Congestion modeling and mitigation in the National Airspace System
Dr. David Lovell Presented at
Portland State University
10.24.13
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1
Outline
• Diffusion models of queueing delays at individual airports (NASA)
• Equitable resource allocation methods for airspace flow program planning (FAA)
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DIFFUSION MODELS OF QUEUEING DELAYS AT INDIVIDUAL AIRPORTS Project Sponsor: NASA
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Single airport queue formulation NEXTOR
0
2
4
6
8
10020
4060
80100
0
0.1
0.2
0.3
0.4
; density of length of queue at time if x t i t
Fokker-Plank equation
2
2
; 1; ; ; ;
2
f x tV x t f x t M x t f x t
t xx
Assumptions
•Continuity
•Markov
•2nd order approximatable
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The Fokker-Plank equation and boundary conditions
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2
2
; 1; ; ; ;
2
i
i i i i
f x tV x t f x t M x t f x t
t x x
PDE:
Boundary Conditions:
0
(0; ) (0; ) ( ; ) ( ; ) 0 0x
f t M t f x t V x t tx
Initial conditions:
( ;0) ( )f x x
Reflecting barrier to prevent negative queue length
Queue empty at the beginning of the day
5
Mesh generation for the finite element method
• Allow for non-uniform finite element widths
• Standard FEM implementationsmight assume uniform element widths when computing stiffness matrix and load vector
1
1
1
1
ei-1 ei ei+1 ei+2
fi
fi+1
fi+2
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Global stiffness matrix assembly
• Goal: solve the linear system
where
• The products fj’fi’, fjfi’, and fjfi
are only non-zero for • Thus, the matrix {Kij} is
tridiagonal • One option is to assemble the
matrix from 2x2 element-wise contributions; however, they are NOT symmetric
K11
e1
K21
e1
K12
e1
K22
e1
K12
K22
K11
K21 K23
K33K32 K34
K44K43 K45
K55K54K22
e2
K32
e2
K23
e2
K33
e2
K33
e3
K43
e3
K34
e3
K44
e3
1
1
NL
j ij i
j
a K R
1 11' ' '
2
1
L L
ij j i j i
j i
K V x x dx M x x dx
x x dxt
f f f f
f f
1i j
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M/M/1, l = 5, m = 40, n = 10,000 MC time = 106.9 sec, diff time = 8.17 sec
0 20 40 60 800
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5Mean queue length
Monte Carlo
Diffusion
0 20 40 60 800
0.05
0.1
0.15
0.2
0.25Variance of queue length
Monte Carlo
Diffusion
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Results from Chicago O’Hare Airport NEXTOR
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0 20 40 60 800
5
10
15
20
25
30
35
time slice
aircra
ft/h
ou
rArrival rate
0 20 40 60 800
5
10
15
20
25
time slice
aircra
ft/h
ou
r
Service rate
0 20 40 60 800
5
10
15
20
time slice
aircra
ft
Mean queue length
Diffusion
Monte Carlo
0 20 40 60 800
20
40
60
80
100
120
140
160
time slice
aircra
ft2
Variance of queue length
Diffusion
Monte Carlo
Contributions
• Less distribution dependence: – Can specify distributions only up to 1st and 2nd moments
• Independent mean and variance: – Important stochastic properties can be evaluated, and can
propagate if these models are chained together to form a network
• Solution time – A complete stochastic profile of the solution can be
generated in a single run of the model (a few seconds) rather than having to run Monte Carlo thousands of times
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Final results
• Lovell, David J., Kleoniki Vlachou, Tarek Rabbani, and Alexandre Bayen (2013). A diffusion approximation to a single airport queue. Transportation Research Part C: Emerging Technologies, vol. 33, pp. 227-237.
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EQUITABLE RESOURCE ALLOCATION METHODS FOR AIRSPACE FLOW PROGRAM PLANNING
Project Sponsor: FAA
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• During adverse weather conditions, reduced en-route capacity leads to reduction in the number of flights that can pass the impacted area
• The available slots at the boundary of the constrained area are less than the flights scheduled to pass that portion of the airspace
Problem Description NEXTOR
• Ground Delay Programs (GDP’s) – A GDP issues departure delay to aircraft expected to arrive
at a constrained airport. These ground delays are less costly and safer than the airborne delays that would result without such actions.
– Ration-By-Schedule
• Flow Constrained Areas (FCA’s) – FCAs are used to show areas where the traffic flow should
be evaluated or where initiatives should be taken due to severe weather or volume constraints.
• Airspace Flow Program (AFP) – AFP combines the power of GDP’s and FCAs to allow more
efficient, effective, equitable, and predictable management of airborne traffic in congested airspace.
Traffic Flow Management (TFM) Tools
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• Two key enabling ideas
– NAS customers submit cost weight sets of trajectory options to the traffic management system
– Traffic managers manage demand on resources by setting capacities on those resources then running allocation algorithms that adjust demand to meet those capacities
Collaborative Trajectory Options Program (CTOP)
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• Characterize preference and cost information provided by flight operators
• Explicit consideration of three performance metrics
– System efficiency (performance criteria such as throughput and flight delay)
– Equity (flight operators are treated fairly)
– User cost (internal flight operator cost function)
Address Specific Problems NEXTOR
• Allocate fairly the reduced number of slots to airlines
• “Proportional random allocation” is used to estimate the fair share of each flight and each airline for each of the slots
Allocation Procedure NEXTOR
Fair Share Computation
Definitions Example
Time of flights: the time each flight ( ) is scheduled to arrive at the boundary of the FCA
Time of slots ( )
Index of which flight corresponds to which airline (Airlines: A1=1, A2=2, A3=3)
358 400 402 403 405 406
400 402 404 406
1 2 1 2 3 3
f1 f2 f3 f4 f5 f6
s1 s2 s3 s4
f1 f2 f3 f4 f5 f6
fi
s j
i 1,2,3,4,5,6 for our example
j 1,2,3,4 for our example
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Fair Share Computation
• Find the earliest slot that each flight can be assigned to (Slots: S1=1,
S2=2, S3=3, S4=4)
• Find the total number of flights that can be assigned to each slot
1 1 2 3 4 4
1 1 1 1
1 1 1 1
0 1 1 1
0 0 1 1
0 0 0 1
0 0 0 1
s1 s2 s3 s4
f1 f2 f3 f4 f5 f6
Ni, j 1, if flight i canbeassigned toslot j
0, otherwise
nm Ni,mi
is the the number of flights that can be assigned to the respective slot nm
Ni, j
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Fair Share Computation
• Find the share of each flight for each slot, where share given by
where is the earliest slot that flight can be assigned to and is the number of flights that can be assigned to the respective slot
Example,
1
2
1
4
1
8
1
24
1
2
1
4
1
8
1
24
01
2
1
4
1
12
0 01
2
1
6
0 0 01
3
0 0 01
3
s1 s2 s3 s4
f1
f2
f3
f4
f5
f6
Sharesjfi Ni, j *
1
(nm mk
j
(m 1))
nm
Shares2f1
1
(2 (11)) (3 (2 1))1
4
m 1,2,3,4 forourexample
k fi
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Fair Share Computation
• Find the total share of each flight for all slots
22
24
22
24
10
12
2
3
1
3
1
3
f1
f2
f3
f4
f5
f6
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Fair Share Computation
• Find the fair share of each airline for all available slots
42
24
38
24
2
3
A1 A2 A3
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Airline preference information
• Priority number and maximum delay (before cancellation or re-route) for each flight:
1 2 3 4 5 6flight
carrier 1 2 1 2 3 3
priority 2 3 4 4 1 1
max delay 35 25 23 32 50 33
f f f f f f
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• Start by considering only fractional shares
– For carriers with large shares, this should be approximately uniformly distributed
– For small carriers with only a fractional share, this allows them not to be systematically disadvantaged
• Once fractional shares are exhausted, revert to integer shares
Preference Based Proportional Random Allocation (PBPRA)
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• For each slot, determine the carriers that have a claim on that slot – Enforce maximum delay constraints
• Allocate the slot randomly, but with probabilities proportional to the magnitude of the claims
• Assign the slot to the flight of the winning carrier with the highest priority number
• Reduce the winning carrier’s claims to subsequent slots where that flight contributed to its fair share
• Repeat until all slots/flights are either assigned or rejected (cancelled or re-routed)
PBPRA 2 NEXTOR
• In a given day the slots allocated to an airline won’t match exactly its fair share
• Over a large number of days the airlines will get what they want on average
• Fair Share – Actual Allocation = Error
Variance in Slot Allocation NEXTOR
PBPRA results
• Total delay can decrease at high levels of congestion because flights are cancelled
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0
500
1000
1500
2000
2500
3000
25 ac/hr 20ac/hr 15ac/hr
We
igh
ted
De
lay
Scenario for FCA Capacity (ac/hr)
Total Weighted Delay
RBS
RBS with Substitutions
PBPRA
PBPRA results 2
• This is weighted average delay amongst only those flights that were assigned slots (delays)
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0.00
5.00
10.00
15.00
20.00
25.00
30.00
35.00
25 ac/hr 20ac/hr 15ac/hr
De
lay
(min
ute
s/fl
igh
t)
Scenario for FCA Capacity (ac/hr)
Weighted Average Delay
RBS
RBS with Sunstitutions
PBPRA
A-PBPRA results
• Two types of airlines: – A) prefer earlier (fewer) slots
– B) prefer more (later) slots
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0.00
5.00
10.00
15.00
20.00
25.00
30.00
35.00
25 ac/hr 20ac/hr 15ac/hr
De
lay
(min
ute
s/fl
igh
t)
Scenario for FCA Capacity (ac/hr)
Weighted Average Delay
RBS
RBS with Substitutions
A-PBPRA
Final results
• Vlachou, K. and David J. Lovell (2013). Mechanisms for equitable resource allocation when airspace capacity is reduced. Transportation Research Record 2325, pp. 97-102.
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Thank you! NEXTOR