
Slot #
1
2
3
4
5
6
7
8
9
10
11
12
∑
1
10/09
10/10
10/08
01/08
10/10
09/05
04/04
02/02
02/03
05/03
02/02
03/04
68
Arrival Flow
Fixes
∑
2
3
4
10/10 00/01 04/04 24
07/10 03/03 04/01 24
10/10 01/02 03/04 24
10/10 05/03 10/05 26
10/10 08/04 00/04 28
08/05 05/09 06/09 28
00/00 04/04 06/06 14
03/03 07/07 08/08 20
02/01 10/10 10/10 24
02/01 08/10 09/10 24
02/05 10/07 10/10 24
10/09 00/01 05/04 18
74
61
75
278
1
07/06
10/06
02/06
03/05
03/02
01/01
04/04
08/07
00/06
09/06
08/06
01/01
56
Departure Flow
Fixes
∑
2
3
4
06/06 04/06 07/07 24
05/06 06/06 03/03 24
08/06 04/06 10/10 24
04/05 09/05 03/03 19
02/02 02/02 01/01 8
03/03 03/03 03/03 10
04/04 04/04 05/05 17
08/07 08/07 03/03 27
08/05 10/07 06/06 24
07/06 02/06 06/06 24
02/06 04/06 10/10 24
01/02 02/00 00/00 4
58
58
57
229
Table 2: Comparison of calculated and quoted flows in various fixes
Note: The first entry indicates calculated value, and the second indicated value quoted in (Gilbo, 1997).
13
Time
Slot #
1
2
3
4
5
6
7
8
9
10
11
12
∑
Arrival Queue
Fixes
∑
1
2
3
4
00/01 01/01 01/00 00/00 2
03/04 08/05 03/02 02/05 16
08/11 10/07 09/07 07/09 34
16/12 15/12 06/06 00/07 37
08/04 07/04 00/04 00/03 15
00/00 00/00 00/00 00/00 0
00/00 00/00 00/00 00/00 0
00/00 00/00 00/00 00/00 0
03/01 00/01 04/04 09/09 16
00/02 00/02 08/06 09/08 17
01/03 04/03 00/01 01/00 6
00/00 00/00 00/00 00/00 0
Total Arrival queue
143
Departure Queue
Fixes
∑
1
2
3
4
02/03 03/03 05/03 02/03 12
00/05 06/05 07/05 07/05 20
00/01 00/01 05/01 00/02 5
01/00 00/00 00/00 00/01 1
00/00 00/00 00/00 00/00 0
00/00 00/00 00/00 00/00 0
00/00 00/00 00/00 00/00 0
01/02 00/01 00/01 05/02 6
09/04 00/04 00/04 07/04 16
05/03 00/05 03/03 06/03 14
00/00 01/02 02/00 00/01 3
00/00 00/00 00/00 00/00 0
Total Departure queue
77
Table 3: Comparison of calculated and quoted queues in various fixes
Note: The first entry indicates calculated value, and the second indicated value quoted in (Gilbo, 1997).
14
Time
Slot#
1
2
3
4
5
6
7
8
9
10
11
12
∑
Assumed distribution of demand through Arrival Fixes
1
2
3
4
A1+A2+A3+A4 A1+A2+A3+A4 A1+A2+A3+A4 A1+A2+A3+A4
5+1+3+1
3+1+6+1
0+1+0+0
2+1+1+0
7+3+2+1
3+3+7+1
1+1+1+2
2+2+2+0
3+3+3+6
4+4+0+4
1+1+1+4
3+2+1+2
1+3+3+2
5+5+0+5
1+0+1+0
3+0+0+0
1+1+0+0
1+1+0+0
0+1+1+0
0+0+0+0
1+0+0+0
0+1+0+0
4+0+1+0
3+2+1+0
2+2+0+0
0+0+0+0
2+1+1+0
3+2+1+0
1+1+0+0
1+0+2+0
3+1+2+1
3+2+1+2
2+2+1+0
1+0+1+0
14+0+0+0
3+4+2+10
1+0+1+0
2+0+0+0
4+3+2+3
3+3+3+0
0+0+0+3
3+0+0+3
2+0+0+0
1+0+1+0
1+0+1+0
3+0+0+3
0+0+0+0
2+2+0+0
25+16+14+13
26+15+16+17
32+9+10+10
28+20+13+14
Table 4: Arrival demand of four airlines at four fixes for 12 time slots.
15
Time
Slot#
1
2
3
4
5
6
7
8
9
10
11
12
∑
Assumed distribution of demand through Departure Fixes
1
2
3
4
A1+A2+A3+A4 A1+A2+A3+A4 A1+A2+A3+A4 A1+A2+A3+A4
6+3+0+0
4+1+3+0
2+1+2+4
4+0+5+0
4+2+2+0
4+3+0+2
3+4+1+0
4+3+1+0
1+1+0+0
0+4+0+1
1+0+1+0
2+1+0+0
2+0+1+1
1+5+1+0
1+2+1+0
2+1+0+0
2+0+0+0
1+1+1+0
1+0+1+0
0+0+0+1
0+0+1+0
1+1+2+0
2+1+0+0
1+2+0+0
2+2+0+0
1+0+2+0
2+1+1+0
2+1+0+2
4+1+1+3
2+0+2+3
5+0+1+2
2+2+1+3
2+2+2+2
2+0+2+2
8+1+0+1
2+1+2+3
1+2+1+1
4+0+0+0
2+1+1+1
2+1+2+0
1+1+0+1
1+0+1+0
1+0+1+1
2+2+0+0
1+0+0+0
0+0+0+0
0+0+0+0
0+0+0+0
26+14+8+8
21+15+14+8
28+11+10+9
23+14+11+9
Table 5: Departure demand of four airlines at four fixes for 12 time slots.
16
Time
Slot#
1
2
3
4
5
6
7
8
9
10
11
12
∑
Optimized Flow for each airline through Arrival Fixes
1
2
3
4
A1+A2+A3+A4 A1+A2+A3+A4 A1+A2+A3+A4 A1+A2+A3+A4
5+1+3+0
3+1+6+0
0+1+0+0
2+1+1+0
7+0+2+0
3+0+6+0
1+0+1+0
2+0+2+0
3+0+3+0
4+4+1+0
1+0+1+0
3+3+1+0
1+6+3+0
5+5+0+0
1+2+1+0
3+1+0+0
1+4+0+5
1+4+0+5
0+1+1+6
0+0+0+0
1+0+0+5
0+1+0+6
4+0+1+0
3+2+1+2
2+2+0+0
0+0+0+0
2+1+1+0
3+2+1+0
1+1+0+0
1+0+2+0
3+1+2+1
3+2+1+2
2+2+1+0
1+0+1+0
10+0+0+0
3+2+2+0
1+0+1+0
2+0+0+0
8+0+2+0
3+4+3+0
0+0+0+3
3+0+0+3
2+3+0+0
1+1+1+7
1+0+1+0
3+0+0+3
0+0+0+3
2+2+0+3
25+16+14+13
26+15+16+17
32+9+10+10
28+20+13+14
Table 6: Arrival flow data (α1= 0.70, α2 =0.10, α3=0.15, α4=0.05, β = 1, λ=0.5)
17
Time
Slot#
1
2
3
4
5
6
7
8
9
10
11
12
∑
Optimized Flow for each airline through Departure Fixes
1
2
3
4
A1+A2+A3+A4 A1+A2+A3+A4 A1+A2+A3+A4 A1+A2+A3+A4
6+0+0+0
4+0+3+0
2+0+0+0
4+0+5+0
4+0+2+0
4+0+0+0
3+0+3+0
4+3+1+0
1+6+0+0
0+5+0+2
1+5+1+0
2+1+0+0
2+0+1+0
1+2+1+1
1+2+1+0
2+1+0+0
2+0+0+1
1+0+1+0
1+0+1+4
0+0+0+1
0+0+1+0
1+0+2+0
2+1+0+0
1+2+0+0
2+2+0+0
1+1+2+0
2+1+1+0
2+1+0+2
4+1+1+0
2+1+2+2
5+0+1+0
2+2+1+3
2+2+2+0
2+1+2+0
8+1+0+0
2+0+2+0
1+2+1+2
4+4+0+0
2+1+1+0
2+2+2+0
1+1+0+5
1+1+1+0
1+0+1+5
2+2+0+3
1+0+0+0
0+0+0+3
0+0+0+0
0+0+0+0
26+14+8+8
21+15+14+8
28+11+10+9
23+14+11+9
Table 7: Departure flow data (α1= 0.70, α2 =0.10, α3=0.15, α4=0.05, β = 1, λ=0.5)
18
Time
Slot #
1
2
3
4
5
6
7
8
9
10
11
12

1
10
10
10
1
10
9
4
2
2
5
2
3
68
Arrival flow at various fixes
β =1
β = 0.75
2
3
4
1
2
3
10
0
4
10
3
0
7
3
4
9
0
2
10
1
3
10
9
3
10
5
10
8
10
1
10
8
0
0
8
10
8
5
6
4
10
6
0
4
6
4
0
2
3
7
8
10
0
1
2
10
10
0
2
0
2
8
9
8
2
0
2
10
10
3
3
0
10
0
5
1
6
7
74
61
75
67
53
32
4
0
8
3
0
10
1
0
10
4
5
10
10
61

β =1

β = 0.75
24
24
24
26
28
28
14
20
24
24
24
18
278
13
19
25
19
28
21
6
21
6
15
16
24
213
Table 8: Optimum Arrival flow for β =0.75,  =0.5
19
Time
Slot #
1
2
3
4
5
6
7
8
9
10
11
12

1
7
10
2
3
3
1
4
8
0
9
8
1
56
Departure flow at various fixes
β=1
β = 0.75
2
3
4
1
2
3
6
4
7
9
3
5
5
6
3
4
10
0
8
4
10
0
0
10
4
9
3
6
0
2
2
2
1
2
6
0
3
3
3
0
9
4
4
4
5
9
4
9
8
8
3
1
4
3
8
10
6
10
8
3
7
2
6
10
0
10
2
4
10
0
1
10
1
2
0
0
10
2
58
58
57
51
55
58
4
6
3
1
9
0
2
8
7
9
1
9
0
55

β =1
24
24
24
19
8
10
17
27
24
24
24
4
229
Table 9: Optimum Departure flow for β =0.75,  =0.5
20

β = 0.75
23
17
11
17
8
15
30
15
30
21
20
12
219
Captions of Figures
Figure 1: Airport Capacity Curve (Gilbo, 1997)
Figure 2: Geographically distributed arrival and departure fixes (Gilbo, 1997)
Figure 3: Modified Airport Capacity Curve with β =0.75
Figure 4: Effect of increase in airline priority on total (Arrival + Departure) queue
Figure 5: Queues in different time slots with equal and differing priorities assigned
to airlines
Figure 6: Cumulative queues in different time slots with equal and differing
priorities assigned to airlines
21
35
0, 30
17, 30
Departures / time slot
30
25
24, 24
20
28,15
15
10
5
28,0
0
0
5
10
15
20
Arrivals / time slot
22
25
30
Arrival fixes
Arrival Queue
Departure fixes
1
1
Arrival Queue
2
2
Runway
system
Arrival Queue
3
3
Departure Queue
Arrival Queue
4
4
Terminal
23
24
Arrival +Departure Queue
220
200
180
160
140
120
100
80
60
40
20
0
Queue for A1
Queue for A2
Queue for A3
Queue for A4
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Airline Priority
25
1
Queue in each time slot
35
30
A1(alpha1=0.70)
A2(alpha2=0.10)
A3(alpha3=0.15)
A4(alpha4=0.05)
A1(alpha1=0.25)
A2(alpha2=0.25)
A3(alpha3=0.25)
A4(alpha4=0.25)
25
20
15
10
5
0
1
2
3
4
5
6
7
Time slot
26
8
9
10
11
12
Cumalative queues for all airlines
160
140
A1(alpha1=0.70)
A2(alpha2=0.10)
A3(alpha3=0.15)
A4(alpha4=0.05)
A1(alpha1=0.25)
A2(alpha2=0.25)
A3(alpha3=0.25)
A4(alpha4=0.25)
120
100
80
60
40
20
0
0
1
2
3
4
5
6
7
Time slot
27
8
9 10 11 12
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