Traffic Theory

Jo, Hang-Hyun (KAIST)

April 9, 2003

Motivations & Aims

• Traffic problems get worse.

- Heavy traffic congestion, Smog, Noise, and Environmental problems, etc.

• To discover the fundamental properties and laws in the transportation systems and make applications to the real world.

Brief history on traffic research

• Greenshields (1935)• Lighthill & Whitham (1955) : macroscopic mode

l based on fluid-dynamic theory

• Prigogine et al. (1960) : gas-kinetic model based on the Boltzmann equation

• Newell (1961) : microscopic, optimal velocity model

• Musha & Higuchi (1976, 1978) : the noisy behavior of traffic flow

(continued)

• O. Biham et al. (1992) : “Self-organization and a dynamical transition in traffic-flow models”, PRA

• K. Nagel & M. Schreckenberg (1992) : “A cellular automaton model for freeway traffic”, J. Phys. I

France

• B. S. Kerner & P. Konhauser (1993) : “Cluster effect in initially homogeneous traffic flow”, PRE

Review Papers

• Chowdhury et al. : Statistical physics of vehicular traffic and some related systems (Physics Reports 2000)

• Helbing : Traffic and related self-driven many-particle s

ystems (RMP 2001) • Kerner : Empirical macroscopic features of spatial-tem

poral traffic patterns at highway bottlenecks (PRE 2002)

• Nagatani : The physics of traffic jams (RPP 2002)

Empirical Findings

Schematic diagram of“Fundamental diagram”

Modeling Approachfor Vehicle Traffic

Models 1 : Microscopic

2*

0

),(1

)(

s

vvs

v

va

dt

tdv

Follow-the-leader model / optimal velocity model : Newell

; relaxation time,

; Optimal velocity function

Intelligent driver model : Treiber et al. (PRE 2000)

; headway

)())((')( tvtdv

dt

tdv e

)()()( 1 txtxtd

cce dddvdv tanh)tanh()2/()(' 0

relative velocity

(continued)

)()))(('))(('())((')( 1 tvtdvtdvtdv

dt

tdv eee

10

The next-nearest-neighbor interaction : Nagatani (PRE 1999)

; interaction strength

Backward looking optimal velocity model :Nakayama (PRE 2001)

)())(())((')( 1 tvtdvtdv

dt

tdv Be

→ Stabilize and enhance the traffic flow

Models 2 : Cellular Automata

))1,1,min(,0max( )(max1

piiii vdvv

mx 5.7

Nagel-Schreckenberg model (J. Phys. I France 1992)

1)Motion : vi cells forward2)Acceleration : vi’=vi+1 if vi<vmax

3)Deceleration : vi’’=di-1 if di≤vi’4)Randomization : vi+1=vi’’-1 with probability p

st 1discretization of time and space :

Models 3 : Macroscopic

0)(

x

v

t

Traffic flow as a 1D compressible fluid :Kerner & Konhauser (PRE 1993)

x

v

x

vV

x

c

x

vv

t

v

1)(20Eq. of motion :

Eq. of continuity : variance of velocity

safe velocity

relaxation time

viscosity

(continued)

Lighthill-Whitham Model (1955)

0),(),(

x

txQ

t

tx),(),(),( txVtxtxQ

)),(()),((),( txVtxQtxQ ee

)/1(),( jam0 VtxVe

, where

Greenshields (1935)

equilibrium velocity-density relation

d

dVV

d

dQC e

ee )()(velocity of propagation of

kinematic waves :

0)(

xC

t

(continued)

x

tx

tx

DtxVtxV e

),(

),()),((),(

To avoid the development of shock fronts, add a diffusion term.

2

2 ),(),(),(

),(

x

txCD

x

txCtxC

t

txC

→ Burgers equation

2

2 ),(),(

x

txD

t

tx

x

tx

tx

DtxC

),(

),(

2),( Cole-Hopf transformation

→ linear heat equation

Models 4 : Gas-kinetic

intacc

~~~~

dt

d

dt

d

xv

t

Prigogine’s Boltzmann-like Model (1960,1971)

)(

~),,(

~

0 vP

tvxP ; distribution function

; desired velocity distribution

; relaxation time

),,(~

),(),,(~ tvxPtxtvx

),;(~

)(~),(~

0acc

txvPvPtx

dt

d

vw

vw

twxtvxwvpdw

tvxtwxvwpdwdt

d

),,(~),,(~)(ˆ1

),,(~),,(~)(ˆ1~

int

(continued)Phase diagram in the presence of inhomogeneities :

Helbing et al. (PRL 1999)

rmp

rmp ),()(),(

L

txQ

x

V

t

tx

)()1(

)()(

)(12

20 vB

VPA

VV

xx

VV

t

V

a

aaa

→ the nonlocal, gas-kinetic-based traffic model

Pedestrian Traffic

Formation of human trail systems

Active walker model : Helbing et al. (Nature 1997, PRE 1997)

Active walker

Environment

Other walkers

Active walker model

)(

),(1)(),(

max rG

trGrItrQ

)(0 rGnatural ground potential

ground potential

))((),(),()()(

1),( 0 trrtrQtrGrGrTdt

trdG

strength of new markings

The moving agents continuously change the environment by leaving markings while moving.

(continued)

rdrU

)(

),()(Norm),,( trVrUtvre tr

),(),( )(/2 trGerdtrV rrrtr

)(2)(),,()( 0

ttvtvrev

dt

tvd

)(

)(tv

dt

trd

desired direction

trail potential or attractiveness

destination potential

motion of a pedestrian α

Active walker model : Results

/IT

For large σ,∇Vbecomes negligible.→ direct way system

For small σ, ∇Ubecomes negligible.→ minimal way system

Otherwise,→ minimal detour system

Alfred Russell / Panic (1961)

Escape panic

• People try to move faster than normal.• Interactions among people become

physical in nature.• Jams build up.• People show a tendency towards mass

behavior to do what other people do.• Alternative exits are often overlooked or

not efficiently used in escape situations.

Modeling

acceleration equation

W

iWij

iji

iiii

ii ff

tvtetvm

dt

vdm

)(

0 )()()(

ijtjiijijijijijiijijiij tvdrgndrkgBdrAf

)()(/)(exp

iWiWiiWiiWiWiiiWiiiW ttvdrgndrkgBdrAf

))(()(/)(exp

repulsive interaction force

body force

sliding friction force

ijijtijjiij tvvvrrr

)(,

Results

ijiiii tepepte )()1(Norm)( 00

Applications & Future Works

• Optimization of byways around ‘Duck Square’ in KAIST using trail formation methods

• Study on the transportation systems that guarantee the safety of pedestrians

Traffic andSelf-Organized Criticality

Fundamental Diagram

Flow-density relations ; empiricalHelbing (RMP 2001)

Schematic diagramNagatani (RPP 2002)

Phase transition in traffic

Traffic Gas-liquid

Freeway traffic Gas

Jammed traffic Liquid

Headway(dist. between cars) Volume

Vehicle density Density

Drivers’ sensitivity Temperature

What happens at critical point?

4gapn

(i) If v=vmax and ngap≥vmax, vt+1=vt

(ii) When it is jammed1) acceleration : vt+1=vt+1 with prob. ½ if ngap≥v+12) slow down due to other cars : vt+1=ngap if ngap≤v-13) overaction : v=max(ngap-1,0) with prob. ½

(iii) Movement : Each vehicle advances v sites

“Emergent traffic jams” : Nagel & Paczuski (PRE 1995)

SOC in traffic : Cellular automata model

CA Model : Results

(continued)

2/3)( ttP

prob. dist. of jams of lifetime t

“phantom traffic jams”-Spontaneous formation of jams with no obvious reason

noise /1)( ftN l

(i) Spontaneous small fluctuations grow to jams of all sizes.(ii) Cruise-control may make prediction more difficult.

Why self-organize?

cp

?

p

Sandpile model

c

Traffic model

)(Q Maximumthroughput

Sandpile Traffic

Subcritical Threshold=0 Low density

Critical 0<Threshold<∞

Critical density

Supercritical

Threshold→∞ High density

How self-organize?

Model of computer network traffic :Sole & Valverde (Physica A 2001)

• Hosts and routers on 2D lattice• Hosts create packets with prob. λ.• Packets are forwarded via routers to destination hosts.• To minimize the communication time, the shortest path

must be taken and the congested links avoided.• The congestion of a link = the amount of packets forward

ed through that link.

Network traffic model : ResultsIf congestion is low, users might increase their levels of activity.

If it is very congested,users decrease or leave the system.

→ The system might self-organize into the critical state.

Applications & Future Works

• How to optimize traffic networks (the Internet)- to minimize cost

• Which is the best path (strategy) for drivers (users)?

• Analogy : particles ↔ vehicles ↔ packets

• However, self-driven many-particle system

agents neuron signals