D lovell 102413

Congestion modeling and mitigation in the National Airspace System

Dr. David Lovell Presented at

Portland State University

10.24.13

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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

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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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• 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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• 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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• 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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• 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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500

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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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5.00

10.00

15.00

20.00

25.00

30.00

35.00


De

lay

(min

ute

s/fl

igh

t)


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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5.00

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20.00

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30.00

35.00


De

lay

(min

ute

s/fl

igh

t)


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

D lovell 102413

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