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