Fundamental Traffic Diagrams : A Maxplus Point of View N. Farhi, M. Goursat & J.-P. Quadrat

Fundamental Traffic Diagrams : A Maxplus Point of View

Introduction LWR Equation Circular Road Height shape Road Extension Bibliography

N. Farhi, M. Goursat & J.-P. Quadrat INRIA-Rocquencourt (France)

14/6/2010

Outline Fundamental Traffic Diagrams : A Maxplus Point of View N. Farhi, M. Goursat & J.-P. Quadrat Introduction LWR Equation

1 Introduction 2 LWR Equation 3 Circular Road 4 Height shape Road

Circular Road Height shape Road

5 Extension

Extension Bibliography

6 Bibliography

Introduction Fundamental Traffic Diagrams : A Maxplus Point of View N. Farhi, M. Goursat & J.-P. Quadrat Introduction LWR Equation Circular Road Height shape Road Extension Bibliography

Main Points • Following Daganzo we discuss the variational formulation

of the Lighthill-Witham-Richards equation describing the traffic on a road. • We extend it to the case of two roads with a junction with

the right priority. • The equation obtained is no more an HJB equation. To

study its eigenvalue we consider its space discretization for which we are able to derive analytically its eigenvalue as function of the car density. This function gives a good approximation of what we call the global fundamental traffic diagram.

Lighthill-Whitham-Richards Model Fundamental Traffic Diagrams : A Maxplus Point of View N. Farhi, M. Goursat & J.-P. Quadrat

The LWR Model expresses the mass conservation of cars : ( ∂t ρ + ∂x ϕ = 0 , ϕ = f (ρ),

Introduction

• ϕ(x, t) denotes the flow at time t and position x on the

LWR Equation

road. • ρ(x, t) denotes the density. • f (ρ) is a given function called the fundamental traffic diagram.

Circular Road Height shape Road Extension Bibliography

For traffic, this diagram plays the role of the gas law for the fluid dynamics. The diagram has been estimated using experimental data, and its behavior is quite different from standard gas at high density.

Variational Formulation of the LWR Model Fundamental Traffic Diagrams : A Maxplus Point of View N. Farhi, M. Goursat & J.-P. Quadrat Introduction LWR Equation Circular Road

Adapting Daganzo in [4]. • qxt denotes the cumulated number of vehicles having reached the point x at time t. • ϕ = ∂t q is the flow. • ρ = −∂x q + a is the car density where a is the initial density. If f is concave, it exists f ∗ convex : f (ρ) = inf u {−uρ + f ∗ (u)}. ( ( ϕ = f (ρ), ∂t q = inf u {u(∂x q − a) + f ∗ (u)}, ⇐⇒ ∂t ρ + ∂x ϕ = 0, ∂tx q = ∂xt q.

Height shape Road Extension Bibliography

Variational Formulation of the Traffic on a Line The LWR model is equivalent to a control model where the control is the drift term and the cost function is the dual of the opposite of the fundamental traffic diagram.

Global Fundamental Diagram Fundamental Traffic Diagrams : A Maxplus Point of View N. Farhi, M. Goursat & J.-P. Quadrat Introduction LWR Equation Circular Road Height shape Road

The HJB equation admits eigen-elements λ and rx satisfying : λ = inf {u∂x r + f ∗ (u))}, ∀x. u

This eigenvalue is equal to the average growth rate : qxT − qx0 , ∀x. T →+∞ T Moreover if the road (R) is closed there is car conservation in the system, that is, there exists a constant d (the global density): Z Z t d = 1/|R| (−∂x q + a)dx = 1/|R| adx, ∀t. χ=

R

lim

R

Extension Bibliography

Global Fundamental Traffic Diagram When it exists f¯ such that χ = f¯(d ) we call f¯ the global fundamental traffic diagram.

Circular Road Fundamental Traffic Diagrams : A Maxplus Point of View N. Farhi, M. Goursat & J.-P. Quadrat

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Introduction LWR Equation

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Figure: Petri net representation of the traffic on a circular roads.

Traffic Modeling on a Circular Road Fundamental Traffic Diagrams : A Maxplus Point of View N. Farhi, M. Goursat & J.-P. Quadrat Introduction LWR Equation Circular Road Height shape Road Extension

• The vehicle number entered in the section x before time t

is denoted qxt . • The initial vehicle quantity [resp. available place] in the

section [x, x + k] is ax k. [resp. ¯ax k with ¯ax = (1 − ax )]. • The index x being modulo 1, the dynamics is given by : t t qxt+h = min{ax−k k + qx−k , ¯ax k + qx+k },

The speed of cars being v = 1 and choosing k = h, with h → 0 we obtain : Circular Road Traffic Equation :

Bibliography

∂t q = min{−∂x q + a, ∂x q + ¯a}.

Fundamental Traffic Diagram Fundamental Traffic Diagrams : A Maxplus Point of View N. Farhi, M. Goursat & J.-P. Quadrat Introduction LWR Equation Circular Road Height shape Road Extension Bibliography

Since in the circular road case : ( 0 when u = −1, f ∗ (u) = 1 when u = 1, we have : Circular Road Fundamental Traffic Diagram: f (ρ) = min{ρ, 1 − ρ}.

Global Fundamental Traffic Diagram Fundamental Traffic Diagrams : A Maxplus Point of View N. Farhi, M. Goursat & J.-P. Quadrat Introduction LWR Equation Circular Road Height shape Road Extension Bibliography

Theorem The global fundamental traffic diagram of the circular road is : χ(d ) = min(d , 1 − d ), where d =

R1 0

ax dx is the average car density on the road.

Proof: There three way to make a circuit on the road using the two possible control {−1, 1} of the traffic HJB equation : • Always backward u = 1 with average cycle cost : R1 0 ax dx = d . • Always forward u = −1 with average cycle cost : R1 0 (1 − ax )dx = 1 − d . • Backward u = 1 during τ , forward during τ : R y +τ R y +τ 1/(2τ )[ y ax dx + y (1 − ax )dx] = 1/2. Then the result follows from the fact 1/2 ≥ min{d , 1 − d }.

A height shape road with right priority Fundamental Traffic Diagrams : A Maxplus Point of View N. Farhi, M. Goursat & J.-P. Quadrat Introduction LWR Equation Circular Road

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Figure: A road having the height shape cut in sections (top-right), its Petri net simplified modeling (middle) and the precise modeling of the junction (top left).

Conflict resolution using negative weights 1 Fundamental Traffic Diagrams : A Maxplus Point of View N. Farhi, M. Goursat & J.-P. Quadrat

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Introduction LWR Equation Circular Road Height shape Road Extension Bibliography

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Figure: The Petri net with conflict, given in the left figure, is made clear by giving top priority to q3 against q4 in the right figure.

Conflict resolution using negative weights 2 Fundamental Traffic Diagrams : A Maxplus Point of View N. Farhi, M. Goursat & J.-P. Quadrat Introduction LWR Equation Circular Road Height shape Road Extension Bibliography

The constraint expressed by the Peti net with conflict is : q4k q3k = aq1k−1 q2k−1 . clearly q3 and q4 are not defined uniquely. To define them uniquely we can clarify the dynamics by : Choose a priority rule (top priority to q3 against q4 ) q3k = aq1k−1 q2k−1 /q4k−1 ,

q4k = aq1k−1 q2k−1 /q3k .

Height Shape Road Dynamics Fundamental Traffic Diagrams : A Maxplus Point of View N. Farhi, M. Goursat & J.-P. Quadrat Introduction LWR Equation Circular Road Height shape Road Extension Bibliography

We precise the dynamics of the traffic system by giving the right priority to enter in the junction.  t+h t t  qx = min{kax−k + qx−k , k¯ax + qx+k }, x 6= z, 0 , t t qzt+h = min{k˜az + qkt + qz+k − q0t+h , kaz−k + qz−k },   t+h t t t t q0 = min{k˜a0 + qk + qz+k − qz , ka1−k + q1−k } , This dynamics is homogeneous of degree 1 but not monotone. When the car speed v = 1, taking k = h, with h → 0 we obtain the height shape road dynamics (HSRD) :  − +  ∂t q = min{a − ∂x q, ¯a + ∂x q}, ∀x 6= 0, z , (∂t q)z = min{az − (∂x− q)z , ˜az + (∂x+ q)0 + (∂x+ q)z − (∂t q)0 },   (∂t q)0 = min{a0 − (∂x− q)0 , ˜a0 + (∂x+ q)0 + (∂x+ q)z }, This equation is not an HJB equation since the monotony is lost.

The Eigenvalue Problem Fundamental Traffic Diagrams : A Maxplus Point of View N. Farhi, M. Goursat & J.-P. Quadrat Introduction LWR Equation Circular Road Height shape Road Extension Bibliography

We want solve the eigenvalue problem find λ real and r (x) such that :  − +  λ = min{a − ∂x r , ¯a + ∂x r }, ∀x 6= 0, z , λ = min{az − (∂x− r )z , ˜az + (∂x+ r )0 + (∂x+ r )z − λ},   λ = min{a0 − (∂x− r )0 , ˜a0 + (∂x+ r )0 + (∂x+ r )z }, In this case the growth rate is not equal to the eigenvalue but the eigenvalue gives good approximation of the growth rate for the discrete model.

Reducing the positive eigenvalue problem to an ergodic HJB equation Fundamental Traffic Diagrams : A Maxplus Point of View N. Farhi, M. Goursat & J.-P. Quadrat Introduction LWR Equation Circular Road Height shape Road Extension

In the traffic problem we have ˜az = ˜a0 = 1 − a0 − az since these two quantities mean the density of free place in the junction. Under the assumption that r is continuous at 0, if λ > 0 we have : λ ≤ ˜az + (∂x r )0 + (∂x r )z − λ < ˜az + (∂x r )0 + (∂x r )z which implies that the research of the positive eigenvalue, when r is regular in 0, is reduced to solve :   λ = min{a − ∂x r , ¯a + ∂x r }, ∀x 6= 0, z , λ = min{az − (∂x r )z , 1/3(¯az + (∂x r )z )},   λ = a0 − (∂x r )0 .

Bibliography

Since this equation is an HJB ergodic equation, looking at the circuit in this system we deduce λ = min{d , 1/4}.

Solving the Eigenvalue of the Discrete Model I Fundamental Traffic Diagrams : A Maxplus Point of View N. Farhi, M. Goursat & J.-P. Quadrat Introduction LWR Equation Circular Road Height shape Road

Using minplus notations we have to solve :   ¯ x rx+k , x 6= z, 0 , µrx = αx−k rx−k ⊕ α µrz = α ¯ rk rz+k /r0 µ ⊕ αz−k rz−k ,   µr0 = α ¯ rk rz+k /rz ⊕ α1−k r1−k , with k = h, µ = λk , α ¯ = (˜az )k = (˜a0 )k and αx = (ax )k . From µrz ≤ α ¯ rk rz+k /r0 µ, µrk ≤ α0 r0 , µrz+k ≤ αz rz ,

Extension Bibliography

we deduce λ ≤ 1/4.

Solving the Eigenvalue of the Discrete Model II Fundamental Traffic Diagrams : A Maxplus Point of View N. Farhi, M. Goursat & J.-P. Quadrat Introduction LWR Equation Circular Road Height shape Road Extension Bibliography

From the first equation using λ ≤ 1/4 we obtain : ( bx rk /λx−k ⊕ b¯x rz /λz−x if 0 < x < z, rx = cx rz+k /λx−z−k ⊕ c¯x r0 /λ1−x if z < x < 1, Q Q Q with bx = x−k αs , b¯x = z−k α ¯ s ,cx = x−k k x z+k αs , Q1−k c¯x = x α ¯s .   rz = α ¯ rk rz+k /r0 λ2k ⊕ bz rk /λz ,    r k ¯z+k r0 /λ1−z , z+k = αz rz /λ ⊕ c r0 = α ¯ rk rz+k /rz λk ⊕ c1 rz+k /λ1−z ,    r = α r /λk ⊕ b¯ r /λz , 0 0 k k z

Solving the Eigenvalue of the Discrete Model III Fundamental Traffic Diagrams : A Maxplus Point of View N. Farhi, M. Goursat & J.-P. Quadrat Introduction LWR Equation Circular Road Height shape Road Extension Bibliography

Searching for (λ > 0) using r0 rz ≤ α ¯ rk rz+k /λ2k < α ¯ rk rz+k /λk the system can be reduced to :   rz = (¯ αλ1−z−2k /c1 ⊕ bz /λz )rk ,    r k cz+k /λ1−z )r0 , z+k = (αz /λ )rz ⊕ (¯  r0 = (c1 /λ1−z )rz+k ,    r = (α /λk )r ⊕ (b¯ /λz )r , 0 0 z k k This system being linear in r we have to find the λ such that the circuits of minimal weight are 0.

Precedence graph Fundamental Traffic Diagrams : A Maxplus Point of View

r0

N. Farhi, M. Goursat & J.-P. Quadrat Introduction

rz rz+k

LWR Equation Circular Road Height shape Road

rk

Extension Bibliography

Figure: Precedence Graph of the Eigenvalue System.

Eigenvalue of the eight shape road Fundamental Traffic Diagrams : A Maxplus Point of View N. Farhi, M. Goursat & J.-P. Quadrat

There are three circuit : • (rk , rz , rk ) with weight : (¯ αλ1−z−2k /c1 ⊕ bz /λz )(b¯k /λz ). • (rz+k , r0 , rz+k ) with weight :

(¯ cz+k /λ1−z )(c1 /λ1−z ).

Introduction LWR Equation Circular Road

• (rk , rz , rz+k , r0 , rk ) with weight :

(¯ αλ1−z−2k /c1 ⊕ bz /λz )(αz /λk )(c1 /λ1−z )(α0 /λk ).

Height shape Road

Then passing to the limit k → 0, we obtain :

Extension

Positive eigenvalue of the eight shape road :

Bibliography

λ = min{d , 1/4, (z − d )/(2z − 1)}.

Global fundamental traffic diagram Fundamental Traffic Diagrams : A Maxplus Point of View N. Farhi, M. Goursat & J.-P. Quadrat

χ

Introduction LWR Equation

χ

Circular Road Height shape Road Extension Bibliography

Figure: Comparison of the global fundamental traffic diagram and the eigenvalue.

Extension to Regular Towns Fundamental Traffic Diagrams : A Maxplus Point of View N. Farhi, M. Goursat & J.-P. Quadrat

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Figure: Roads on a torus of 4 × 2 streets with its authorized turn at junctions (left) and the asymptotic car repartition in the streets on a torus of 4 × 4 streets obtained by simulation.

Light Controlled Fundamental Diagram Fundamental Traffic Diagrams : A Maxplus Point of View N. Farhi, M. Goursat & J.-P. Quadrat Introduction

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LWR Equation Circular Road Height shape Road

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Figure: Light policies comparison for a regular town on a torus. (1) right priority, (2) open loop, (3) local feedback, (4) global feedback obtained by LQ methods.

Fundamental Traffic Diagrams : A Maxplus Point of View N. Farhi, M. Goursat & J.-P. Quadrat Introduction LWR Equation

M. Akian, S. Gaubert, C. Walsh : The Max-Plus Martin Boundary, ArXiv.math/04124080v2, 2005. F. Baccelli, G. Cohen, G.J. Olsder, and J.P. Quadrat: Synchronization and Linearity, Wiley (1992). N. Farhi, M. Goursat, J.-P. Quadrat: Derivation of the fundamental traffic diagram for two circular roads and a crossing using minplus algebra and Petri net modeling, in Proceedings IEEE-CDC, 2005, Seville (2005).

Circular Road Height shape Road Extension

C. F. Daganzo, A variational formulation of kinematic waves: Basic theory and complex boundary conditions. Transportation Research part B, 39(2), 187-196, 2005.

Bibliography

C. F. Daganzo, N. Geroliminis, An analytical approximation for the macroscopic fundamental diagram of urban traffic. Transportation Research part B, 42(9), 771-781, 2008.

Fundamental Traffic Diagrams : A Maxplus Point of View N. Farhi, M. Goursat & J.-P. Quadrat Introduction LWR Equation Circular Road Height shape Road Extension Bibliography

N. Farhi, M. Goursat, J.-P. Quadrat: Fundamental Traffic Diagram of Elementary Road Networks algebra and Petri net modeling, in Proccedings ECC-2007, Kos, Dec. 2007. N. Farhi, M. Goursat, J.-P. Quadrat: About Dynamical Systems Appearing in the Microscopic Traffic Modeling, ArXiv 2009 : http://arxiv.org/abs/0911.4672. N. Farhi: Modélisation minplus et commande du trafic de villes régulière, thesis dissertation, University Paris 1 Panthéon - Sorbonne, 2008. M. Fukui, Y. Ishibashi: Phase Diagram for the traffic on Two One-dimensional Roads with a Crossing, Journal of the Physical Society of Japan, Vol. 65, N. 9, pp. 2793-2795, 1996. S. Gaubert and J. Gunawerdena: The Perron-Frobenius theorem for homogeneous monotone functions, Transacton of AMS, Vol. 356, N. 12, pp. 4931-4950, 2004. B. Hassenblatt and A. Katok: A first course in Dynamics, Cambridge University Press, 2003.

Fundamental Traffic Diagrams : A Maxplus Point of View N. Farhi, M. Goursat & J.-P. Quadrat Introduction LWR Equation Circular Road Height shape Road Extension Bibliography

J. Lighthill, J. B. Whitham: On kinetic waves: II) A theory of traffic Flow on long crowded roads, Proc. Royal Society A229 p. 281-345, 1955.