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Congestion Pricing and

Queuing Theory

Giovanni Andreatta and Guglielmo Lulli

Dip. di Matematica Pura ed Applicata - Università di Padova

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Demand versus CapacityFast and steady increase of demand

(up to 11 September 2001 ...)

Modest increase of capacity

Need to address demand

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Demand Management Strategies should

Limit demand for access to busy airfields and/or congested airspace

Modify temporal (and/or spatial) distribution of demand

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LGA demand before and after the lottery

*** from Odoni & Fan; November 2000 as a representative profile prior to slot lottery at LaGuardia; August 2001 as a representative after slot lottery; Source: Official Airline Guide

Scheduled operations per hour on weekdays

Time of day, e.g. 5 = 0500 - 0559

Scheduled operations reduced by 10% (from 1,348 to 1,205/day)

0

10

20

30

40

50

60

70

80

90

100

5 7 9 11 13 15 17 19 21 23 1 3

Nov, 00

Aug, 01

75 flt/hour

Capacity of 75/hr does not include allocation of six slots for g.a. operations

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Small reduction in demand may lead to dramatic reduction in delays

Minutes of delay per operation

Average delay reduced by >80% during evening hours

Lottery was critical in improving operating conditions at LGA

Capacity = 75 operations/hr

Time of day

0

20

40

60

80

100

120

5 7 9 11 13 15 17 19 21 23 1 3

Nov, 00

Aug, 01

*** from Odoni & Fan

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Objective of this presentation

Use queue theory models to show the possible benefits of the demand management approach

Highlight fairness/equity issuesInvestigate different approaches (mix of

administrative and market-based measures)Provide a demonstration of the approaches

through an example

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What has already been donePeak period pricing in general (widely

investigated)Applications to congestion-pricing of

transportation facilities (more recent)Applications to air transportation (fewer)

Concentrated on airport congestion

Very limited work (unpublished) on airspace side

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Airport environment:Illustrative example

Parameter Type 1 Type 2 Type 3

Service rate (movements per hour)

80 90 100

Standard deviation of service time (seconds)

10 10 10

Cost of delay time ($ per hour)

$2,500 $1,000 $400

40 50 600,001 0,003 0,01

0,00001 0,00002 0,00008x lambda 1 lambda 2 lambda 3

0 40 50 60100 39,8 49,5 58,2200 39,4 48,6 54,8300 38,8 47,3 49,8400 38 45,6 43,2500 37 43,5 35600 35,8 41 25,2700 34,4 38,1 13,8800 32,8 34,8 0,8900 31 31,1 -13,8

1000 29 27 -301100 26,8 22,5 -47,81200 24,4 17,6 -67,21300 21,8 12,3 -88,21400 19 6,6 -110,81500 16 0,5 -1351600 12,8 -6 -160,81700 9,4 -12,9 -188,21800 5,8 -20,2 -217,21900 2 -27,9 -247,82000 -36 -280

Deman d Fun ct ion s f or t hr ee t ypes of user s

-500

0

500

0 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000

T o ta l c o st ($ )

Type 1

Type 2

Type 3

Demand Functions for three types of users

0

10

20

30

40

50

60

70

Total cost ($)

Arr

ival

rat

e (U

sers

/uni

t tim

e)

Type 1

Type 2

Type 3

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Case 1: No congestion feeParameter Type 1 Type 2 Type 3

Delay cost (DC) per aircraft ($) 1802 721 288

Congestion fee (CF) ($) 0 0 0

Total cost of access (DC + CF) ($) 1802 721 288

Demand (no. of movements per hour) 5.7 37.4 50.5

PST (percentage of service time) 7.2 41.9 51.9

Total demand (no. of movements per

hour)

93.6

Expected delay per aircraft 43 minutes 15 seconds

Utilization of the airport

(% of time busy)

99.2%

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Congestion pricing

(One) Objective of congestion pricing (or auctions): operators should pay a price for using a slot that is at least equal to the marginal cost of using that slot

flights scheduled during high demand periods will be high revenue flights, e.g. large passenger loads, high paying customers or …

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Optimal congestion feeA congestion fee on a user is optimal when it is

equal to the external costs that the user imposes on the other users.

For a M/G/1 queue:

Marginal Internal External cost cost cost

d

dWcWc

d

dCMC q

q

= +

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MC = Marginal Cost

c = (delay) cost per unit time per

customer

Wq = Expected queuing time per customer

= demand rate

d

dWcWc

d

dCMC q

q

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System of non-linear equation

)(

)(

)(

hEC

ECCF

fDC

CFDCg

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Optimization Model2)(min inout

)(

)(

)(

in

in

out

hEC

ECCF

fDC

CFDCg

Case 2: Optimal congestion fee

Optimal Congestion Fee Type 1 Type 2 Type 3

Delay cost (DC) per aircraft ($) 135 54 22

Congestion fee (CF) ($) 853 750 670

Total cost of access (DC+CF) ($) 988 804 692

Demand (no. of movements per hour) 29.2 34.6 14.9

PST (Percentage of Service Time) 40.6 42.8 16.6

Total demand (no. of movements per

hour)

78.7

Expected delay per aircraft 3 minutes 15 seconds

Utilization of the airport (% of time

busy)

89.9%

Demand Functions for three types of users

0

10

20

30

40

50

60

70

Total cost ($)

Arr

iva

l ra

te (

Us

ers

/un

it t

ime

)

Type 1

Type 2

Type 3++

+

o

o

oo No Fee

+ With Fee

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What is fair?

No formal definition available in the literature

Subjective measure Up to the Airport Authority

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Alternative Approaches Two-phase (choose PST)

No economic interpretation

Constrained market-based Bounds on the minimum PST are imposed

Intra-class congestion fee Reduced external costs

Implement different concepts of fairness

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Comparison of the cases

Percentage of Service Time

0

10

20

30

40

50

60

Large

Medium

Small

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Comparison of the cases (ctd.)

Average Delay

05

101520253035404550

No CFMkt based CF

2-stage

Constrained (MIX3 = 30%)

Mixed

Subjective

Approaches

(Min

ute

s)

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Comments

We analyze other pricing structuresConstrained market-based provides balanced PST

Market-based mechanismWhen demand is dynamic, use DELAYS instead of

Queuing TheoryEstimation of demand functions i(x): (challenging

problem!)MbDM approaches are as much political and

institutional as they are technical: the proposed analysis can provide significantly more quantitative details.

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Thanks !

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Comparison between the two cases

By charging a congestion fee equal to the external delay costs, we have:

Reduced the utilization of the runway system (89.9% vs. 99.2%)

Greatly reduced the average delay per aircraft (3’15’’ vs. 43’15’’)

Greatly reduced the delay costs per aircraft ($135 from $1802, $54 from $721, $22 from $288)

Augmented the no. of pax per hour (9600 vs. 5900)

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Equity Metrics aka Measures of Dispersion

The following measures are suggested for measuring the equity of the distribution of funds to school districts:

Variance: squared deviation from the mean; related measure -- coefficient of variation: square root of variance divided by mean

Gini coefficient: average difference between each pair of values divided by two times the mean.

McLoone coefficient -- assesses equity in the lower half of a distribution – average of the difference between the median and the value of each element below the median (oriented toward distribution of money assumes lower half is worse half – should change to upper half for delay allocation).

Assumption: perfect equity each claimant receives same allocation

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Reducing dispersion and pair-wise comparison principle

1st solution can be “improved” using the following type of exchange:oag(f1) = 4:00; eta(f1) = 5:00; D(f1) = 60 moag(f2) = 4:30; eat(f2) = 4:50; D(f2) = 20 mExchange: oag(f1) = 4:00; eta(f1) = 4:50; D(f1) = 50 moag(f2) = 4:30; eta(f2) = 5:00; D(f2) = 30 mAverage delay is same: 80/2 = 40 m but dispersion is less

Note that this exchange represent a pair of flights that do not satisfy the pair-wise comparison principle:

if flight f has been assigned t* units of delay, it should not be possible to reduce the delay assigned to f without increasing the delay assigned to another flight a value of t* or higher.

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Airline CommentsPriority based on accrued delay rewards poor airline

performance!! airlines that have late departures (due to their own

inefficiencies) are given priority later.

Devise systems that allows airlines to compete by rewarding better performance and better internal management systems

But: RBS has this same propertyWhat about encouraging provision of up-to-date flight status information??

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Resource Allocation Concept: Balance Major Traffic Flow Categories

Traffic classes, e.g. IAD inboundtraffic; ascending traffic fromCLE;E to W NRP traffic.

r1

r6

r

r2

• Need to balance major flow categories• Possible balance criterion: proportional to

historical traffic flows• Can be throughput/fairness tradeoff