| |

Prof. Dr. rer. nat. Dirk Helbing, ETH Zurich, Computational Social Science with Anders Johansson, Martin Treiber, Arne Kesting, Stefan Lämmer, Martin Schönhof, and others

Complex Traffic Dynamics on Freeways and in Urban Road Networks

| |

§  Empirical findings §  Time-dependent phenomena

§  Traffic flow modeling §  Phase diagram of congested traffic states

§  Overcoming freeway congestion §  Traffic light control

§  Further reading

Content

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§  Traffic sticks to lanes (quasi 1-dimensional) §  Phase changes to stop-and-go traffic etc. §  Behaves very different in different countries §  In some countries, traffic is self-organized, here we have

top-down control §  Traffic congestion during peak hours §  Safety distance depends on speeds and relative velocities

Discussion: What Do You Know About Traffic?

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§  What’s the most efficient way to organize traffic? §  Why the hype about roundabouts? §  What’s the safest way to organize traffic? §  What rules are universal, which ones are country-

specific? §  How is environmental pollution related to traffic? §  How will car2car communication change the operation of

traffic? §  Do self-driving cars behave more rational than humans? §  How resilient is traffic to the percentage of “bad drivers”?

Discussion: What Else Might Be Interesting?

| |

Counter-Intuitive Behavior of Traffic Flow

Perturbing traffic flows and, paradoxically, even decreasing them may sometimes cause congestion.

| |

Congestion Patterns

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Analytical Theory of Traffic Flows

OCT,SGW

FT

Q

(v

eh

icle

s/h

/la

ne

)up

!Q (vehicles/h/lane)

HCT

=Qm

ax

Qtot

!Q

=Q

-Q

c3out

Small Perturbation

| |

Self-Driving Cars

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Traffic Light Control

| |

Empirical Findings

| |

Speed-Density Relationship

0

20

40

60

80

100

120

140

0 20 40 60 80 100 120 140 160

Density (veh/km/lane)

Spe

ed (k

m/h

)

| |

Flow-Density Relationship

0

500

1000

1500

2000

2500

3000

0 10 20 30 40 50 60 70 80 90 100

Density (veh/km/lane)

Vehi

cle

flow

(veh

/h/la

ne)

| |

Dynamical Phenomena

| |

Speed and Density as a Function of Time

0102030405060708090100110120

6:30 7:00 7:30 8:00 8:30 9:00 9:30 10:00

Speed (km/h)

Density (veh./km/lane)

Time (h)

| |

Time Gap Distribution

00.010.020.030.040.050.06

0 2 4 6 8 10 12

Rela

tive

Freq

uenc

y

(a)

Δt (s)

ρ=10-20 veh./km

left (f)right (f)both (f)

00.010.020.030.040.050.060.070.080.09

0 2 4 6 8 10 12Δt (s)

ρ=20-30 veh./km

(b)

freecongested

| |

Oscillatory Adaptation of Relative Speed and Distance

0

10

20

30

40

50

60

-4 -3 -2 -1 0 1 2 3 4 5

s (m

)

Δv (m/s)

| |

Wide Scattering of Congested Traffic Flows

0

500

1000

1500

2000

2500

3000

0 20 40 60 80 100 120 140 160

Density (veh/km/lane)

Flow

(veh

./h/la

ne)

| |

Capacity Drop

0500

10001500200025003000

0 20 40 60 80 100

Traf

fic F

low

Q (v

eh./h

/lane

)

Density ρ (veh./km/lane)

(a) Capacity Drop

ρcr

Qcr Left LaneρVsep

0

500

1000

1500

2000

2500

6:00 7:00 8:00 9:00 10:00 11:00 12:00

Traf

fic F

low

Q (v

eh./h

)

t (h)

(b)Discharge Flows

Right LaneLeft (Fast) Lane

| |

Emergence of

Congestion Patterns

| |

Instability of Traffic Flow: Stop-and-Go Waves

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„Phantom Traffic Jams“

Source: Sugiyama et al.

| |

Vehicle Trajectories Showing “Phantom Traffic Jam”

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Complex Congestion Patterns

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Surprising Variety of Congestion Patterns

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Elementary Traffic Patterns: Examples of Pinned Localized Clusters

The congested area remains localized as long as it does not reach the upstream end of the off-ramp.

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Elementary Traffic Patterns: Example of a Moving Localized Cluster

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Elementary Traffic Patterns: Examples of Stop-and-Go Waves

| |

Wide Moving Jams (by B.S. Kerner et al.)

V (km/h)

I1I2

I3

I1I2

I3

I1I2

I3

TimeTime

(a) (b)

Moving Jams Moving JamsFree Flow

Synchroni-zed Flow

Q (v

eh./h

)

Location (km)

Location (km)

| |

Elementary Traffic Patterns: Oscillating Congested Traffic

| |

Elementary Traffic Patterns: Homogeneous Congested Traffic

Homogeneous congested traffic requires large bottleneck strengths exceeding about 700 vehicles per kilometer and lane, which usually occur after serious accidents only. That is, if cases of accidents are excluded from the data set, one cannot find homogeneous congested traffic (if street capacities are properly dimensioned).

| |

Empirical Phase Diagram

| |

Examples of the “Boomerang Effect”

| |

The underlying dynamics of this transition is a “boomerang effect”

The boomerang effect was observed in 18 out of 245 cases of traffic breakdowns.

Transitions from Free to Congested Traffic

| |

Boomerang Effects Are due to Overtaking Trucks

| |

Traffic Flow Modeling

| |

§  “Traffic is a fluid medium.” §  Relevant variables:

§  Conservation of vehicles §  Continuity equation:

Fluid-Dynamic Traffic Equations

| |

§  “Flow Q and density ρ are empirically correlated.”

§  Fundamental diagram Qe(ρ)

Specification of the Flow as a Function of Density

Lighthill-Whitham(-Richards) (LWR) model (describes a conservation of the number of vehicles)

derivative plays an important role

| |

Kinematic Waves

0

2

4

6

8

10

0 2 4 6 8 10

05

10 0

5

10

0

50

100

Density

Location Time

Loca

tion

Time

| |

Fundamental Diagram

0500

10001500200025003000

0 20 40 60 80 100 120 140 160Equi

libriu

m F

low

(veh

./h)

Density (vehicles/km)

| |

Intelligent-Driver Model (IDM)

►  Equations of motion:

►  Dynamic desired distance

| |

Speed-Distance Relationship in the Bando-Sugiyama Car-Following Model

v0

dd0

1/ρ( jam , 0)

1/ρ v0( ),out

00

v

| |

Emergent Waves = “Phantom Traffic Jams”

0

50

100

150

200

250

300

350

400

0 200 400 600 800 1000

| |

Instability of Traffic Flow

| |

Metastability of Traffic Flow

| |

Unstable and Metastable Traffic States

! ! !

"!c

(b)

(a)

(c)

!c1 !c2 !c3 !c4 !max

!c1 !c2 !c3 !c4 !max

!c1 !c2 !c3 !c4 !max

#

#

#

"!(x) "!(x)

x x0 0

Diagram of states of traffic flow

States of homogeneous traffic flow:

Types of structures in traffic flow:Complex sequences of traffic

jams or of "dipole-layers"

Sta-ble

State

Sta-ble

State

"Metastable" State

"Metastable" State

Unstable State

Localized Structures

Nonlocalized Structures

"Dipole Layers"

Traffic Jams

xx x

Source: B.S. Kerner

| |

Breakdown of Traffic due to a Supercritical Reduction of Traffic Flow

Perturbing traffic flows and, paradoxically, even decreasing them may sometimes cause congestion.

| |

Phase Diagram of Congested Traffic States

| |

Congested Traffic States Simulated with a Macroscopic Traffic Model

Phys. Rev. Lett. 82, 4360 (1999).

Similar congested traffic states are found for several other traffic models, including “microscopic” car-following models.

| |

Schematic Instability Diagram

Grey areas = metastable regimes, where result depends on perturbation amplitude

| |

Theoretical Phase Diagram of Traffic States and Universality Classes

Phase diagrams are not only important to understand the conditions under which certain traffic states emerge. They also allow one to categorize the more than 100 traffic models into different universality classes. From the universality class that reproduces the empirically observed stylized facts, one may choose any representative, e.g. the simplest or the most accurate one, depending on the purpose.

OCT,SGW

FT

Q

(v

eh

icle

s/h

/la

ne

)up

!Q (vehicles/h/lane)

HCT

=Qm

ax

Qtot

!Q

=Q

-Q

c3out

Small Perturbation

| |

Empirical Phase Diagram

| |

The Outflow Depends on the Weather Conditions

1400

1500

1600

1700

1800

1900

2000

0 10 20 30 40

Out

flow

Qou

t (ve

h/h/

lane

)

Range of Visibility (km)

D

DD

DD

D D

D

D

DD

D

D

DD

D

DW

W

(w)W

W

W

(w)(w)

(w)(w)

(w)

W

W

WW

W: wet road(w): affected by showers

D: dry road

| |

Empirical Phase Diagram for Scaled Flows

A scaling by the outflow, that varies from day to day, gives a clearer picture.

| |

“General Pattern” and “Pinch Effect”

According to Kerner, in the “generalized pattern”, synchronized traffic upstream of a bottleneck breads wide moving jams based on the “pinch effect”. That is, upstream of a section with “synchronized” congested traffic close to a bottleneck, a so-called “pinch region” gives spontaneously birth to narrow vehicle clusters. These perturbations should be growing while traveling further upstream. Eventually, wide moving jams form by the merging or disappearance of narrow jams. Once formed, wide jams suppress the occurrence of new narrow jams in between.

| |

Design of the German Freeway A5

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(Intermittent) Activation of an Off-Ramp Bottleneck

Milder form of congested traffic upstream of off-ramp expected, e.g. OCT or SGW instead of HCT. Appearance would correspond to “generalized pattern”.

| |

Combination of an Off-Ramp with an On-Ramp

offramp

onramp

| |

Wide Scattering as Effect of Heterogeneous Traffic

The jam line with variable parameters can explain the observations quantitatively! Scattering and stochasticity do not contradict models with a fundamental diagram, just models with identical driver-vehicle units.

| |

“Synchronized Traffic” Considering Cars+Trucks

Homogeneous-in-speed states

Time series Fundamental diagram

| |

Homogeneous-In-Speed States

0500

1000150020002500300035004000

0 20 40 60 80 100

Traf

fic F

low

Q (v

eh./h

/lane

)

Density l (veh./km/lane)

| |

Traffic Congestion is Predictable

| |

An Analytical !eory of Tra"c Flow

The European Physical Journal

A selection of articles by Dirk Helbing reprinted from The European Physical Journal B

!e European Physical Journal B

A selection of articles by Dirk Helbing

your physics journal

EPJ .org

Printed on acid free paper

Società Italianadi Fisica

An Analytical Theory of Traf!c Flow

D. HelbingDerivation of non-local macroscopic traf!c equations and consistent traf!c pressures from microscopic car-following modelsDOI: 10.1140/epjb/e2009-00192-5

D. Helbing and A.F. JohanssonOn the controversy around Daganzo's requiem for and Aw-Rascle's resurrection of second-order traf!c "ow modelsDOI: 10.1140/epjb/e2009-00182-7

D. Helbing and M. MoussaidAnalytical calculation of critical perturbation amplitudes and critical densities by non-linear stability analysis of a simple traf!c "ow modelDOI: 10.1140/epjb/e2009-00042-6

D. Helbing, M. Treiber, A. Kesting and M. SchönhofTheoretical vs. empirical classi!cation and prediction of congested traf!c statesDOI: 10.1140/epjb/e2009-00140-5

M. Treiber and D. HelbingHamilton-like statistics in one dimensional driven dissipative many-particle systemsDOI: 10.1140/epjb/e2009-00121-8

D. Helbing and B. TilchA power law for the duration of high-"ow states and its interpretation from a heterogeneous traf!c "ow perspectiveDOI: 10.1140/epjb/e2009-00092-8

D. HelbingDerivation of a fundamental diagram for urban traf!c "owDOI: 10.1140/epjb/e2009-00093-7

D. Helbing and A. MazloumianOperation regimes and slower-is-faster effect in the control of traf!c intersectionsDOI: 10.1140/epjb/e2009-00213-5

A selection

of articles by D

irk Helb

ing rep

rinted

from T

he E

uropean

Physical Journ

al B

Traffic Physics

| |

Living Costs and Public Transport Determine Travel Time Budget Take Home Messages §  The fundamental diagram describes the flow-density and average

speed-density relationships as well as the speed of kinematic waves and shock waves (jam fronts)

§  The understanding of the emergence of “phantom traffic jams” and other congestion patterns requires a model with delayed adaptation and unstable as well as metastable density areas

§  The instability of traffic flow comes with a capacity drop §  There is a dynamic self-organization of characteristic traffic constants

such as the outflow from congested traffic and the speed of the downstream jam front

§  To derive the phase diagram of traffic states, one must also consider the characteristic constants of traffic flow, i.e. the outflow from congested traffic and dissolution speed of traffic jams

§  While the moment of traffic breakdowns is only probabilistically predictable, travel times before and after are

| |

Freeway Traffic Control

| |

Cooperative Driving Based on Autonomous Vehicle Interactions

“Attention: Traffic jam”

In: Transportation Research Record (2007)

§  On-board data acquisition („perception“)

§  Inter-vehicle communication

§  Cooperative traffic state determination (“cognition”)

§  Adaptive choice of driving strategy (“decision-making”)

§  Driver information

§  Traffic assistance (higher stability and capacity of traffic flow)

| |

Data Fusion for Dynamic Traffic State Detection

| |

Design of Traffic State Adaptive Cruise Control Invent-VLA: Intelligent Adaptive Cruise Control (IACC) for the avoidance of traffic breakdowns and a faster recovery from congested traffic

| |

Overcoming Congestion by Real-Time Feedback

| |

Enhancing Traffic Performance by Adaptive Cruise Control

§  Traffic breakdowns delayed §  Faster recovery to free traffic §  High impact on travel times §  Reduced fuel consumption

and emissions

ρ(veh./km/lane)

10% Equipped Vehicles

20% Equipped Vehicles

“Mechanism design”

| |

How to Detect the Spatiotemporal Dynamics of a Traffic Jam?

Downstream jam fronts

| |

Jam Front Detection – Intervehicle Communication

Congested traffic

Change: Position,Time

Free traffic

Message core: Floating car (ACC)

Inter-Vehicle-Communication (V2V):

| |

Example: Information about a Stop-and-Go Wave

| |

Traffic-Adaptive Driving Strategy for ACC

Color of trajectories ≡

ACC operating state:

-  Free traffic -  Approaching to jam -  Jam -  Downstream jam front

| |

Statistics of Message Transmission

Distance between communicating vehicles exponentially distributed → Distributions for T1, T2, and T3, e.g.

| |

Urban Planning vs. Enabling Self-Organization

| |

Zurich‘s Laudable Goals

§  Better mobility §  Less pollution

§  Less noise §  Less traffic

The Noble Goals of Traffic Planning

| |

But This Is Often the Reality …

| |

… and This: Urban Gridlock

| |

Flows towards a Center with and without Congestion

| |

Traffic Light Control

| |

Adaptive Traffic Light Control §  for complex street networks §  for traffic disruptions (building sites, accidents, etc.) §  for particular events (Olympic games, pop concerts, etc.)

| |

§  Directed links are homogenous road sections §  Traffic dynamics: congestion, queues

§  Nodes are connectors between road sections §  Junctions: merging, diverging

§  Intersections §  Traffic lights: control, optimization

§  Traffic assignment §  Route choice, destination flows

Road Network as Directed Graph

Local Rules, Decentralization, Self organization

| |

§  Propagation velocity c(ρ)

§  Free traffic: §  c(ρ) = V0

§  Congested traffic: §  c(ρ) ≈ -15 km/h (universal)

Triangular Flow-Density Diagram

| |

Characteristic Velocities

| |

A Queueing-Theoretical Traffic Model The continuity equation for the vehicle density ρ(x,t) at place x and time t in road section i is

Terms Source =∂

∂+∂

∂xtxQ

ttx i ),(),(ρ

We assume the fundamental flow-density relation

⎪⎩

⎪⎨⎧

−=<=

=otherwise if

jamcong

crfree

TQVQ

Qi

iii /)/1()(

)()(

0

ρρρρρρρ

ρ

0iV = free speed on road section i T = safe time gap jamρ = jam density

The number Ni of vehicles in section i changes according to

)()()()()()(11 tQtQtQtQtQ

dttdN

iiiiii deprampdepdeparr −+=−= −−

arriQ = arrival rate of vehicles depiQ = departure rate of vehicles

Treatment of ramp flows at downstream section ends: )()()(1 tQtQtQ iii

rampdeparr +=+

| |

A Queueing-Theoretical Traffic Model The traffic-state dependent departure rate is given by

⎪⎩

⎪⎨

⎧

=−−≠≠=≠−

=

++

+

+

1)( if)()(0)( and 1)( if)(0)( and 1)( if)(

)(

1rampcong

1dep

1cap

1freearr

dep

tStQTtQtStStQtStSTtQ

tQ

iiii

iii

iiii

i

The capacity of congested road section is: ]0),(,)(),(max[)()( 1 tQQIItQtQItQ iiiiii Δ−−= + out

rampout

cap

Ii = number of lanes outQ = (1- ρcr/ρjam)/T = outflow per lane from congested traffic

The maximum capacity in free traffic is: ]0),(,)(),(max[)( 0

10 tQVIItQVItQ iiiiiiii Δ−−= + cr

rampcr

max ρρ

Definition of free (Si = 0), fully congested (Si = 1) and partially congested (Si = 2) traffic states:

⎪⎩

⎪⎨

⎧−≤−−=−<−−=

=otherwise2

)()( and )( if1)()( and 0)( if0

)( arrcongdep

maxfreearr

dttQTdttQLtldttQTdttQtl

tS iiiii

iiii

i

Li = length of road section i, li = length of congested road section

| |

A Queueing-Theoretical Traffic Model

Growth of the length li of congested traffic according to shock wave theory:

)/)/)]([(()/)/)((()/)]([()/)((

0

0

iiiiiiiiii

iiiiiii

IVtlLtQIctltQVtlLtQctltQ

dtdl

−−−−−−−−−= arrfreedepcong

arrdep

ρρ

jamcong

free

ρρρ

)1()(

,/)( 0

iii

iiii

TQQVQQ−=

=

with densities

Delay-differential equation for the travel time Ti on section i, when entered at time t:

1))(()()(1

))(()()( 11 −

++=−

+= −−

tTtQtQtQ

tTtQtQ

dttdT

ii

ii

ii

iidep

rampdep

dep

arr

| |

§  Movement of congestion

§  Number of vehicles

§  Travel time

Network Links: Homogenous Road Sections

| |

§  Side Conditions

§  Conservation

§  Non-negativity

§  Upper boundary §  Branching

§  Goal function

Network Nodes: Connectors

| |

§  1 to 1:

§  1 to n: Diverging with branch weight αij

§  n to 1: Merging

Network Nodes: Special Cases

| |

Simulation of Diverges and Merges Diverges and Merges

| |

Network Representation of Intersections

| |

§  Traffic light §  Additional side condition

§  Intersection §  Is only defined by a set of

mutually excluding traffic lights §  Each intersection point gives

one more side condition

Intersections: Modelling

| |

Interdependence of Subsequent Intersections

Green wave

Non-optimal phase shift

Different cycle times

Stochastic switching

| |

Self-Organized Oscillations at Bottlenecks and Synchronization §  Pressure-oriented, autonomous,

distributed signal control: §  Major serving direction alternates, as

in pedestrian flows at intersections §  Irregular oscillations, but

‘synchronized’ §  In huge street networks:

§  ‘Synchronization’ of traffic lights due to vehicle streams spreads over large areas

| |

Travel time minimization, “homo economicus”

Central control, “benevolent dictator”

Same, but other-regarding coordination with neighbors

Comparing 3 Ways to Organize a Complex System

| |

Top-down regulation

Clearing longest queue

Stefan Lämmer and Dirk Helbing

Bottom-Up Self-Regulation Can Outsmart Optimal Top-Down Control

| |

Top-down regulation

Clearing longest queue

Selfish optimization,

self-organization

Stefan Lämmer and Dirk Helbing

Bottom-Up Self-Regulation Can Outsmart Optimal Top-Down Control

| |

Top-down regulation

Clearing longest queue

Selfish optimization,

self-organization

Adam Smith’s invisible hand works

Stefan Lämmer and Dirk Helbing

Bottom-Up Self-Regulation Can Outsmart Optimal Top-Down Control

| |

Top-down regulation

Clearing longest queue

Selfish optimization,

self-organization

Stefan Lämmer and Dirk Helbing

Adam Smith’s invisible hand fails

Bottom-Up Self-Regulation Can Outsmart Optimal Top-Down Control

| |

Top-down regulation

Clearing longest queue

Selfish optimization,

self-organization

Other-regarding optimization,

self-regulation

Stefan Lämmer and Dirk Helbing

Bottom-Up Self-Regulation Can Outsmart Optimal Top-Down Control

| |

Top-down regulation

Clearing longest queue

Selfish optimization,

self-organization

Other-regarding optimization,

self-regulation

Stefan Lämmer and Dirk Helbing

Invisible hand works

Bottom-Up Self-Regulation Can Outsmart Optimal Top-Down Control

| |

Towards Self-Organized Traffic Light Control in Dresden

| |

The Measurement and Control Area

| |

Disturbance of Traffic Coordination by Bus and Tram Lines

| |

Comparison of Current and Self-Organized Traffic Light Control

| |

Comparison of Current and Self-Organized Traffic Light Control

| |

Synchronize Traffic by Green Waves or Use Gaps as Opportunities?

| |

Performance in Dependence of the Traffic Volume

| |

Red Time Distribution for Pedestrians

| |

Gain in Performance

| |

A New Paradigm

| |

| |

| |

§  Advantage of centralized control is large-scale coordination

§  Disadvantages are due to §  vulnerability of the network

§  information overload

§  wrong selection of control parameters

§  delays in adaptive feedback control

§  Decentralized control can perform better in complex systems with heterogeneous elements, large degree of fluctuations, and short-term predictability, because of greater flexibility to local conditions and greater robustness to perturbations

Centralized Control and Its Limits

Leve

l of g

oal

achi

evem

ent

Level of autonomy of control

(Windt, Böse, Philipp, 2006)

| |

Living Costs and Public Transport Determine Travel Time Budget Take Home Messages §  Traffic flow can be improved using self-organization

concepts based on local interactions §  Traffic assistance systems, based on adaptive cruise

control (ACC) systems using radar sensors, reduce time gaps and increase acceleration in a traffic-state-dependent way to increase capacity and stabilize the flow

§  Self-organized oscillations of pedestrian flows at bottlenecks have inspired the principle of a self-organizing traffic light control (self-control, self-regulation)

§  The self-control combines local minimization of waiting times at each intersection with a immediate clearing of queues, which would otherwise create spill-over effects

§  Self-organization outperforms top-down control in case of largely variable, hardly predictable complex systems

| |

§  D. Helbing (2001) Traffic and related self-driven many-particle systems. Reviews of Modern Physics 73, 1067-1141.

§  D. Helbing and M. Schreckenberg (1999) Cellular automata simulating experimental properties of traffic flows. Physical Review E 59, R2505-R2508.

§  D. Helbing, A. Hennecke, and M. Treiber (1999) Phase diagram of traffic states in the presence of inhomogeneities. Physical Review Letters 82, 4360-4363.

§  D. Helbing (2005) Traffic flow. In: Alwyn Scott (ed.) Encyclopedia of Nonlinear Science (Routledge, New York).

Further Reading

| |

§  D. Helbing and K. Nagel (2004) The physics of traffic and regional development. Contemporary Physics 45(5), 405-426.

§  D. Helbing, A. Hennecke, V. Shvetsov, and M. Treiber (2002) Micro- and macro-simulation of freeway traffic. Mathematical and Computer Modelling 35(5/6), 517-547.

§  D. Helbing (2001) Traffic and related self-driven many-particle systems. Reviews of Modern Physics 73, 1067-1141.

Further Reading

| |

§  D. Helbing, A. Hennecke, V. Shvetsov, and M. Treiber (2001) MASTER: Macroscopic traffic simulation based on a gas-kinetic, non-local traffic model. Transportation Research B 35, 183-211.

§  C. Leduc, K. Padberg-Gehle, V. Varga, D.Helbing, S. Diez, and J. Howard (2012) Molecular crowding creates traffic jams of kinesin motors on microtubules. PNAS 16, 6100-6105

§  D. Helbing and A. Mazloumian (2009) Operation regimes and slower-is-faster effect in the control of traffic intersections. European Physical Journal B 70(2), 257–274.

Further Reading

| |

§  D. Helbing (2009) Derivation of non-local macroscopic traffic equations and consistent traffic pressures from microscopic car-following models. European Physical Journal B 69(4), 539-548.

§  D. Helbing and A. Johansson (2009) On the controversy around Daganzo’s requiem for and Aw-Rascle’s resurrection of second-order traffic flow models. European Physical Journal B 69(4), 549-562.

§  D. Helbing and M. Moussaid (2009) Analytical calculation of critical perturbation amplitudes and critical densities by non-linear stability analysis of a simple traffic flow model. European Physical Journal B 69(4), 571-581.

Further Reading

| |

§  D. Helbing, M. Treiber, A. Kesting, and M. Schönhof (2009) Theoretical vs. empirical classification and prediction of congested traffic states. European Physical Journal B 69(4), 583-598.

§  D. Helbing and B. Tilch (2009) A power law for the duration of high-flow states and its interpretation from a heterogeneous traffic flow perspective. European Physical Journal B 68(4), 577-586.

§  M. Treiber and D. Helbing (2009) Hamilton-like statistics in onedimensional driven dissipative many-particle systems. European Physical Journal B 68(4), 607-618.

Further Reading

| |

§  D. Helbing (2009) Derivation of a fundamental diagram for urban traffic flow. European Physical Journal B 70(2), 229-241.

§  S. Lämmer and D. Helbing (2008) Self-control of traffic lights and vehicle flows in urban road networks. JSTAT P04019.

§  M. Treiber, A. Kesting, and D. Helbing (2007) Influence of reaction times and anticipation on stability of vehicular traffic flow. Transportation Research Record 1999, 23-29.

§  D. Helbing, J. Siegmeier, and S. Lämmer (2007) Self-organized network flows. Networks and Heterogeneous Media 2(2), 193-210.

Further Reading

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§  M. Krbalek and D. Helbing (2004) Determination of interaction potentials in freeway traffic from steady-state statistics. Physica A 333, 370-378.

§  D. Helbing (2003) A section-based queueing-theoretical traffic model for congestion and travel time analysis in networks. Journal of Physics A: Mathematical and General 36, L593-L598.

§  R. Kölbl and D. Helbing (2003) Energy laws in human travel behaviour. New Journal of Physics 5, 48.1-48.12.

§  V. Shvetsov and D. Helbing (1999) Macroscopic dynamics of multilane traffic. Physical Review E 59, 6328-6339.

Further Reading

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§  M. Treiber, A. Hennecke, and D. Helbing (1999) Derivation, properties, and simulation of a gas-kinetic-based, non-local traffic model. Physical Review E 59, 239-253.

§  D. Helbing (1996) Gas-kinetic derivation of Navier-Stokes-like traffic equations. Physical Review E 53, 2366-2381.

Further Reading