Stochastic Modeling and Simulation of Highway Traffic

Alexandros Sopasakis

Dept. of Mathematics and Statistics,

University of North Carolina

Charlotte

Outline

• The traffic model: properties and dynamics• Calibration and parameter estimation• Simulations and comparisons (one-lane highway)• Deterministic closures and macroscopic models

• Multi-lane extensions

• Conclusions & References

NSF Focus , 2009

Proposed Traffic Model Properties/Attributes

• Asymmetric Simple Exclusion Process (ASEP)• Arrhenius microscopic stochastic dynamics• One directional flow• Look-ahead interaction potential• Retarded acceleration• Timely braking• Conservation of vehicles (assuming no entrances or exits)• Numerical simulations via Kinetic Monte Carlo (KMC)• Extensions to macroscopic traffic flow models and PDEs

NSF Focus , 2009

We let Λ denote a lattice of N cells.

We also denote by t (x) the spin configuration at x.

We introduce the microscopic stochastic Ising process

A spin configuration is an element of the configuration space and we write

The stochastic process is a continuous time jump Markov process on with generator

Σ={0,1}Λ

σ={σ x : x∈Λ}{σ t }t≥0

L∞ Σ ,R

Mf σ =∑x∈Λy≠x

c σ [ f σ ¿− f σ ]

0 11 10 σ x

Main Statistical Mechanics Concepts

{σ t }t≥0

NSF Focus , 2009

The corresponding energy Hamiltonian is

where the interaction potential is given by

where J denotes the local interaction potential

Here and L denotes the interaction radius.

H σ =12 ∑x∈Λ

U x σ x

U x =∑z≠xz∈Λ

J x , z σ z −h

J x , y =γ V γ∣x− y∣

γ=1 /2L1

Main Statistical Mechanics Concepts

NSF Focus , 2009

Equilibrium states of the stochastic model are described by the Gibbs measure at the prescribed temperature T,

where and

and

μβ ,N dσ =1Z

e−βH σ PN dσ

PN dσ =∏x∈ Λ

ρdσ x

ρ σ x =0 =12, ρσ x =1 =

12

β=1kT

μβ ,N

Main Statistical Mechanics Concepts

NSF Focus , 2009

The Mathematical Model

where

and denotes a new lattice configuration

Mf σ =∑x∈Λ

c σ [ f σ∗− f σ ]

ddx

Ef σ =EMf σ

σ ¿

NSF Focus , 2009

Microscopic ArrheniusSpin-Exchange Dynamics

The Arrhenius spin-exchange rate c(x,y,σ )

With exchange rate constant,

Here denotes the characteristic time of the stochastic process.

Again recall that

cd e−U x , if σ x =1, and σ y =0,

cd e-U y , if σ x =0, and σ y =1,

0 , otherwise¿

c x , y , σ =¿ {¿ {¿ ¿¿¿

cd=1

τ0

τ 0

U x =∑z≠xz∈Λ

J x , z σ z

NSF Focus , 2009

The Arrhenius spin-flip rate c(x,σ ) at lattice site x and spin configuration t is given by

With adsorption/desorption constants,

Here denotes the characteristic time of the stochastic process.

cd e−U x , when σ x =0

ca , when σ x =1¿

c x ,σ =¿ {¿ ¿ ¿¿

ca=cd=1τ I

τ I

Microscopic ArrheniusSpin-Flip Dynamics

NSF Focus , 2009

We consider short vehicle potential interactions J,

where as usual ordains the range of microscopic Interactions. Here via,

Which enforces:• Exclusion princile• Vehicles do not go backward in traffic• Local effect of the interactions (thus once again, more realistic traffic conditions)

J x , y =V γ x− y , x , y∈Λ

V r ={J 0 , if 0r10, otherwise

V : RRγ=1 /L

NSF Focus , 2009

The Traffic Model

ddt

Ef σ =EMf σ

ddt

Ef σ =E ∑x∈Λy≠x

c x , y , σ [ f σ¿−f σ ]

ddt

Ef σ =E ∑x∈Λ

c 0 σ x 1−σ x1 e−U x ,σ [ f σ x , x1

− f σ ]

or in more detail,

which based on the spin-exchange rate c(x,y,σ ) for y=x+1 gives,

The probability of a spin-exchange between x and y=x+1 during time [t, t+Dt] is

c x , y , σ ΔtO Δt2

NSF Focus , 2009

A simple schematic describing the traffic model dynamics

LL

U

LL

U

LL

U

LL

U

L L L L

NSF Focus , 2009

Free Parameters and Calibration

The model is characterized by the following three undetermined parameters:

• - the characteristic time of the stochastic process• - the strength of the interactions• - the interaction potential range

Cell length is assumed to be 22 feet (average vehicle size plus safe distance).

LJ 0

τ 0

Δt cell=22 feet65 miles/hour

≈14

sec .

NSF Focus , 2009

NSF Focus , 2009

NSF Focus , 2009

NSF Focus , 2009

Flow-Density DiagramExperimental Data

NSF Focus , 2009

NSF Focus , 2009

NSF Focus , 2009

0

10

20

30

40

50

60

70

80

0 500 1000 1500 2000 2500 3000 3500

Flow (veh/hr/lane)

Spe

ed (m

ph)

Velocity-Flow Diagram (real freeway data from San Antonio) [Halkias & Mahmassani (MTI 2005)]

NSF Focus , 2009

NSF Focus , 2009

NSF Focus , 2009

Recall the traffic flow model is, or

where . This becomes,

Exact but not yet closed for

Suppose that J has uniform (J=Jo), weak long interactions.

ddt

Ef σ =EMf σ

ddt

Ef σ =E ∑x∈Λ

c 0 σ x 1−σ x1 e−U x ,σ [ f σ x , x1

− f σ ]

Deterministic Closures

U x , σ =∑z≠xz∈Λ

J x , z σ z

NSF Focus , 2009

ddt

Eσ t z =−Ec0 σ z 1−σ z1 e−U z , σ Ec0σ z−1 1−σ z e−U z−1, σ

Eσ t z =Prob σ t z =1

Finite Difference Scheme

The LLN formally applies and the fluctuations of about their mean will be small.

Then in the long range interaction limit we have,

Let then we obtain an approximate semi-discrete finite difference scheme:

where

For a periodic lattice this scheme is conservative.

u z ,t =Eσ t z

∑y≠x

J y−x σ y

ddt

u z ,t F z1, t −F z ,t =0

F z , t =c0u z−1, t 1−u z , t e−J °u z−1, t

Ee−U x , σ =Ee

−∑ J y−x σ y ¿

N , L∞

e−∑ J y− x Eσ y oN 1

NSF Focus , 2009

Comparisons between semi-discrete scheme and microscopic stochastic model

NSF Focus , 2009

NSF Focus , 2009

PDE model

Expanding in Taylor we obtain,

Rescaling time via in order to absorb h and omiting the term, we obtain the following macroscopic transport equation:

where the PDE flux is F u=c0u1−u e−J °u

ddt

u[ hc0u1−u e−J °u]z=O h 2

t th−1Oh2

utFu z=0

NSF Focus , 2009

NSF Focus , 2009

Hierarchical Comparisons

Expanding the convolution,

we can approximate the exponential via,

The traffic model PDE therefore becomes…

e−J °u≈e−J 0u [ 1−J1 uz−J2 uzz ]

J °u=∫z

∞

V y−z u y dy =x= y−z

∫0

∞

V x u xz dx=J 0uJ 1uzJ 2uzz

u tcu 1−u e−J °u=0

NSF Focus , 2009

The traffic model PDE,

Note:• No interactions (J=0): Lighhill-Whitham/Burger’s eq.

utc0[ u1−u e−J 0u ]z=c0 [ J1 u1−u e

−J 0uuz ]zc0 [ J2 u1−ue

−J 0uu zz ]z

u tc0 [u 1−u ]z=0

NSF Focus , 2009

Lighthill-Whitham / Burgers flux

NSF Focus , 2009

The traffic model PDE,

Note:• No interactions (J=0): Lighhill-Whitham/Burger’s eq.

• Long range (L=N) uniform (J=Jo) interactions: Non-local flux

• Including terms up to Jo in the convolution, Non-convex flux

• Terms up to Nonlinear diffusive LWR type

• Full model is higher order dispersive (KDV type?) with nonlinear coefficients

utc0[ u1−u e−J 0u ]z=c0 [ J1 u1−u e

−J 0uuz ]zc0 [ J2 u1−ue

−J 0uu zz ]z

u tc0u 1−u =0

u tc0 [u 1−u e−J 0 u

]z=0

u tc0 [u 1−u e−J 0u ]z =0

J 1

NSF Focus , 2009

Multi-lane extensions

• We assume a two-dimensional domain (multi-lane highway).• We introducing preferred direction in lane-changing via an anisotropy

type potential. Thus our total interaction potential now consists of:

where with

• Calibrate parameters:

U x =U eU ah

U a x =∑y=nn

ψ x , y ψ x , y ={k l if y=x1

kr if y=x-1k f if y=xn

NSF Focus , 2009

NSF Focus , 2009

NSF Focus , 2009

Test case toy problem: an incident

• Assume a 2-lane highway• Block lane 1 due to an accident

NSF Focus , 2009

Conclusions

Presented a novel modeling approach based on microscopic Arrhenius spin-exchange dynamics

Extended method to multi-lane traffic

Studied deterministic closures of the microscopic stochastic model at different length scales

Obtained formal hierarchical comparisons with other well-known models of traffic

Presented Kinetic Monte Carlo simulations which allows for comparisons with actual traffic data as well as other PDE models

NSF Focus , 2009

ReferencesP.Athol. Interdependence of certain operational characteristics within a moving traffic stream. Highway Research

Record, 58-97, 1972

A.B.Bortz, M.H.Kalos and J.L.Lebowitz. A new algorithm for Monte Carlo simulations of Ising spin systems. J.Comput. Phys. 17:10, 1975

C.Chowdhury, L.Santen and A.Schadschneider. Phys. Rep. 329:199, 2000

E.F.Codd. Cellular Automata. Academic Press. New York. 1968

M.Cremer and J.Ludwig. Mathematics and Computers in Simulation, 28:297, 1986

C.F.Daganzo. Requiem for second-order fluid approximations of traffic flow. Transportation Research B, 29:277, 1995

C.F.Daganzo, M.J.Cassidy and R.L.Bertini. Some traffic features at freeway bottlenecks. Transportation Research B, 33:25-42, 1999

N.Dundon and A.Sopasakis. Stochastic modeling and simulation of multi-lane traffic, in progress

L.Gray and D.Griffeath. The ergodic theory of traffic jams. J. Statist. Ohys., 105(3-4):413, 2001

F.L.Hall. Traffic Flow Theory, Chapter 2, pg2-34. Washington, D.C.: US Federal Highway Administration, 1996

CCM. Highway Capacity Manual. Technical Report, Transportation Research Board, 1985. Special Report 209

D.Helbing. Gas-Kinetic derivation of Navier-Stokes-like traffic equations. Physical Review E, 53(3):2366, 1995

D.Helbing, A.Hennecke, V.Shvetsov and M.Treiber Micro and macro simulation of freeway traffic. Math. Comp. Modelling, 35:517, 2002

D.Helbing and M.Treiber. Gas-kinetic-based traffic model explaining observed hysteretic phase transition. Phys. Rev. Let., 3042-3045, 1998

R.Illner, A.Klar and T.Materne. Vlassov-Fokker-Planck models for multilane traffic flow. Commun. Math. Sci., 1:1, 2003

S.Jin and J.G.Liu. Relaxation and diffusion enhanced dispersive waves. Proc. Roy. Soc. London A., 446:555-563, 1994

B.S.Kerner. Dependence of empirical fundamental diagram on spatial-temporal traffic patterns features.

B.S.Kerner. S.L.Klenov and D.E.Wolf. Cellular automata approach to three-phase traffic theory. J.Phys.A:Math. Gen., 35:9971, 2002

B.S.Kerner and P.Konhauser. Structure and parameters of cluster in traffic flow. Phys. Rev. E., 50:54-83, 1994

B.S.Kerner and H.Rehborn. Experimental properties of phase transitions in traffic flow. Phys.Rev.Let., 49:4030, 1997

A.Klar and R.Wegener. Kinetic derivation of macroscopic anticipation models for vehicular traffic. SIAM J.Appl.Math.60:1749-1766, 2000

W.Knospe, L.Santen, A.Schadschneider and M.Schreckenberg. J.Phys.A:Math.Gen, 33:L477,2000

W.Knospe,L.Santen,A.Schnadschneider and M.Schreckenberg. Phys.Rev.E,65:015101®,2002

J.Krug and P.A.Ferrari. Phase transitions in driven diffusive systems with random rates. J.Phys. A, 29:L465,1996

J.Krug and H.Spohn. Universality classes for deterministic surface growth. Phys.Rev.A, 38:4271, 1988

R.D.Kunhe. Macroscopic freeway model for dense traffic-stop-start wave and incident detection. Ninth international sysmposium on Transportation and Traffic Theory, 21-42,1984

D.A.Kurtze and D.S.Hong. Traffic jams, granular flow and solitons selections. Phys.Rev.E, 52:218-221, 1995

C.Landim and C.Kipnis. Scaling limits of interacting particle systems. Springer-Berlin, 1999

T.Li. Nonlinear dynamics of traffic jams. Physica D, May 2005. to appear

T.Li and H.Liu. Stability of a traffic flow model with non-convex relaxation. Comm. Math. Sciences, 3:101-118, 2005

T.M.Liggett. Interacting Particle Systems. Springer 1985

M.J.Lighhill and G.B.Whitham. On kinematic waves II-A theory of traffic flow on long crowded roads. Proc. Roy. Soc London, A229:317, 1955

M.Muramatsu and T.Nagatani. Phys. Rev. E, 60:180, 1999

T.Nagatani. The physics of traffic jams. Rep. Prog. Phys. 65:1331, 2002

K.Nagel and M.Paczuski. Phys. Rev. E, 51:2909, 1995

K.Nagel and M.Schreckenberg. A cellular automaton model for freeway traffic. J.Phys. I 2:2221, 1992

K.Nagel, D.E.Wolf, P.Wagner and P.Simon. Two-lane traffic rules for cellular automata: A systematic approach. Phys. Rev. E, 58(2):1425, 1998

P.Nelson. Phys.Rev.E, 61:383, 2000

P.Nelson, On two-regime flow, fundamental diagrams and kinematic-wave theory

G.F.Newell. Nonlinear effects in theory of car following. Operations Research, 9:209-229, 1961

J.C.Munoz and C.F.Daganzo. Structure of the transition zone behind freeway queues. Transportation Science, 37,312, 2003

O.Penrose. A mean-field equation of motion for the dynamic Ising model. J.Stat.Phys., 63(5/6):975-986,1991

A.Schadschneider. Traffic flow: a statistical physics point of view. Physica A, 312:153, 2002

A.Sopasakis. Unstable flow theory and modeling. Math. Comput. Modelling, 35(5-6):623, 2002

A.Sopasakis. Formal asymptotic models of vehicular traffic. Model closures. SIAM J.Appl.Math.,63:1561,2003

A.Sopasakis. Stochastic noise approach to traffic flow modeling. Physica A, 342:741, Nov. 2004

D.Stauffer. Computer simulations of cellular automata. J.Phys. A, 24:909-927, 1991

M.Takayasu and H.Takayasu. Phase transition and 1/f noise in one dimensional asymmetric particle dynamics. Fractals 1, 860-866, 1993

M.Treiber, A.Hannecke and D.Helbing. Derivation, properties and simulation of a gas-kinetic based non-local traffic model. Phys.Rev.E, 59:239-253, 1999

J.Treiterer and J.A.Myers. The hysteresis phenomenon in traffic flow. In D.J.Beckley, editor, Proc. 6th ISTTT. Sydney, Australia, 13-38. A.W.Reed Pty Ltd 1974

D.G.Vlachos and M.A.Katsoulakis. Derivation and validation of mesoscopic theories for diffusion of interacting molecules. Phys.Rev.Let., 85(18):3898,2000

J.D.Wardrop. Some theoretical aspects of road traffic research. Proc. Of the institution of civil engineers. Part II, 325,1952

G.B.Whitham. Linear and nonlinear waves. Wiley, 1974

R.Wiedemann. Simulation des stravenverkehrsflusses. Schriftenreihe des instituts fur verkenhrswesen der universitat karlsruhe. 8, Germany, 1974

S.Wolfram. Cellular automata and coplexity. Addison-Wesley, Reading, MA 1994