1 trip-timing decisions with traffic incidents in the bottleneck model mogens fosgerau (technical...
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1
Trip-timing decisions with traffic incidents
in the bottleneck model
Mogens Fosgerau
(Technical University of Denmark; CTS Sweden; ENS Cachan)
Robin Lindsey
(University of British Columbia)
Tokyo, March 2013
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Outline
1. Literature review
2. The model
3. No-toll user equilibrium
4. System optimum
Quasi-system optimum
Full optimum
5. Numerical examples
6. Conclusions/further research
3
Literature on traffic incidents
Simulation studies: many
Analytical static modelsEmmerink (1998), Emmerink and Verhoef (1998) …
Analytical dynamic models(a) Flow congestion. Travel time has constant and exogenous
variance.Gaver (1968), Knight (1974), Hall (1983), Noland and Small (1995), Noland (1997).
(b) Bottleneck model. Travel time has constant, exogenous and independent variance over time. No incidents per se.
Xin and Levinson (2007).
(c) Bottleneck model with incidents
Arnott et al. (1991, 1999), Lindsey (1994, 1999), Stefanie Peer and Paul Koster (2009).
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Bottleneck model studies
Timing of incidents
Capacity during travel
period
Trip-timing preferences
System optimization
Tolling
ADL (1991, 1999) Pre-trip Constant α--
No No Lindsey (1994, 1999)
Yes Yes
Peer & Koster (2009)
During trip. Exogenous
Temporarily zero
α-- ? ?
THIS PAPER During trip. Caused by
a driver
Temporarily zero or reduced
Scheduling utility
Optimal or quasi-optimal
Quasi-optimal
Scheduling utility approach
Vickrey (1973), Ettema and Timmermans (2003), Fosgerau and Engelson (2010),Tseng and Verhoef (2008), Jenelius, Mattsson and Levinson (2010).
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Outline
1. Literature review
2. The model
3. No-toll user equilibrium
4. System optimum
Quasi-system optimum
Full optimum
5. Numerical examples
6. Conclusions/further research
6
The model
Demand
N drivers, 0...n N t departure time from origin (e.g., home) a arrival time at destination (e.g., work) Scheduling utility over the day [-H,W]:
At home In car At work
, 0t a W
H t au t a u du du u du
where
0, 0u u ,
0, 0u u .
If a t , utility maximized with *a t t :
* *t t .
*tH u
Scheduling utility: Zero travel time
W
u
u
Move from H to W at t*
As in Engelson and Fosgerau (2010)
*tH u
Scheduling utility: Positive travel time
Wt a
u
u
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The model (cont.)
Supply
Bottleneck capacity: s with no incident k with incident, 0,k s
Major incident: 0k Minor incident: 0k
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The model (cont.)Incidents At most one incident per day
f n probability that driver n causes incident
F n cumulative probability of incident
(Pre-trip incidents model is degenerate case with constant .)F n
duration of incident Identity of driver who causes incident is exogenous, but timing is endogenous. Good day: no incident. Bad day: incident occurs.
Behaviour
Drivers maximize expected utility.
0 , Nt t departure period
t departure rate
R t cumulative departures
q t queuing time
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Outline
1. Literature review
2. The model
3. No-toll user equilibrium
4. System optimum
Quasi-system optimum
Full optimum
5. Numerical examples
6. Conclusions/further research
*t u
No-toll user equilibrium in deterministic model
s
0et 0 /e e
Nt t N s
R t
N
Queuing time
cumulative departures
cumulative arrivals
No-toll user equilibrium with major incidents
N
m
mt ma ma eNt u
cumulative departures
cumulative arrivalsDriver m causes incident
0et
eNt
R t
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No-toll user equilibrium properties
Property E1: [major & minor incidents]
*0e e
Nt t t .
On good days there is no queue at eNt .
Property E2: [major & minor incidents]
On good days a queue persists until eNt if
1
eN
eN
tF N
t
.
Similar to pre-trip incidents model.
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No-toll user equilibrium (cont.)
Property E3: [major incidents]
The departure equilibrium is unique. Property E4: [major incidents]
If f n is constant, the equilibrium departure rate e t
is decreasing, hence eR t is concave.
Similar to pre-trip incidents model.
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Outline
1. Literature review
2. The model
3. No-toll user equilibrium
4. System optimum
Quasi-system optimum
Full optimum
5. Numerical examples
6. Conclusions/further research
17
System optimum
Deterministic SO
Maximize aggregate utility.
Optimal departure rate = s (design capacity)
Stochastic SO
Maximize aggregate expected utility.
What is optimal departure rate?
Property W1: [major & minor incidents]
*0w w
Nt t t . w t s .
Questions
1) When is w t s ?
If probability & duration of incident large enough. 2) Should w t be reduced enough to eliminate queuing in
some incident states?
Possibly. Suppose f(m) is very large and is small. 3) Is w t necessarily decreasing over time?
No. Suppose f(m) is very large and is small.
If m is an early driver the model is similar to the pre-trip incidents model for which the optimal departure rate is weakly increasing over time. (Lindsey, 1999)
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General properties of system optimum (w)
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System optimal approaches
Quasi-system optimum (x)
Departure rate
Choose optimal
Full optimum (w)
Choose optimal and
0 ,x xNt t
w t s 0 ,w wNt t
x t s
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Outline
1. Literature review
2. The model
3. No-toll user equilibrium
4. System optimum
Quasi-system optimum (QSO)
Full optimum
5. Numerical examples
6. Conclusions/further research
21
Quasi-system optimum (QSO)
Property X1 [major incidents]:
Departures begin later than in the NTE: 0 0x et t .
Intuition: Suppose only the last driver can cause an incident … .
Property X2 [major incidents]:
If the QSO is decentralized using a non-negative toll, drivers are worse off.
Proof: The last driver travels later than in the NTE and thus has a lower scheduling utility.
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Outline
1. Literature review
2. The model
3. No-toll user equilibrium
4. System optimum
Quasi-system optimum
Full optimum
5. Numerical examples
6. Conclusions/further research
Natural to formulate the SO with time as the running variable, and departure rate
t as control variable.
Technical difficulties
1. Dependence of Hamiltonian on lagged values of control variable, t
2. Dependence of Hamiltonian on lagged values of state variable, R t
3. Equation of motion for queuing time is not differentiable at 0q t
Resolutions
1. Use n rather than t as running variable, and time headway between drivers (h(n)) rather than departure rate as control variable.
2. Introduce state-dependent queuing times: ,q n , mq n m n .
3. Impose constraint 0q n if solution calls for 1/h n s .
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Full system optimum (SO)
System optimum with queue persistence
wR tN
m
mt nt mt n
cumulative departures
cumulative potential arrivalsDriver m causes incident
n mm
n mt t
sq n
mq nn
Departure rate
n̂
s
n0
t
NInterval 1 Interval 2
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System optimum (major incidents, queue persistence)
0 0 1 , ,Max
N n
n mh
mn
q n q nF n U t n t n f m U t n t n dm dn
Interval 1: , ˆ0n n , ˆ 0, choice variablen N
Departure rate maintained at capacity headway 1/h n s .
Constraints:
1 costate 0dt n
h ndn
n
0 multiplier 0n nq
1= costate 0
dq nh n
dn sn
1= , costates 0m
m
dq nh n nm n
dn s
Initial and terminal conditions:
0 freet
0 0q , mq m ,
0 0 .
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System optimum (major incidents, queue persistence)
Interval 2: ˆ,n n N .
Departure rate held below capacity headway 1/h n s .
Constraints:
2 costate 0dt n
h ndn
n
0 multiplier 0q n n
1= , costates 0m
m
dq nh n m n n
dn s
Continuity condition
2 1ˆ ˆn n .
Terminal conditions:
2 m0, 0, 0...N N m N .
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System optimum (major incidents, queue persistence)
Optimality conditions
Interval 1: ˆ0,n n
10
0
1 , ,
1 11 1
n
mm
n
mm
H F n U t n t n q n f m U t n t n q n dm n h n
F n n q n F n n h n f m n dm h ns s
1 0Opportunity cost Reduced queuing Reduced queuingof arrival time slots with no incident with incident
1 0.n
mm
Hn F n n f m n dm
h
1
0
More time at home Less time at work Less time at workwith no incident with incident
1n
mm
n Ht n F n t n q n f m t n q n dm
n t n
n Ht n q n n
n q n
, m
mm
n Ht n q n m n
n q n
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System optimum (major incidents, queue persistence)
Interval 2: ˆ ,n n N
0
2 0
1 , ,
1
n
mm
n
mm
H F n U t n t n f m U t n t n q n dm
n h n h n f m n dms
2 0Opportunity cost Reduced queuingof arrival time slots with incidents
0.n
mm
Hn f m n dm
h
1
0
More time at home Less time at work Less time at workwhen no incident when incident
1n
mm
n Ht n F n t n f m t n q n dm
n t n
, m
mm
n Ht n q n m n
n q n
Delays incident, extendsMore time at home Less time at workqueuing time forwhen no incidentremaining travelers
1 nt n F n t n f n n
Differentiate with respect to n:
1
.1
nf n f nt n t n nh n
s
n
F n t n t n
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Outline
1. Literature review
2. The model
3. No-toll user equilibrium
4. System optimum
Quasi-system optimum
Full optimum
5. Numerical examples
6. Conclusions/further research
31
Calibration of schedule utility functions
Source: Tseng et al. (2008, Figure 3)
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Calibration of schedule utility functions (cont.)
Source: Authors’ calculation using Tseng et al. mixed logit estimates for slopes.
40 8.86 , 40 25.42 .t t t t
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Calibration of incident duration
Mean incident duration estimates
Golob et al. (1987): 60 mins. (one lane closed)
Jones et al. (1991): 55 mins.
Nam and Mannering (2000): 162.5 mins.
Select: 30
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Other parameter values
N = 8,000; s = 4,000
f(n)=f, fN = 0.2
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Results: Major incidents
fN=0 fN=0.2 Difference NTE QSO=SO NTE QSO=SO NTE QSO=SO Initial dept time ( 0t [hr.])
-1.00 -1.00 -1.101 -1.037 -0.101 -0.037
Trip cost €17.14 €5.71 €20.78 €8.43 €3.64 €2.72
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Results: Major incidents
Total cost of incident in NTE
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Results: Major incidents
Total cost of incident in QSO
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Results: Major incidents
Total cost of incident in QSOTotal cost of incident in NTE
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Results: Major incidents, individual costs
NTE, no incident occurred NTE, incident occurred
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Results: Major incidents, individual costs
NTE, no incident occurred
QSO, no incident occurred
NTE, incident occurred
QSO, incident occurred
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Minor incidents
Complication:
NTE departure rate depends on lagged values of itself •No closed-form analytical solution.
•Requires fixed-point iteration to solve.
•Results reported here use an approximation.
42
Results: Minor incidents
Initial departure time
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Results: Minor incidents
Increase in expected travel cost due to incidents
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Modified example with SO different from QSO
s = 4,000
N = 3,000
fN = 0.4
Explanation for parameter changes:
• Shorter peak: Lower cost from moderating departure rate
• Higher incident probability and duration: Greater incentive to avoid queuing by reducing departure rate
0.8
45
Results: Modified example
Departure rates for QSO and SO
46
Results: Modified example
Quasi system optimum
Optimum
Initial departure time ( 0t ) [hr.] -0.49 -0.65
Last departure time ( Nt ) [hr.] 0.26 0.70
Departure duration ( 0Nt t ) [hr.] 0.75 1.35
Trip cost €9.10 €8.75
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6. Conclusions (partial)
• Properties of SO differ for endogenous-timing and pre-trip incidents models
• Plausible that QSO is a full SO: optimal departure rate = design capacity (same as without incidents), but with departures beginning earlier
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Future research
Theoretical
1. Further properties of NTE and QSO for minor incidents.
2. SO for minor incidents.
3. Stochastic incident duration
Caveat: Analytical approach becomes difficult!
Empirical
4. Probability distribution of capacity during incidents
5. Dependence of incident frequency on level of traffic flow, time of day, etc.
49
Thank you
mf@transport.dtu.dk
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