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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