traffic theory jo, hang-hyun (kaist) april 9, 2003
TRANSCRIPT
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)
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
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
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
)(,
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
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
(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.