Trip-timing decisions with traffic incidents in the bottleneck model

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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

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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


2. The model


4. System optimum


Full optimum



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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


2. The model


4. System optimum


Full optimum



*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 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


2. The model


4. System optimum


Full optimum



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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


2. The model


4. System optimum

Quasi-system optimum (QSO)

Full optimum



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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


2. The model


4. System optimum


Full optimum



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 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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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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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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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


2. The model


4. System optimum


Full optimum



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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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Total cost of incident in NTE

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Total cost of incident in QSO

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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.

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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

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Results: Modified example

Departure rates for QSO and SO

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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.

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Thank you

[email protected]

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Trip-timing decisions with traffic incidents in the bottleneck model