Traffic stream models

Tom Mathew

2

Traffic stream models

• Macroscopic

– Expression of the average behavior of the

vehicles at the specific location and time

• Mesoscopic

– Small group of traffic entities with activities

and interactions

• Microscopic

– Space-time behavior of the systems’ entities

(i.e. vehicle and drivers)

3

Traffic stream models

4

Traffic stream models

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Traffic stream models

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Traffic stream models

• Macroscopic Stream Models

– Greenshield's linear

– Greenberg's logarithmic

– Underwood's exponential

– Pipe's generalized

– Multi regime models

• Two and Three regime

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Greenshield's model

• Linear speed-density relationship

Relation between speed and density

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Greenshield's model

• Description

– v = mean speed

– k = density

– 𝒗𝒇 = free flow speed

– 𝒌𝒋= jam density

– When density becomes zero, speed

approaches free flow speed

Two parameter model

𝒗 = 𝑣𝑓 −𝑣𝑓

𝑘𝑗𝒌

0

10

20

30

40

50

60

70

80

90

0 50 100 150 200 250

Sp

ee

d (

v)

Density (k)

vf=80, kj=200

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Greenshield's model

• Relation between flow and density

𝒗 = 𝑣𝑓 −𝑣𝑓

𝑘𝑗𝒌

𝒒 = 𝒌 𝒗

𝒒 = 𝒌 𝑣𝑓 −𝑣𝑓

𝑘𝑗𝒌

𝒒 = 𝑣𝑓𝒌 −𝑣𝑓

𝑘𝑗𝒌2

0

500

1000

1500

2000

2500

3000

3500

4000

4500

0 50 100 150 200

Flo

w (

q)

Density (k)

vf=80, kj=200

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Greenshield's model

• Boundary conditions

– Maximum flow qmax

– Density corresponding to max. flow ko

– Speed corresponding to max. flow vo

• Model parameters

– Jam density kj

– Free flow speed vf

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Greenshield's model

• Derivation of ko

– We have

– Differentiating

𝒒 = 𝑣𝑓𝒌 −𝑣𝑓

𝑘𝑗𝒌2

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Greenshield's model

• Derivation of qmax

• Derivation of vo

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Greenshield's model

• Relation between speed and flow

𝒗 = 𝑣𝑓 −𝑣𝑓

𝑘𝑗𝒌

𝒒 = 𝒌 𝒗

𝒌 = 𝑘𝑗 −𝑘𝑗

𝑣𝑓𝒗

𝒒 = 𝑘𝑗𝒗 −𝑘𝑗

𝑣𝑓𝒗2

0

10

20

30

40

50

60

70

80

90

0 1000 2000 3000 4000 5000

Speed (v

)

Flow (q)

vf=80, kj=200

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Greenshield's model

• Calibration

– Determination of model parameters

– Free flow speed (vf)

– Jam density (kj)

where x is density and y denotes speed

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Greenshield's model

• Calibration

– Using linear regression method

OR

a is

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Greenshield's model

• Example

– Calibrate Greenshields model

using the data give in the

table

– Find the maximum flow

– Find the density

corresponding to a speed of

30 km/hr

No K v

1 171 5

2 129 15

3 20 40

4 70 25

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Greenshield's model

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Model Comparison

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Model Comparison

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Greenberg's model

• Logarithmic relation

– Advantage

• Analytical derivation

• Good at congestion

– Drawbacks

• Infinite speed

• Poor at low densities

𝒗 = 𝑣𝑜𝑙𝑛𝑘𝑗

𝒌

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Underwood's model

• Exponential Model

– Advantage

• Good at low speed

– Drawbacks

• Speed is zero only at

infinity density

• Poor at high densities

𝒗 = 𝑣𝑓𝒆−𝒌𝑘𝑜

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Pipes' model

• Generalized Model

– When n is 1 Pipe’s model resembles

Greenshield’s model

𝒗 = 𝒗𝒇 1 −𝒌

𝒌𝒋

𝒏

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Comparison of Models

0

10

20

30

40

50

60

70

80

90

100

0 50 100 150 200 250

Sp

eed

V(k

mp

h)

Density k (veh/km)

k = 200, vf = 90, ko = 100, v0 = 45

Liner n = 1

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Comparison of Models

0

10

20

30

40

50

60

70

80

90

100

0 50 100 150 200 250

Sp

eed

V(k

mp

h)

Density k (veh/km)

k = 200, vf = 90, ko = 100, v0 = 45

Liner n = 1

Log

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Comparison of Models

0

10

20

30

40

50

60

70

80

90

100

0 50 100 150 200 250

Sp

eed

V(k

mp

h)

Density k (veh/km)

k = 200, vf = 90, ko = 100, v0 = 45Liner n = 1

Log

Exp

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Comparison of Models

0

10

20

30

40

50

60

70

80

90

100

0 50 100 150 200 250

Sp

eed

V(k

mp

h)

Density k (veh/km)

k = 200, vf = 90, ko = 100, v0 = 45 Gen n = 0.5

Liner n = 1

Gen n = 2

Log

Exp

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Model Comparison

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Model Comparison

Model Comparison

• Limitations

– Assumes uniform behaviour

– Poor predictability

– Generalization

• Need for multi-regime models

– Traffic states divided into various regimes

– Behaviour depends on the regime

– Separate models for each regime

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Multiregime model

• Eddie’s Two Regime Model

– Based on field data (Chicago)

Regime 1:

Exponential

Regime 2:

Logarithmic

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Multiregime model

• Eddie’s Two Regime Model

– Based on field data (Chicago)

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Multiregime model

• Eddie’s Two Regime Model

Greenshields

Model

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Multiregime model

• Three Regime Model

– Free flow

– Normal

– Congested

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Multiregime model

Free flow

CongestedNormal

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Multiregime model

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Multi-regime models

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Speed –Flow: Effect of location

• Simultaneous treatment of q k v

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Conclusion

• Concerns

– The current status of mathematical models for speed-

flow concentration relationships is in a state of flux

– The models that dominated for nearly 30 years are

incompatible with the data currently being obtained

– but no replacement models have yet been developed

Lieu 1999, Traffic-Flow Theory

US DOT, Federal Highway Administration

http://www.tfhrc.gov/pubrds/janfeb99/traffic.htm

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Conclusion

• Trends

– Despite those words of caution, it is important

to note that there have been significant

advances in understanding traffic stream

behavior since 1980’s leading to a better

understanding of traffic operation

– Efforts to implement ITS will provide

challenges for applying this improvement

– Equally important, ITS will likely provide the

opportunity for acquiring more and better data

to advance understanding of traffic operations

Thank You

Questions?