session 20 joel p. franklin
TRANSCRIPT
13 January 2010, 1
Congestion and Travel Time Reliability:
Comparing a Random Bottleneck to Empirical Data
Joel P. FranklinKungliga Tekniska Högskolan
13 January 2010, 2
Overview
• Investigate the Role of Travel Time Variability in Costs/Benefits Estimation
• Case of Interest:• Future demand is known• Traveler preferences are known• Future travel time variability is unknown
• What tools can we use to predict future variability?
13 January 2010, 3
Background
• Travel Time Variabilty Matters to Travelers
• It Matters in a Particular Way – Implications for Measurement
• Lateness matters more than Earliness• i.e. standard deviation can be misleading
• We can assume people try to optimize departure time
• Hence, total costs are a mixture of:• Early departure from previous activity*• Travel time in itself*• Expected late arrival to next activity: ”Mean Lateness”*• *Given optimized departure time
13 January 2010, 4
Hour of Day
Tra
vel T
ime
/ M
ea
n T
rave
l Tim
e
1
2
3
5 6 7 8 9 10 11
What does our congestion look like?
Hour of Day (morning)
Bergslagsvägen, Inbound
13 January 2010, 5
5 6 7 8 9 10 11
20
04
00
60
08
00
(a)
Hour of Day
Tra
vel T
ime
(se
c.)
How does it vary?
Hour of Day (morning)
Bergslagsvägen, Inbound
13 January 2010, 6
What features are important to us?
• Take a person who wants to arrive on-time 20% of the time• Value of Lost Time at Home: 1/hr• Value of Lost Time at Work: 5/hr• Additional Cost of Travel Time: 1/hr
• Where: ”mean lateness” is the average time late• Mean Travel Time – predicted by standard demand models• Mean Lateness…?
1 1 Mean Travel Time 5 Mean LatenessU
13 January 2010, 7
Theoretical Framework(Fosgerau & Karlström, 2009)
ChosenDeparture Time
PreferredArrival Time
Actual Arrival Time
15
1/5
Marginal Utilities
CDF of Travel Time
Area = Mean Lateness
13 January 2010, 8
1.0 1.2 1.4 1.6 1.8 2.0
0.0
50
.10
0.1
50
.20
0.2
50
.30
Mean / Freeflow Travel Time
La
ten
ess
K /
Fre
eflo
w T
rave
l Tim
e
56
7
89
1011121314
15
1617
18192021
How do those features vary?
Mean Time / Freeflow Time
Bergslagsvägen, Inbound
13 January 2010, 9
Results
1.0 1.2 1.4 1.6 1.8 2.0 2.2
0.1
0.2
0.3
0.4
(c)34 Centralbr.--Slussen-Klarastr.svia.
Mean / Freeflow Travel Time
La
ten
ess
K /
Fre
eflo
w T
rave
l Tim
e
56
7
89
10
111213
14
15
1617
18
19
20
21
Static Approach (for comparison):• Scale is 50% low• Looping pattern is absent
(mostly)• No Peak at Shoulders
Effect:• Role of Variability is
understated by about 50%• Difference in Variability at
Times of Day is Lost
Conclusion:• May need a Dynamic
approach
Results of Static Approach
Centralbron, Inbound
13 January 2010, 10
Research Question
• How well can a random bottleneck model predict mean lateness?
• Why?• Could be a very simple way to forecast
improvements in travel time variability• Incorporates time-dependent congestion
dynamics
13 January 2010, 11
Bottleneck Approach
• Fixed capacity
• Demand surpasses capacity at time ”0”
• Demand later subsides below capacity
• For an arrival at t:• Queue Length
measured by ”Q(t)”• Queue Time measured
by ”q(t)”Time of Day0
Q(t)
q(t)
t
13 January 2010, 12
Random Bottleneck
• Demand in each period is random
• Queue time ”q” is random, depending on random variation in demand up to time ”t”
Time of Day0
Q(t)
q(t)
t
13 January 2010, 13
Procedure
1. Selected Stockholm Highway Segments:• Example shown here: Centralbron—Northbound
2. Observed Delay Distributions by 15-Min Periods
3. Simulate Bottleneck Delay Distributions• Started with observed flow by 15-minute periods• Simulate random deviations in each period, using exponential distribution
• Also tested log-normal, poisson, negative-binomial• Manually Calibrate for Capacity, Random Dispersion
4. Compute Optimal Expected Lateness by 15-Min Periods• Also, Mean and Standard Devation
13 January 2010, 14
20 40 60 80
02
46
81
0
Time Period
Re
lativ
e L
eve
l
Results
5 10 15 20
0.0
0.2
0.4
0.6
0.8
1.0
Hour of Day
Tra
vel T
ime
/ Mea
n T
rave
l Tim
e
Observed Travel Times (Segment)Random Bottleneck Travel Times (Point)
Centralbron, Northbound Centralbron, Northbound
13 January 2010, 15
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Mean Travel Time
Sta
nd
ard
De
via
tion
Results
Bottleneck Approach:
• Looping effect exaggerated
• Peaks not separated• No peaks at shoulders
• Scale different by nature (point delay vs. link delay)
Centralbron, Northbound
9:00
16:45
1.0 1.2 1.4 1.6 1.8 2.0 2.2
0.1
0.2
0.3
0.4
(c)34 Centralbr.--Slussen-Klarastr.svia.
Mean / Freeflow Travel Time
La
ten
ess
K /
Fre
eflo
w T
rave
l Tim
e
56
7
89
10
111213
14
15
1617
18
19
20
21
13 January 2010, 16
Main Conclusions
• Dynamic approach of the bottleneck reproduces the cyclical behavior behind ”mean lateness”
• Scale issues need to be better-calibrated• Maximum Capacity• Freeflow Travel Time• Random Characteristics of Traffic Demand
• Pure Bottleneck may be too simple• E.g. Demand just under capacity gives zero delay
• Overall:• The Bottleneck doesn’t give better predict than a static approach (yet)• But better random-demand data can give better forecasts
13 January 2010, 17
Other Observations
• Under the Random Bottleneck Model:• Mean Lateness tracks very
closely with Standard Deviation
For facilities that truly operate as a bottleneck, standard deviation (multiplied by a constant) may be a good approximation to mean lateness
20 40 60 80
01
23
45
Time Period
Re
lativ
e L
eve
l
Mean FlowMean Travel TimeStd. Dev. Travel TimeMean LatenessMean Lateness / Std. Dev.
13 January 2010, 18
Future Directions
• Mixed congestion models:• Uncongested volume-to-capacity relationship of static models• Congested time-dependency of bottleneck models
• Data on traffic volume variations:• Needed to appropriately simulate randomness• Potential Source: California highway data (day-to-day flows and travel times)
• Theoretical Exploration of Bottleneck Model• Why do Standard Deviation and Mean Lateness often track so closely?