Transcript A Decomposition Method for Stochastic Mixed Integer Programs

Dip. di Matematica Pura ed Applicata - Università di Padova
Congestion Pricing
and
Queuing Theory
Giovanni Andreatta and Guglielmo Lulli
1
Demand versus Capacity
Fast and steady increase of demand
(up to 11 September 2001 ...)
Modest increase of capacity
Need to address demand
2
Demand Management
Strategies should
Limit demand for access to busy airfields
and/or congested airspace
Modify temporal (and/or spatial)
distribution of demand
3
LGA demand before and after the lottery
Scheduled operations per
hour on weekdays
100
Nov, 00
90
Aug, 01
80
75 flt/hour
70
60
50
40
30
 Scheduled
operations
reduced by
10% (from
1,348 to
1,205/day)
20
10
0
5
7
9
11
13
15
17
19
21
Time of day, e.g. 5 = 0500 - 0559
23
1
3
Capacity of 75/hr does not
include allocation of six slots
for g.a. operations
*** 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
4
Small reduction in demand
may lead to dramatic reduction in delays
Minutes of delay per operation
120
Nov, 00
Aug, 01
100
80
60
 Average delay
reduced by
>80% during
evening hours
 Lottery was
critical in
improving
operating
conditions at
LGA
40
20
0
5
7
9
11
13
15
17
19
21
23
1
3
Time of day
Capacity = 75 operations/hr
*** 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 issues
Investigate different approaches (mix of
administrative and market-based measures)
Provide a demonstration of the approaches
through an example
6
What has already been done
Peak 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Very limited work (unpublished) on airspace side
7
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
$400
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Case 1: No congestion fee
Parameter
Type 1
Type 2
Type 3
1802
721
288
0
0
0
1802
721
288
Demand (no. of movements per hour)
5.7
37.4
50.5
PST (percentage of service time)
7.2
41.9
51.9
Delay cost (DC) per aircraft ($)
Congestion fee (CF)
($)
Total cost of access (DC + CF) ($)
Total demand (no. of movements per
93.6
hour)
Expected delay per aircraft
Utilization of the airport
43 minutes 15 seconds
99.2%
(% of time busy)
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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 …
11
Optimal congestion fee
A 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:
dWq
dC
MC 
 c Wq  c
d
d
Marginal = Internal + External
cost
cost
cost
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dWq
dC
MC 
 c Wq  c
d
d
MC = Marginal Cost
c
= (delay) cost per unit time per customer
Wq = Expected queuing time per customer

= demand rate
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System of non-linear equation
  g ( DC  CF )
DC  f ( )
CF  EC
EC  h ( )
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Optimization Model
2
min ( out  in )
out  g ( DC  CF )
DC  f (in )
CF  EC
EC  h (in )
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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)
($)
988
804
692
Demand (no. of movements per hour)
29.2
34.6
14.9
PST (Percentage of Service Time)
40.6
42.8
16.6
Total demand (no. of movements per
78.7
hour)
Expected delay per aircraft
Utilization of the airport (% of time
busy)
3 minutes 15 seconds
89.9%
70
o
o+
40
30
20
Type 1
Type 2
Type 3
+
+
10
o
Total cost ($)
00
18
00
16
00
14
00
12
00
10
0
80
0
60
0
40
20
0
0
00
50
20
60
0
Arrival rate (Users/unit time)
Demand Functions for three types of users
o
+
No Fee
With Fee
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
60
50
40
Large
Medium
Small
30
20
10
0
CF=
0
CF=
EC
Cons
Intra
Redu
-clas
train
ced E
s CF
ed m
C
kt- ba
sed
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Comparison of the cases (ctd.)
(Minutes)
Average Delay
50
45
40
35
30
25
20
15
10
5
0
No
CF
Mk
tb
as
ed
2- s
t
ag
e
CF
Co
ns
tra
ine
Mi
d(
xe
d
MI
X3
Su
bje
c
tiv
e
=3
0%
)
Approaches
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Comments
 We analyze other pricing structures
 Constrained market-based provides balanced PST
Market-based mechanism
 When demand is dynamic, use DELAYS instead of
Queuing Theory
 Estimation 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 m
oag(f2) = 4:30; eat(f2) = 4:50; D(f2) = 20 m
Exchange:
oag(f1) = 4:00; eta(f1) = 4:50; D(f1) = 50 m
oag(f2) = 4:30; eta(f2) = 5:00; D(f2) = 30 m
Average 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
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t* or higher.
Airline Comments
 Priority 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 property
What about encouraging provision of up-to-date flight
status information??
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Resource Allocation Concept: Balance
Major Traffic Flow Categories
r
6
r
1
r
2
Traffic classes, e.g. IAD inbound
traffic; ascending traffic from
CLE;
E to W NRP traffic.
r major flow categories
• Need to balance
• Possible balance criterion: proportional to
historical traffic flows
• Can be throughput/fairness tradeoff
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