unlocking some mysteries of traffic flow theory robert l. bertini portland state university...

Unlocking Some Mysteries of Traffic Flow Theory

Robert L. BertiniPortland State University

University of Idaho, February 22, 2005

Introduction

Objective: learn how to think (avoid recipes) and visualize. Tools of the trade:

Time space diagram Input output diagram Spreadsheets, probability, statistics, simulation,

optimization Transportation operations

Multimodal Fleets: control routes and schedules Flows: streams whose routes and schedules are beyond

our control Transportation Systems

Moving parts: containers, vehicles, trains Fixed parts: networks, links, nodes, terminals Intangibles: “software”

Focus

Travel time Component of transportation cost Measure delays Prediction desirable Facilitates cost minimization/optimization Cost effectiveness: trade off travel time vs. construction

+ operating cost Common elements in transportation

Rush hours/peaking Seasonal variation Long run trends in demand

Two tools Peak demand

Can’t accommodate Zero benefit for investment in last increment of capacity

A model Transportation system as a network of channels connected by

bottlenecks (flow restrictions) The time space plane

Study how vehicles overcome distance Study vehicular movement between bottlenecks

Queueing theory Estimate delays at facilities when demand exceeds capacity Study bottlenecks

Impacts to non-users Safety Noise Energy consumption Air pollution

Some Reminders

Dimensional Analysis

Triangles

km/hr 96.6m 1000

km 1

cm 100

m 1

in 1

cm 2.54

ft 1

in 12

mi 1

ft 5280

hr

mi 60

pax/hour 3000car

passengers 50

train

cars 10

hour

trains 6

Rise

Run

Slope=Rise/Run

Some basic meaurements.Consider a single vehicle at one point.

Stand at a point. Establish a line across road. Record passage time of each vehicle. Do this over a specific time interval (15 min, 1 hour, 1 day,

1 year)

Vehicle Time1 9:02:092 9:04:343 9:06:444 9:08:125 9:09:376 9:11:227 9:12:498 9:13:339 9:14:20

0

1

2

3

4

5

6

7

8

9

10

9:00:00 9:05:00 9:10:00 9:15:00

Time

Veh

icle

Nu

mb

er


Stand at a point. Establish a line across road. Record passage time of each vehicle. Do this over a specific time interval (15 min, 1 hour, 1 day,

1 year)

Vehicle Time1 9:02:092 9:04:343 9:06:444 9:08:125 9:09:376 9:11:227 9:12:498 9:13:339 9:14:20

FlowTime

things of Number

veh/hr 36hr 1

min 60

min 15

veh 9


Still standing at one point. Imagine you are at a bus stop. Count number of buses per unit time = Frequency We might be interested in the actual or average time

between buses – WHY?

HeadwayFrequency

1

buses of Number

Time

FrequencyTime

buses of Number


Parameter UnitsFlow q vehicles/time #/time

Frequency buses/time #/time

Headway h time/vehicle time/#


You can also measure the point speed of a vehicle, for example using a radar gun.

Parameter Units

Speed vt distance/time mi/hr

If you collect a set of vehicle speeds over a time interval and compute the arithmetic mean of these speeds, you have measured the Time Mean Speed for one point and over one time interval:

n

vn

i ti 1

Some basic meaurements.Consider a section of straight road.

Imagine an aerial photograph. If road section is one mile long, we can count

the number of vehicles on the segment at one instant in time.

1 mi Densityveh/mi 6mi 1

veh 6

Distance

vehicles of Number


Imagine an aerial photograph. If road section is one mile long, we can count

the number of vehicles on the segment at one instant in time:

We can now think about the average distance between vehicles on this segment at one instant in time:

1 mi Densityveh/mi 6mi 1

veh 6

Distance

vehicles of Number

Spacingft/veh 880mi 1

ft 5280

veh 6

mi 1

veh of No

Dist


Now imagine two aerial photographs, taken at two times t1 and t2.

1 mi

t

1

t2

11

x1

x2

t1 t2

12

121 tt

xxv


Now imagine two aerial photographs, taken at two times t1 and t2.

If you collect a set of vehicle speeds measured over space and compute the mean, you have measured the Space Mean Speed for this segment over a time interval:

1 mi

t

1

t2

11

x1

x2

t1 t2

12

121 tt

xxvs

n

vn

i si 1

Some basic meaurements.Time mean vs. Space mean speed

Time mean speed: speeds measured at one point averaged over time.

Space mean speed: speeds measured over a segment averaged over space.

The inverse of speed is known as Pace

distance

time

mi

hr

hrmi11

Pace v

Putting together some parametersConsider dimensional analysis.

Parameter UnitsFlow q vehicles/time #/t

Frequency buses/time #/t

Headway h time/vehicle t/#

Density k vehicles/distance #/x

Spacing s distance/vehicle x/#

Speed v distance/time x/t

Pace p time/distance t/x


Frequency buses/time #/t

Headway h time/vehicle t/#


Spacing s distance/vehicle x/#


Pace p time/distance t/x


q #/t

k #/x

v x/t


q=#/t

k=#/x

v=x/t


q=#/tk=#/xv=x/t



t

x

xt

##q=#/tk=#/xv=x/t


t

x

xt

##q=#/tk=#/xv=x/t

q=kv

Putting together some parameters.Consider dimensional analysis.

q=#/t

k=#/x

qmax

kmax


q=#/t

k=#/x

qmax

kmax

Traffic state 1 (k1,q1)

k1

q1


q=#/t

k=#/x

qmax

kmax

Traffic state 1 (k1,q1)

Slope = rise/run = q1/k1 = (#/t)/(#/x) = x/t = v

k1

q1

A straight highwaySome basic traffic flow principles

Consider a 22’ vehicle traveling at 30 mph


Consider a 22’ vehicle traveling at 30 mph How “close together” might we expect two vehicles to travel

comfortably? Maybe 3 vehicle lengths spacing (66 ft) is comfortable.


Consider a 22’ vehicle traveling at 30 mph How “close together” might we expect two vehicles to travel

comfortably? Maybe 3 vehicle lengths spacing (66 ft) is comfortable. What is the headway (a point measurement)? First what are headway units?

seconds/vehicle passing a point

Time to travel 4 vehicle lengths:

hr

veh 1800

hr 1

sec 3600

sec 2

veh 1

sec 2ft 5280

mile 1

hr 1

sec 3600

mi/hr 30

ft 88

An intersectionAdd a cross street

Now add a cross street. Two interrupted traffic streams must now share

the right-of-way. Assume a simple 60 sec cycle with 30 sec

phases for each approach. What is the capacity of the approach now? 1/2*1800 vph = 900 vph Compare to a freeway lane (>2400 vhp

observed)


Think about the value 1800 veh/hr Based on a “minimum” spacing? Is this value useful for anything? Minimum spacing Maximum density? Minimum headway Maximum flow?

It might be useful to think about what the word “capacity” means in this context.

Applicable at a signalized intersection when we are trying to pump through a tightly packed platoon.

Maybe applicable on a freeway if conditions downstream are unconstrained.

An ExampleConsider a 1-mile long elliptical racetrack, with five fast cars that always travel at 80 mph and four slow trucks that always travel at 50 mph.

What is the proportion of slow vehicles as seen from an aerial photograph (in percent)?

What is the space mean speed (mph) on the track, as seen from a series of aerial photographs?

Will the proportion of slow vehicles that would be seen by a stationary observer over time who is positioned somewhere along the track be higher or lower than that observed from an aerial photo?





%5.5554

5





%4.4454

4

mph 66.6

54

)50(4805





%4.4454

4

mph 66.6

54

)50(4805

Lower!

An ExampleNow, what is the proportion (in percent) of slow vehicles seen by a stationary observer who is positioned somewhere along the track?

Will the time means speed on the track (the arithmetic average of the speeds that would be measured by the stationary observer) be higher or lower than that observed from a series of aerial photos?

%3.33600

200

hr

veh 600

hr

mi 80

mi

veh 5

hr

mi 50

mi

veh 4

hr

veh 200

hr

mi 50

mi

veh 4

q

q

vkq

q

truck

ii

truck

An ExampleNow, what is the proportion (in percent) of slow vehicles seen by a stationary observer who is positioned somewhere along the track?

Will the time means speed on the track (the arithmetic average of the speeds that would be measured by the stationary observer) be higher or lower than that observed from a series of aerial photos?

%3.33600

200

hr

veh 600

hr

mi 80

mi

veh 5

hr

mi 50

mi

veh 4

hr

veh 200

hr

mi 50

mi

veh 4

q

q

vkq

q

truck

ii

truck

Higher!

An ExampleNow, what is the time mean speed (in mph) on the track?

70mph600

)80(400)50(200hr

veh 400 ,

hr

veh 200

t

cartruck

v

qq

Some basic meaurements.Consider a series of aerial photographs.

t1


t1 t2


t1 t2 t3


t1 t2 t3 t4

Time-Space DiagramFundamental tool for transportation evaluation

x

t

Dis

tance

Time


Construct from aerial photos. Study movement and interaction from point to point. One vehicle: plot trajectory, one x for every t Speed = dx/dt (slope), acceleration = d2x/dt2 (curvature) Several vehicles: vehicle interactions Intersecting trajectories: passing


x

t

Dis

tance

Time


Headway: time between vehicles passing a point.

Spacing: front to front distance at a given time.

x

t

Dis

tance

Time

Spacing

Headway




Flow (q): number observed at a point divided by time interval.

q=N/T (horizontal slice)

x

t

Dis

tance

Time

Spacing

Headway

Tx0





q=N/T (horizontal slice)

x

t

Dis

tance

Time

Spacing

Headway

Tx0

q at x0=2/T





q=N/T (horizontal slice) Density (k): number observed on

a segment at a given time divided by the segment length.

k=N/L (vertical slice)

x

t

Dis

tance

Time

Spacing

Headway

L

t0







k=N/L (vertical slice)

x

t

Dis

tance

Time

Spacing

Headway

L

t0

k at t0=6/L







k=N/L (vertical slice) N=qt=kL

x

t

Dis

tance

Time

Spacing

Headway

T

L

t0

x0

Time-Space DiagramPoint Measures

Time

Dist

ance

Point Measures(a)

FreeFlowSpeed

x

t = Measurement Interval

d =

Segm

ent D

ista

nce

Speed

i

v -ve f = Delayve = Extrapolated Travel Time

vf = Free Flow Travel Time

Time-Space DiagramSpatial Measures

Time

Dist

ance

Spatial Measures(b)

id

= Se

gmen

t Dis

tanc

ev-vi f = Delayvf = Free Flow Travel Time

vi = Actual Travel Time

j

Queueing TheoryStudy of Congestion Phenomena

Objects passing through point with restriction on maximum rate of passage

Input + storage area (queue) + restriction + output Customers, arrivals, arrival process, server, service

mechanism, departures

Airplane takeoff, toll gate, wait for elevator, taxi stand, ships at a port, water storage in a reservoir, grocery store, telecommunications, circuits…

Interested in: maximum queue length, typical queueing times….

Input Storage

Restriction

Output

Queueing TheoryConservation Principle

Customers don’t disappear Arrival times of customers completely characterizes arrival

process. Time/accumulation axes

N(x,t)

t1 t2 t3 t4

1234

j=A(t)

Time, t @ x

Queueing TheoryArrival Process

j=A(t) increases by 1 at each tj

Observer can record arrival times Inverse t=A-1(j) is time jth object arrives (integers) If large numbers, can draw curve through midpoints of stair

steps….continuous curves (differentiable).

N(x,t)

t1 t2 t3 t4

1234

j=A(t)

Time, t @ x

Queueing TheoryDeparture Process

Observer records times of departure for corresponding objects to construct D(t).

Time, t @ x

N(x,t)

t1 t2 t3 t4

1234

A(t)

t1′ t2

′ t3′ t4

′

D(t)

Queueing TheoryAnalysis

If system empty at t=0: Vertical distance is queue length at time t: Q(t)=A(t)-D(t) A(t) and D(t) can never cross! For FIFO horizontal distance is waiting time for jth

customer.

Time, t @ x

N(x,t)

t1 t2 t3 t4

1234

A(t)

t1′ t2

′ t3′ t4

′

D(t)

Q(t)

Wj


Horizontal strip of unit height, width Wj

Time, t @ x

N(x,t)

t1 t2 t3 t4

1234

A(t)

t1′ t2

′ t3′ t4

′

D(t)

W2


Add up horizontal stripstotal delay Total time spent in system by some number of vehicles

(horizontal strips)

Time, t @ x

N(x,t)

t1 t2 t3 t4

1234

A(t)

t1′ t2

′ t3′ t4

′

D(t)

Total Delay=Area


Add up horizontal stripstotal delay Total time spent in system by some number of vehicles

(horizontal strips) Total time spent by all objects during some specific time

period (vertical strips)

Time, t @ x

N(x,t)

t1 t2 t3 t4

1234

A(t)

t1′ t2

′ t3′ t4

′

D(t)

Total Delay=Area

Queueing Theory Total delay = W Average time in queue: w = W/n Average number in queue: Q = W/T W = QT = wn Q = wn/T say n/T = arrival rate λ Q = λw Average queue length = avg. wait time avg. arrival rate

Time, t @ x

N(x,t)

t1 t2 t3 t4

1234

A(t)

t1′ t2

′ t3′ t4

′

D(t)

Combination

Time space diagram looks at one or more objects, many points

Queueing theory looks at one point many objects. Combining the two results in a three-dimensional

surface Use care when distinguishing between queuing

diagrams and time space diagrams!

Combination

Take vertical “slices” at t1 and t2

Construct vehicle counting functions N(x,t1) and N(x,t2)

Can observe distances traveled and numbers passing a particular point.

Combination

Take horizontal “slices” at x1 and x2

Construct vehicle counting functions N(t,x1) and N(t,x2)

Can observe accumulations and trip times between points.

Combination

Inductive loop detectors.Basic introduction.

Meaurements over space.Consider a single vehicle on a straight road.

Measurement over space.Represent on time-space plane.

x

t

Dis

tance

Time

Measurements over space.Vehicle trajectory on time-space plane.

x

t

Vehicle trajectory.Slope at any time is vehicle velocity.

Slope = distance/time = VELOCITY

x

t

Vehicle trajectory.Represent front and rear of vehicle.

x

t

Lveh

Vehicle trajectory.Single inductive loop detector of fixed length.

x

t

Lloop

Single inductive loop detector.Sends binary on/off signal to controller.

x

t

t

ton toff

Single inductive loop detector.Counting function via arrival time record.

x

t

t

ton toff

Individual vehicle arrival time can be plotted.

i

Single inductive loop detector.Speed estimation possible with vehicle length.

x

t

t

toff

Lloop

Lvehvi

i

onoff

vehloopi

tt

LLv

ton

ton toff

Single inductive loop detector.Measurement of other parameters.

x

t

t

i j k l m n o p

Single inductive loop detector.Usually pre-defined time intervals.

x

t

t

i

1 min

j k l m n o p

Single inductive loop detector.Interval count – number of rising edges.

x

t

t

i

1 min

n=2

j k l m n o p

Single inductive loop detector.Occupancy is percent of time interval “occupied.”

x

t

t

i

1 min

n=2, occupancy (%)= /1 min

j k l m n o p

Double inductive loop detector—speed trap.Directly measure speed—on times.

x

t

t

t on1

t off

1t o

n2

t off

2

Loop 1

Loop 2

Lveh

Lloop

Lint

Lint

Lloop

von

ton1

ton2

12

int

onon

loopon tt

LLv

Lloop

Double inductive loop detector—speed trap.Directly measure speed—off times.

x

t

t

t on1

t off

1t o

n2

t off

2

Loop 1

Loop 2

Lveh

Lloop

Lint

Lint

Lloop

voff

toff1

toff2

12

int

offoff

loopoff tt

LLv

Lloop

Double inductive loop detector—speed trap.Directly measure vehicle length.

x

t

t

t on1

t off

1t o

n2

t off

2

Loop 1

Loop 2

Lveh

Lloop

Lint

Lint

Lloop

voff

ton2

toff2

loopoffoffoffveh LttvL )( 12

Lloop

Lveh

Freeway bottlenecks.Definition and previous studies.

Introduction.Bottleneck diagnosis.

An “active” bottleneck is a restriction that separates upstream queued traffic from downstream unqueued traffic.

An active bottleneck is deactivated when there is either a decrease in flow or when a queue spills back from a downstream bottleneck.

Queued Unqueued

Bottleneck

Detectors

Speed contour plot.Provides temporal and spatial resolution.

Previous studies.Bivariate plot—little information.

0

20

40

60

80

100

120

140

160

180

0 1000 2000 3000 4000 5000 6000 7000 8000Flow (vehicles/hour)

Vel

oci

ty (

kilo

met

ers/

ho

ur)

Previous studies.Time series count data—1 min resolution.

0

1000

2000

3000

4000

5000

6000

7000

8000

0:00 4:00 8:00 12:00 16:00 20:00 0:00Time

Flo

w (

veh

icle

s/h

ou

r)


0

1000

2000

3000

4000

5000

6000

7000

8000

0:00 4:00 8:00 12:00 16:00 20:00 0:00Time

Flo

w (

veh

icle

s/h

ou

r)

Proposed innovative graphical method.Developed at U.C. Berkeley.

Takes advantage of ubiquitous sensor data to inform theoretical underpinning.

Process data without losing resolution.

Reveal parametric changes over time.

Can be used for count (flow), speed and other parameters.

Proposed method.Oblique plotting technique for two hours’ data.

-1,000

0

1,000

2,000

3,000

4,000

5,000

6,000

7,000

8,000

9,000

10,000

11,000

14:00 14:15 14:30 14:45 15:00 15:15 15:30 15:45 16:00

Time

N(x

,t)

Cu

mu

lati

ve C

ou

nt

Motorway A9, Station 340, July 4, 2002

N(x,t)

-1,000

0

1,000

2,000

3,000

4,000

5,000

6,000

7,000

8,000

9,000

10,000

11,000

14:00 14:15 14:30 14:45 15:00 15:15 15:30 15:45 16:00

Time

N(x

,t)

Cu

mu

lati

ve C

ou

nt


N(x,t)


q0=5180 vph

-1,000

0

1,000

2,000

3,000

4,000

5,000

6,000

7,000

8,000

9,000

10,000

11,000

14:00 14:15 14:30 14:45 15:00 15:15 15:30 15:45 16:00

Time

N(x

,t)

Cu

mu

lati

ve C

ou

nt



N(x,t)-q0t´

N(x,t)

q0=5180 vph


N(x,t)-q0t´-1,000

0

1,000

2,000

3,000

4,000

5,000

6,000

7,000

8,000

9,000

10,000

11,000

14:00 14:15 14:30 14:45 15:00 15:15 15:30 15:45 16:00

Time

N(x

,t)

Cu

mu

lati

ve C

ou

nt


14:00 14:15 14:30 14:45 15:00 15:15 15:30 15:45 16:00

Time

N(x

,t)

Cu

mu

lati

ve C

ou

nt

0

1,000

2,000

3,000

4,000

5,000

6,000

7,000

8,000

9,000

10,000

11,000


14:00 14:15 14:30 14:45 15:00 15:15 15:30 15:45 16:00

Time

N(x

,t)

Cu

mu

lati

ve C

ou

nt

-300

-250

-200

-150

-100

-50

0

N(x,t)-q0t´

Proposed method.Plot sensor data cumulatively using oblique axis to reveal details in trends.

0

1,000

2,000

3,000

4,000

5,000

6,000

7,000

8,000

14:00 14:15 14:30 14:45 15:00 15:15 15:30 15:45 16:00

Time

Flo

w (

veh

icle

s/h

ou

r)

-300

-250

-200

-150

-100

-50

0

N(x

,t)-

q0t’

, q

0=

51

80

veh

icle

s/h

ou

r

Proposed method.Oblique plot reveals times at which pronounced flow changes occurred.

-300

-250

-200

-150

-100

-50

0

14:00 14:15 14:30 14:45 15:00 15:15 15:30 15:45 16:00

Time

N(x

,t)-

q0t’

, q

0=

51

80

veh

icle

s/h

ou

r

Proposed method.Plot sensor data cumulatively at one point.

x

Time, t @ x

Travel DirectionN(x,t)6:3

0

6:3

1

6:3

2

6:3

3

6:3

4

6:3

5

6:3

6

6:3

7

6:3

8

6:3

9

6:4

0

6:4

1


x

Time, t @ x

Travel DirectionN(x,t)6:3

0

6:3

1

6:3

2

6:3

3

6:3

4

6:3

5

6:3

6

6:3

7

6:3

8

6:3

9

6:4

0

6:4

1

Equal Time Intervals (1 min)

Interval Count


x

Travel DirectionN(x,t)N(x,t)

Time, t @ x

6:3

0

6:3

1

6:3

2

6:3

3

6:3

4

6:3

5

6:3

6

6:3

7

6:3

8

6:3

9

6:4

0

6:4

1


x


Time, t @ x

6:3

0

6:3

1

6:3

2

6:3

3

6:3

4

6:3

5

6:3

6

6:3

7

6:3

8

6:3

9

6:4

0

6:4

1

Slope = number/time = FLOW


x


Time, t @ x

6:3

0

6:3

1

6:3

2

6:3

3

6:3

4

6:3

5

6:3

6

6:3

7

6:3

8

6:3

9

6:4

0

6:4

1


Flow Increase


x


Time, t @ x

6:3

0

6:3

1

6:3

2

6:3

3

6:3

4

6:3

5

6:3

6

6:3

7

6:3

8

6:3

9

6:4

0

6:4

1


Flow Increase

Flow Decrease

Queueing diagram.Use two oblique plots in series to see queueing and resulting delay.

x1

Time, t

Travel DirectionN(xj,t)

N(x1,t)


x1 x2

Ref. Veh. Trip TimeTime, t


N(x1,t)

N(x2,t)


x1 x2

Ref. Veh. Trip Time

Number

Time, tt1


N(x1,t)

N(x2,t)


x1 x2

Ref. Veh. Trip Time

Number

Trip Timej

Time, t

j

t1


N(x1,t)

N(x2,t)

Queueing diagram.Shift upstream curve to reveal...

x1 x2

Time, t


N(x1,t)

N(x2,t)

Queueing diagram.Shift upstream curve to reveal excess accumulation...

ExcessAccumulation

x1 x2

Time, t


N(x1,t)

N(x2,t)

t2

Queueing diagram.Shift upstream curve to reveal excess accumulation and delay.

ExcessAccumulation

x1 x2

Time, t


Excess Travel Time=Delay

N(x1,t)

N(x2,t)

t2

k

Empirical Analysis of Traffic Sensor Data Surrounding a Bottleneck on a German Autobahn.

Robert L. BertiniSteven HansenPortland State University

Klaus BogenbergerBMW Group

TRB Annual MeetingJanuary 10, 2005

126

Introduction.Objectives.

Empirical analysis of features of traffic dynamics and driver behavior on a German autobahn.

Understand details of bottleneck formation and dissipation.

Improved travel time estimation and forecasting: Traffic management Traveler information Driver assistance systems.

Contribute to improved traffic flow models and freeway operational strategies.

127

Background.

Previous empirical research (U.S., Canada, Germany) Active bottleneck definition:

Queue upstream Unrestricted traffic downstream

Temporally and spatially variable, static and dynamic, merges and diverges.

Activation/deactivation times. Bottleneck outflow features and possible triggers. Opportunity to compare with previous findings using data

from German freeways.

Queued Unqueued

Bottleneck

D e te c to rs

128

Study Area.Data.

14-km section of northbound A9, Munich 17 dual loop detector stations (labeled 280–

630) One-minute counts & average speeds

Cars Trucks

Six days in June–July 2002 Focus on June 27, 2002

Clear weather Variable speed limits and traffic information

(VMS)

630

129

Methodology.Analysis Tools.

Cumulative curves (Newell, Cassidy & Windover): Vehicle count Average speed

Transformations to heighten visual resolution: Oblique axis Horizontal shift with vehicle conservation

Retain lowest level of resolution (one-minute) Identify bottleneck activations and deactivations.

130

Speeds Northbound A9June 27, 2002

630

131

Speeds Northbound A9June 27, 2002

1

132

Bottleneck Activation June 27, 2002Station 380

N(x,t)-q 0t′,

q0=

51

70

ve

h/h

r

Station 380 + Off Ramp

-50

50

150

250

350

450

14:45 14:50 14:55 15:00 15:05 15:10 15:15 15:20 15:25 15:30 15:35 15:40 15:45 15:50

Time

420

390

380off

630

133

Bottleneck Activation June 27, 2002Stations 380–390–420

-50

50

150

250

350

450

14:45 14:50 14:55 15:00 15:05 15:10 15:15 15:20 15:25 15:30 15:35 15:40 15:45 15:50

Time @ station 420

N(x

,t)

- q

0t

420

390

380off

N(x,t)-q 0t′,

q0=

51

70

ve

h/h

r

Station 420 + On Ramp

Station 390

Station 380 + Off Ramp

Time630

134

-50

50

150

250

350

450

14:45 14:50 14:55 15:00 15:05 15:10 15:15 15:20 15:25 15:30 15:35 15:40 15:45 15:50

Time @ station 420

N(x

,t)

- q

0t

420

390

380off

N(x,t)-q 0t′,

q0=

51

70

ve

h/h

r

15:21@ Station 390

15:21@ Station 380

Flow

Reduction

@380


Time630

135

380

15:15 15:20 15:25 15:30

Time

15:21

89 km/h

70 km/h

V(3

80,t

)-b

0t′,

b0=

3300

km

/hr2

Bottleneck Activation June 27, 2002Station 380 Speed

630

136

-50

50

150

250

350

450

14:45 14:50 14:55 15:00 15:05 15:10 15:15 15:20 15:25 15:30 15:35 15:40 15:45 15:50

Time @ station 420

N(x

,t)

- q

0t

420

390

380off

N(x,t)-q 0t′,

q0=

51

70

ve

h/h

r

15:24@ Station 420

15:21@ Station 390

15:21@ Station 380

Flow

Reduction

@380

380

55200

55400

55600

15:15 15:20 15:25 15:30

Time

15:2189

70

V(3

80,

t)-b

0t′

, b0=

3300

km

/hr2

V(3

90,

t)-b

0t′

, b0=

4335

km

/hr2

390

98120

98170

98220

15:15 15:20 15:25 15:30Time

15:21

80 41

420

95010

95060

95110

15:15 15:20 15:25 15:30Time

V(4

20,

t)-b

0t′

, b0=

4850

km

/hr2

15:2492

65


Time630

137

Bottleneck Activation June 27, 2002

630

540

29400

29600

29800

15:15 15:20 15:25 15:30 15:35 15:40 15:45

Time

V(5

40,

t)-b

0t′,

b0=

4550

km

/hr2

15:349739

560

36420

36520

36620

15:30 15:35 15:40 15:45 15:50 15:55 16:00

Time

V(5

60,

t)-b

0t′,

b0=

3550

km

/hr2

15:4183 43

580

34180

34280

34380

34480

15:30 15:35 15:40 15:45 15:50 15:55 16:00

Time

V(5

80,

t)-b

0t′,

b0=

3650

km

/hr2

15:4287 43

600

32400

32500

32600

32700

15:30 15:35 15:40 15:45 15:50 15:55 16:00

Time

630

41450

41650

41850

15:45 15:50 15:55 16:00 16:05 16:10 16:15

Time

V(6

00,

t)-b

0t′

, b0=

3450

km

/hr2

15:47

72 38

V(6

30,

t)-b

0t′

, b0=

3100

km

/hr2

15:5893 20

138


15:21

15:24

15:34

15:4115:42

15:47

15:58 630

1

540

29400

29600

29800

15:15 15:20 15:25 15:30 15:35 15:40 15:45

Time

V(5

40,

t)-b

0t′,

b0=

4550

km

/hr2

15:349739

560

36420

36520

36620

15:30 15:35 15:40 15:45 15:50 15:55 16:00

Time

V(5

60,

t)-b

0t′,

b0=

3550

km

/hr2

15:4183 43

580

34180

34280

34380

34480

15:30 15:35 15:40 15:45 15:50 15:55 16:00

Time

V(5

80,

t)-b

0t′,

b0=

3650

km

/hr2

15:4287 43

600

32400

32500

32600

32700

15:30 15:35 15:40 15:45 15:50 15:55 16:00

Time

630

41450

41650

41850

15:45 15:50 15:55 16:00 16:05 16:10 16:15

Time

V(6

00,

t)-b

0t′

, b0=

3450

km

/hr2

15:47

72 38

V(6

30,

t)-b

0t′

, b0=

3100

km

/hr2

15:5893 20

139

-300

-200

-100

0

100

200

14:45 14:50 14:55 15:00 15:05 15:10 15:15 15:20 15:25 15:30 15:35 15:40 15:45 15:50

Time

N(x

,t)

- q

0t

380

350

340

320

N(x,t)-q 0t′,

q0=

51

70

ve

h/h

r

Station 350

Station 320 + Off-Ramp

Station 340

Station 380 + On-Ramp

Time @ station 380

Bottleneck Activation June 27, 2002Stations 320–340–350–380

Time630

140

-300

-200

-100

0

100

200

14:45 14:50 14:55 15:00 15:05 15:10 15:15 15:20 15:25 15:30 15:35 15:40 15:45 15:50

Time

N(x

,t)

- q

0t

380

350

340

320

N(x,t)-q 0t′,

q0=

51

70

ve

h/h

r

15:21@ Station 380

Flow Reduction@380

Time @ station 380


Time630

141

-300

-200

-100

0

100

200

14:45 14:50 14:55 15:00 15:05 15:10 15:15 15:20 15:25 15:30 15:35 15:40 15:45 15:50

Time

N(x

,t)

- q

0t

380

350

340

320

N(x,t)-q 0t′,

q0=

51

70

ve

h/h

r

15:23@ Station 350

15:27@ Station 320

15:26@ Station 340

15:21@ Station 380

Flow Reduction@380

Time @ station 380

350

40500

40600

40700

15:15 15:20 15:25 15:30Time

15:23

84

84

V(3

50,

t)-b

0t′

, b0=

4200

km

/hr2

340

22320

22370

22420

15:15 15:20 15:25 15:30 15:35 15:40 15:45

Time

15:26

99

97

V(3

40,

t)-b

0t′

, b0=

5600

km

/hr2

320

19690

19740

19790

15:15 15:20 15:25 15:30 15:35 15:40 15:45

Time

15:27

105

104

V(3

20,

t)-b

0t′

, b0=

6000

km

/hr2


Time630

142

Active bottleneck located between detectors 380 and 390.

Activated at 15:21. Queue propagated as far as detector

630. Unrestricted traffic downstream.


630

143

Bottleneck ActivationJune 27, 2002

Direction of Travel

15:21 19:40

17:40

18:44

17:28

17:35

17:38

15:24

15:34

15:4115:42

15:47

15:58

19:18

1

630

144


Direction of Travel

15:21 19:40

17:40

18:44

17:28

17:35

17:38

15:24

15:34

15:4115:42

15:47

15:58

19:18

1

2

630

145

Direction of Travel

15:21 19:40

17:40

18:44

17:28

17:35

17:38

15:24

15:34

15:4115:42

15:47

15:58

19:18


1

2

3

630

146

Direction of Travel

15:21 19:40

17:40

18:44

17:28

17:35

17:38

15:24

15:34

15:4115:42

15:47

15:58

19:18


1

2

34

630

147


N(420,t)

N(390,t)

N(380,t)0

100

15:00 16:00 17:00 18:00 19:00

Time

N(x,t)-q0t

, q0=

51

78

ve

h/h

r

630

148


N(420,t)

N(390,t)

N(380,t)0

100

15:00 16:00 17:00 18:00 19:00

Time

N(x,t)-q0t

, q0=

51

78

ve

h/h

r

15:2

1 17:3

5

17:4

0

18:4

5

19:1

8

1

630

149


N(380,t)0

100

15:00 16:00 17:00 18:00 19:00

Time

N(x,t)-q0t

, q0=

51

78

ve

h/h

r

15:2

1 17:3

5

17:4

0

18:4

5

19:1

8

1

630

150


N(380,t)0

100

15:00 16:00 17:00 18:00 19:00

Time

N(x,t)-q0t

, q0=

51

78

ve

h/h

r

15:2

1 17:3

5

17:4

0

18:4

5

19:1

8

5510

vph

5370 vph

1

630

151


N(380,t)0

100

15:00 16:00 17:00 18:00 19:00

Time

N(x,t)-q0t

, q0=

51

78

ve

h/h

r

15:2

1 17:3

5

17:4

0

18:4

5

19:1

8

5510

vph

5370 vph

5410 vph

1 4

630

152

Bottleneck Activation Northbound A9Outflow Summary at 380

Day

Date

Pre-

Queue Flow

Pre-Queue

Standard Deviation

Bottleneck

Outflow Duration

Bottleneck Outflow

Bottleneck Outflow

Standard Deviation

Flow Drop

Wed 6/27/2002 5510 10.53 2:13 5370 5.8 -3% 6/27/2002 0:33 5410 6.5

Thu 6/28/2002 5800 11.22 1:41 5545 6.3 -4% 6/28/2002 0:55 5266 6.3 6/28/2002 0:30 5359 4.6 6/28/2002 1:46 5393 6.8

Mon 7/2/2002 5972 5.58 1:34 5177 6.2 -13% Tue 7/3/2002 5835 8.31 0:11 5485 4.9 -6%

7/3/2002 1:40 5017 5.2 Wed 7/4/2002 3:30 5551 6.8 Thu 7/5/2002 5527 9.43 1:22 5429 7.0 -2%

Mean: 5620 5370 -5% Standard Deviation: 320 160

630

153

Bottleneck Activation Northbound A9On-Ramp DynamicsJune 27, 2002 Station 420

2280 veh/hour

2630

1830 2370

2660 (+12%)

18502330

2630

15:16

15:20

15:2

1@

390

0

100

200

300

400

500

600

14:30 14:45 15:00 15:15 15:30 15:45 16:00

Time

N(4

20 o

n r

amp

,t)

- q

0t'

630

154

Bottleneck Activation Northbound A9Station 420 Truck Flow Dynamics June 27, 2002Station 420 Trucks

0

50

100

150

14:00 14:15 14:30 14:45 15:00 15:15 15:30 15:45 16:00

Shoulder Trucks (q0=220)

Median Trucks (q0=22)

Right Ramp (q0=385)

Left Ramp (qo=15)

RampRightq0=385 veh/hour

400 veh/hour

750 (+190%)

15:2

1@

390

MainlineRightq0=220veh/hour

RampLeftq0=15veh/hour

MainlineLeftq0=22 veh/hour

550 320

580

320

530

260

490 330

510

230

390

240 170330

160350 (+120%)

230

50

100

50

270 (9 trucks in 2 minutes)

24

20

3040

Trucks Only

N(x,t)-q 0t

630

155

Bottleneck 1 Activation Northbound A9Station 390 Truck Flow Dynamics June 27, 2002

0

50

100

150

14:00 14:15 14:30 14:45 15:00 15:15 15:30 15:45 16:00

Shoulder Trucks

Center Trucks

Median Truck

N(x,t)-q 0t

Shoulderq0=560 veh/hour

Centerq0=65 veh/hour

Medianq0=15 veh/hour

630 veh/hour 400

740

470670

770

420820 (+95%)

570

740

220

20200

80210

100 210 (+120%)

15:2

1@

390

14080

180

2070

20180 (6 trucks in 2 minutes, +680%)

7030

Trucks Only

630

156

Conclusion.

Method for diagnosing active bottlenecks. 11 bottleneck activations on 6 days at one location. Measured bottleneck outflows appear stable:

Day to day (contrary to other research) Preceded by queueing or not

Pre-queue flows measurably higher than bottleneck outflow.

Precursors to queue formation some distance downstream of merge: Rise in on-ramp flow (total) Surges in truck counts

Research continuing at this and other sites in Germany.

157

Thank you for your attention.

AcknowledgementsBMW GroupOregon Engineering Technology Industry CouncilPortland State UniversitySteven Boice

unlocking some mysteries of traffic flow theory robert l. bertini portland state university...

Documents

average time

unit time

single vehicle

specific time interval

record passage time

time mean speed

point speed

timeheadway htimevehicle