Modeling Demand Uncertainties During Ground Delay Programs Narender Bhogadi M.S. in Systems Engineering University of Maryland Ground Delay Programs (GDPs) Q To balance arrival demand and capacity at the afflicted airport Q Transfer costly airborne delay to less expensive ground delay source source GDP airport source Q Input parameters - Airport Capacity and Arrival Demand Q Capacity and demand , both stochastic in nature Demand Uncertainties Q Three main sources of demand uncertainties z Flight Drifts, z Flight Cancellations, and z Pop-up Flights Q Combined Effects z z z Under-Utilization of the airport resources (slots) Unpredictable arrival sequence Increased airborne holding Flight Cancellations Q Flight Cancellations without notices in advance, cause “holes” in the arrival sequence Q Timed Out (TO) Cancellations almost always result in slots being unused Destination Cancelled flight Sources Pop-up Flights Q Any flight that arrived during a GDP and that first appeared in the ADL after the GDP model time Pop-up Destination Sources Pop-up Q Pop-ups add to the arrival demand and displace the actual arrival sequence Flight Drifts Q Flight Drifts are results of : z z 1645z CTD non-compliance , where ARTD > or < CTD CTA non-compliance , where AETE > or < OETE CTD=1700z 1715z 1730z Ground Drift 1900z CTA = 1915z 1930z CTD = 1700z Enroute Drift Q Net Drift = Ground Drift + Enroute Drift •ARTA - Actual Runway Time of Arrival •ARTD - Actual Runway Time of Departure •AETE - Actual Enroute Time •CTA - Control Time of Arrival •CTD - Control Time of Departure •OETE - Original Estimated Enroute Time Modeling Demand Uncertainties Q Stochastic Mixed Integer Optimization (SMIO) Model z Incorporates only flight cancellations and pop-ups z Generates Optimal Planned Arrival Rates (PAARs) for any GDP scenario Q Simulation Model z Incorporates flight cancellations, pop-ups and drifts z Validates the SMIO model by generating Pareto Optimal PAARs for a large set of scenarios Details on SMIO Model Q Objective Function : Minimize the expected airborne queue Q Variables : Xpaar(k,t) = 1 if PAAR = k in time period t; else 0 Y(k,j,t) : probability that at the end of time period “t” an airborne queue of size “j” exists and PAAR = k in time period “t” Q Main Input Parameters : AAR(t), P cnx , Ppop , and Utilization Parameter “ε” Q Main Constraints z Markovian Constraint : ∑ k Y(k,j,t) = ∑i q(k,i,j,t) Y(k,i,t-1), where q(k,i,j,t) = Pr{i + number of arrivals in t - AAR(t) = j / PAAR = k} z Utilization Constraint : Expected Number of unutilized slots ≤ ε Q Outputs : Optimal PAARs and Optimal Expected Queue Size Formulation of SMIO Model Minimize Σt Σk Σj j Y(k,j,t) Subject to: Σk Xpaar(k,t) = 1 ∀ t = 1, 2,…P Σj Y(k,j,t) ≤ Xpaar (k,t) ……….………… (1) ∀ j = 1,2,…MaxQ …………….…… (2) Σk* Y(k*,j,t) ≤ Σk Σi q(k,i,j,t) Y(k,i,t-1) ∀ j, ∀ t Σt Σk Σi Qe(k,i,t) Y(k,i,t-1) ≤ ε …………….…… (4) Xpaar(k,t) ε {0,1} ...……. (3) Details on Simulation Model Q Single-Server Queuing Model Q Input Distributions z Geometric distribution for flight cancellations z Empirical distribution for drifts z Exponential distribution for pop-ups Q Performance Measures z Ground delay z Airborne delay z Utilization Empirical Analysis of Drifts Frequency Distribution of Ground Drifts (1999 SFO) 450 100.00% 400 90.00% 350 80.00% 70.00% 300 60.00% 250 50.00% 200 40.00% 150 30.00% 100 20.00% 50 10.00% 0 .00% -90 -75 -60 -45 -30 -15 Frequency 0 15 30 Bin (minutes) 45 60 75 90 More Cumulative % ( Ground Drift = ARTD - CTD ( Mean is shifted to the right - more forward drifts Empirical Analysis of Drifts (contd..) Distribution of Enroute Drifts (1999 SFO) 1000 120.00% 900 100.00% Frequency 800 700 80.00% 600 500 60.00% 400 40.00% 300 200 20.00% 100 0 .00% -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 Bin(minutes) Frequency Cumulative % Q Enroute Drift = AETE - OETE Q Actual Enroute Time less than expected Q Enroute Drifts confined to a small window 80 More Empirical Analysis of Cancellations Distribution of Cancellations (1999 SFO) 250 120.00% 100.00% Frequency 200 80.00% Mean = 18.37 Variance = 402.35 150 60.00% 100 40.00% 50 20.00% 12 0 11 0 10 0 90 80 70 60 50 40 30 20 .00% 10 0 0 Bin Frequency Cumulative % Q Cancellations follow a geometric distribution during GDP Empirical Analysis of Pop-up Flights GDP Avg Popup per Hour SFO 180 162 160 140 Mean = 1.70 Stdv = 1.93 Number of GDPs 120 106 100 80 70 60 49 40 24 15 20 4 4 6 7 1 0 2 2 8 9 10 More 0 0 1 2 3 4 5 Avg Popups per Hour (Courtesy : Bob Hoffman) Results for SMIO Model Q Capacity Scenario : (30,30,30,30,30,30,30) on 05/01/98 SFO 34 33 32 31 30 29 28 27 26 1 2 Time Period (hour) 3 4 5 6 ATC PAARs 7 1 2 3 4 5 SMIO PAARs 6 7 Results for SMIO Model (contd.) Per-Flight AirBorne Delay(in min) Airborne Delay vs. Cancellation Probability 7 6 5 4 3 2 1 0 Exp Num of Open Slots in Program 4.5 5 6 7 0 0.1 0.2 Cancellation Probability 0.3 Results for Simulation Model Q Capacity Scenario : (30,30,30,30,30,30) Q Tested the scenarios for all PAARs in the interval [28 34] Q Used Pareto Optimality with Airborne Delay and Utilization as Objective functions z Pareto Optimality A state A (a set of parameters) is said to be Pareto optimal, if there is no other state B dominating the state with respect to a set of objective functions. A state A dominates a state B , if A is better than B in at least one objective function and not worse with respect to all other objective functions. Results for Simulation Model(contd.) Pareto Optimal PAARs 1 1 - (29,30,30,30,30,30) 2 - (32,30,30,30,30,30) 3 - (33,30,30,30,30,28) 4 - (33,30,30,30,30,29) 5 - (32,30,30,30,28,28) 6 - (33,30,28,30,30,29) 7 - (33,30,28,30,28,30) 8 - (33,30,28,30,28,29) 9 - (34,30,28,28,28,30) 10 - (34,30,28,29,28,29) 11 - (34,30,32,30,29,28) 12 - (34,28,28,32,28,32) 13 - (34,34,28,28,32,32) 13 0.98 12 11 0.96 10 9 Utilization 0.94 8 7 0.92 6 5 0.9 4 0.88 3 2 0.86 1 0.84 0 5 10 Airborne Holding (Min) per Flight 15 Summary Q Significant stochasticity in airport arrival demand Q Demand Uncertainties lead to under-utilization, and excessive airborne holding Q Two models - SMIO and Simulation Model - are developed Q Models recommend policy changes in setting of PAARs - substituting staggered patterns for flat patterns