A multibuffer model for LWR road networks Mauro Garavello∗ Dipartimento di Scienze e Tecnologie Avanzate, Universit`a del Piemonte Orientale “A. Avogadro”, viale T. Michel 11, 15121 Alessandria (Italy). Benedetto Piccoli† Department of Mathematical Sciences and Center for Computational and Integrative Biology Rutgers University - Camden, 311 N 5th Street, Camden, NJ 08102, USA November 28, 2011

Abstract This paper introduces a new model for describing intersections in road networks, whose load dynamics is governed by the LighthillWhitham-Richards model. More precisely we define a solution for intersections using a multibuffer, i.e. a set of buffers, one for each outgoing road. We compare the obtained dynamics with those of some models previously introduced in the literature. In particular, we are able to respect the preferences of drivers and to not block the intersection when only one outgoing road is full. This improve some weaknesses of previous models.

Key Words: scalar conservation laws, traffic flow, Riemann solvers, multibuffer. AMS Subject Classifications: 90B20, 35L65. ∗

E-mail: [email protected]. Partially supported by Dipartimento di Matematica e Applicazioni, Universit`a di Milano Bicocca. † E-mail: [email protected].

1

1

Introduction

The modelling of traffic at a macroscopic level is nowadays a well established approach in the transportation engineering community, which has its roots in the fundamental model proposed independently by Lighthill, Whitham [19] and Richards [21]. Their work introduced to the traffic community the kinematic wave theory, which enables one to reconstruct macroscopic features of traffic flow, in particular tracing backward queues propagation. The model, consisting in a single conservation law for car density, is referred to as LWR model and is based on expressing the average velocity as function only of the car density. Greenshields [13] empirically measured a relation between the density and the flow of vehicles, now known as the fundamental diagram. There is a wide literature of studies on the fundamental diagram, for a review see [8]. The numerics for such model was addressed in various papers, see Lebacque [18]. The supply demand approach there proposed is equivalent to the classical Godunov numerical scheme for general conservation laws [10]. Notice also that such approach is intimately related to the cell transmission model of Daganzo [4], which can be seen as a discretization of the LWR model. More recently, a growing attention was devoted to extensions of the same model to networks; see for instance [2, 3, 7, 14, 17]. The interest was also motivated by other applications: data networks [6], supply chain [5, 11], air traffic management [22]. Here we focus on the LWR model on a network, but the results are of use to other research domains. To define a dynamics on the whole network, one first considers Riemann problems at nodes, which are Cauchy problems with constant initial data on each arc. Notice that the only conservation of cars is not sufficient to determine a unique dynamics. Thus one has to prescribe solutions for every initial data and we call the relative map a Riemann solver at nodes. Then it is possible to construct approximate solutions, via wave front tracking (see [9]), using classical self-similar entropic solutions for Riemann problems inside arcs and an assigned Riemann solver at nodes. Various ways to define solutions at intersections were proposed in the literature; see for instance [2, 3, 8, 16, 20]. This paper introduces a new model for describing dynamics at junctions in road networks. Due to finite speed of waves, we can reduce to the case of a simple road network, composed by a single junction with an arbitrary number of incoming and outgoing roads. On each road, the evolution of the car traffic is governed by the LWR model. In the same spirit as [12, 15, 16], we suppose that, inside the crossroad, there are some buffers, with finite size. More precisely, we assume that there 2

is a buffer in front of each outgoing road, so that the number of buffers equals that of outgoing roads. The basic idea behind this construction is that a car exiting an incoming road enters the buffer associated to its desired destination and then it passes to the corresponding outgoing road by a FIFO policy. Since the number of buffers are equal to the number of outgoing roads, our model is able to capture the preferences of drivers. On the contrary, the model proposed in [16] contains only one buffer inside the junction and so all the cars enter the same buffer loosing the information about their origins and destinations. After introducing in Section 2 the basic assumptions and definitions about conservation laws, we recall the construction of the Riemann solver introduced in [3]. The latter is defined for a model without buffers and makes use of a traffic distribution matrix together with maximization of the through flux. In Section 4, we describe in detail the Riemann solver with multibuffer. A simple example is then given, which illustrates the main differences of our approach with respect to those introduced in [3, 16]. Moreover we provide an analytic comparison of our model with an ODE-PDE ones defined in [15] for supply chains and networks and with that of [16]. The paper ends with Section 6, which contains the conclusions.

2

Basic definitions and notations

Consider a junction J with n incoming roads I1 , . . . , In and m outgoing roads In+1 , . . . , In+m . We model each incoming road Ii (i ∈ {1, . . . , n}) of the junction with the real interval Ii =] − ∞, 0]. Similarly we model each outgoing road Ij (j ∈ {n + 1, . . . , n + m}) of the junction with the real interval Ij = [0, +∞[. On each road Il (l ∈ {1, . . . , n + m}) we consider the partial differential equation (ρl )t + f (ρl )x = 0, (1) where ρl = ρl (t, x) ∈ [0, ρmax ], is the density of cars, vl = vl (ρl ) is the velocity of cars and f (ρl ) = vl (ρl ) ρl is the flux. Hence the datum is given by a finite collection of functions ρl defined on [0, +∞[×Il . For simplicity, we put ρmax = 1. On the flux f we make the following assumption (F ) f : [0, 1] → R is piecewise smooth, concave (i.e. almost everywhere f ′′ ≤ 0), f (0) = f (1) = 0 and there exists a unique a point of maximum σ ∈]0, 1[. Definition 2.1 A function ρl ∈ C([0, +∞[; L1loc (Il )) is an entropy-admissible solution to (1) in the road Il if the following holds. 3

1. For every function ϕ : [0, +∞[×Il → R smooth with compact support in ]0, +∞[× (Il \ {0}) Z +∞ Z  ∂ϕ ∂ϕ  ρl dxdt = 0. (2) + f (ρl ) ∂t ∂x 0 Il 2. For every k ∈ R and every ϕ˜ : [0, +∞[×Il → R smooth, positive with compact support in ]0, +∞[× (Il \ {0}) Z +∞ Z  ∂ ϕ˜ ∂ ϕ˜  |ρl − k| dxdt ≥ 0. (3) + sgn(ρl − k)(f (ρi ) − f (k)) ∂t ∂x 0 Il We now want to describe preferences of drivers. This is done defining a traffic distribution matrix, whose coefficients represents percentages of incoming fluxes which distribute to each outgoing road. Consider the set   0 < αji < 1 ∀i, j,   n+m P A := A = {αji} i=1,...,n : . (4) αji = 1 ∀i   j=n+1,...,n+m j=n+1

Here the coefficient αji indicates the portion of cars coming from incoming road Ii which goes to outgoing road Ij . Let {e1 , . . . , en } be the canonical basis of Rn . For every i = 1, . . . , n, we denote Hi = {ei }⊥ . If A ∈ A, then we write, for every j = n + 1, . . . , n + m, αj = (αj1 , . . . , αjn ) ∈ Rn and Hj = {αj }⊥ . Let K be the set of indices k = (k1 , ..., kℓ ), 1 ≤ ℓ ≤ n − 1, such that 0 ≤ k1 < k2 < · · · < kℓ ≤ n + m and for every k ∈ K define Hk =

ℓ \

Hk h .

h=1

Writing 1 = (1, . . . , 1) ∈ Rn and following [3] we define the set  N := A ∈ A : 1 ∈ / Hk⊥ for every k ∈ K .

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Notice that, if n ≥ m, then N = ∅. The matrices of N will give a unique solution to the Riemann problem at J. Remark 1 If n ≥ m, or more generally A ∈ / N, one can resort to right of way parameters to determine a unique solution. The construction is similar to the one used later in Section 4. For a detailed description we refer the reader to [2]. An alternative approach, for incoming roads which share junction area but not necessarily lead to same outgoing roads as it happens in T -junctions, we refer to [20]. Finally, for traffic data with source-destination patterns, a complete theory is available in [7]. 4

Define also the set: ( P=

P = {pji }

i=1,...,n j=n+1,...,n+m

: 0 < pji < 1 ∀i, j,

n X

pji = 1 ∀j

i=1

)

.

A matrix P ∈ P represents priority coefficients among incoming roads to enter each outgoing road.

3

The Riemann Problem at J without buffers

In this section, we recall the concept of Riemann problem at the junction and the solution, proposed for traffic, in [3]. Fix ρ1,0 , . . . , ρn+m,0 ∈ [0, 1], then the corresponding Riemann problem at J is given by: ( ∂ ∂ ρ + ∂x f (ρl ) = 0, ∂t l l ∈ {1, . . . , n + m}, (6) ρl (0, ·) = ρl,0 , namely the Cauchy problem with initial data constant on each road. To define the dynamics on the whole network, we need to determine solutions at junctions. In particular, we want to describe the solution to Riemann problems at J. This is achieved by describing solutions via a map which to initial conditions associates boundary data for all roads of the junction. More precisely, we define: Definition 3.1 A Riemann solver RS is a function RS :

[0, 1]n+m −→ [0, 1]n+m (ρ1,0 , . . . , ρn+m,0 ) 7−→ (¯ ρ1 , . . . , ρ¯n+m )

satisfying 1. for every i ∈ {1, . . . , n}, the classical Riemann problem  x ∈ R, t > 0,   ρt + f (ρ)x = 0,  ρi,0 , if x < 0,   ρ(0, x) = ρ¯ , if x > 0, i is solved with waves with negative speed;

2. for every j ∈ {n + 1, . . . , n + m}, the classical Riemann problem  x ∈ R, t > 0,   ρt + f (ρ)x = 0,  ρ¯j , if x < 0,   ρ(0, x) = ρ , if x > 0, j,0 is solved with waves with positive speed; 5

3.

Pn

ρi ) = i=1 f (¯

Pn+m

ρj ). j=n+1 f (¯

Remark 2 In the above definition the first two conditions ensure that boundary value problems on each road are solved in a strong sense. This means that weak solutions will indeed achieve the prescribed boundary value as a trace. This is not the case for general weak solutions to boundary value problems, see for instance [1]. Condition 3. then guarantees conservation of cars through the junction, imposing equality between total incoming and outgoing fluxes. We need another property to ensure that a Riemann solver is well defined. Indeed, it may happen that a value attained by RS is not a fixed point for RS itself. Therefore, one may need to reapply RS thus not giving rise to a well defined procedure. We then define: Definition 3.2 We say that a Riemann solver RS satisfies the consistency condition if, for every (ρ1 , . . . , ρn+m ) ∈ [0, 1]n+m, then RS (RS(ρ1 , . . . , ρn+m )) = RS(ρ1 , . . . , ρn+m ). We are now ready to give the definition of solution at the junction: Definition 3.3 Given a Riemann solver RS, a solution to the Riemann problem (6) is a collection of functions (ρ1 , . . . , ρn+m ) such that 1. for every l ∈ {1, . . . , n + m}, the function ρl is an entropy-admissible solution to (1) in the road Il , in the sense of Definition 2.1; 2. for every l ∈ {1, . . . , n+m} and for a.e. t > 0, the function x 7→ ρl (t, x) has a version with bounded total variation; 3. for every l ∈ {1, . . . , n + m}, ρl (0, x) = ρl,0 for a.e. x ∈ Il ; 4. for a.e. t > 0, it holds RS (ρ1 (t, 0−), . . . , ρn+m (t, 0+)) = (ρ1 (t, 0−), . . . , ρn+m (t, 0+)) . There are some general properties which hold for all Riemann solvers. To describe the latter, introduce the following sets 1. for every i ∈ {1, . . . , n} define ( [0, f (ρi,0 )], Ωi = [0, f (σ)], 6

if 0 ≤ ρi,0 ≤ σ, if σ ≤ ρi,0 ≤ 1;

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2. for every j ∈ {n + 1, . . . , n + m} define ( [0, f (σ)], if 0 ≤ ρj,0 ≤ σ, Ωj = [0, f (ρj,0)], if σ ≤ ρj,0 ≤ 1;

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3. for every l ∈ {1, . . . , n + m} define γlmax = max Ωl .

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Proposition 3.1 The following statements hold. 1. For every i ∈ {1, . . . , n}, an element γ¯ belongs to Ωi if and only if there exists ρ¯i ∈ [0, 1] such that f (¯ ρi ) = γ¯ and point 1 of Definition 3.1 is satisfied. 2. For every j ∈ {n + 1, . . . , n + m}, an element γ¯ belongs to Ωj if and only if there exists ρ¯j ∈ [0, 1] such that f (¯ ρj ) = γ¯ and point 2 of Definition 3.1 is satisfied. The proof is trivial and hence omitted. Here we recall the construction of the Riemann solver, introduced for traffic in [3]. For simplicity in this paper we denote it with the symbol RS CGP . 1. Fix a matrix A ∈ N and consider the closed, convex and not empty set ( ) n n+m Y Y Ω = (γ1 , · · · , γn ) ∈ Ωi : A · (γ1 , · · · , γn )T ∈ Ωj . (10) i=1

j=n+1

2. Find the point (¯ γ1 , . . . , γ¯n ) ∈ Ω which maximizes the function E(γ1 , . . . , γn ) = γ1 + · · · + γn ,

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and define (¯ γn+1 , . . . , γ¯n+m )T := A · (¯ γ1 , . . . , γ¯n )T . Since A ∈ N, the point (¯ γ1 , . . . , γ¯n ) is uniquely defined. 3. For every i ∈ {1, . . . , n}, set ρ¯i either by ρi,0 if f (ρi,0 ) = γ¯i , or by the solution to f (ρ) = γ¯i such that ρ¯i ≥ σ. For every j ∈ {n+1, . . . , n+m}, set ρ¯j either by ρj,0 if f (ρj,0) = γ¯j , or by the solution to f (ρ) = γ¯j such that ρ¯j ≤ σ. Finally, define RS CGP : [0, 1]n+m → [0, 1]n+m by RS CGP (ρ1,0 , . . . , ρn+m,0 ) = (¯ ρ1 , . . . , ρ¯n , ρ¯n+1 , . . . , ρ¯n+m ) . 7

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γ2 3 γ 4 1

+ 31 γ2 =

3 4

1

Ω

γ1

3 4

1

Figure 1: The set Ω of Example 1. Let us now illustrate with an example the Riemann solver RS CGP in the case of two incoming and two outgoing roads. Example 1 Let J be a junction with two incoming roads I1 and I2 and two outgoing ones I3 and I4 . Fix a distribution matrix A ∈ N such that α31 = 1/4, α32 = 2/3, α41 = 3/4 and α42 = 1/3. Let us assume that the flux f (ρ) is equal to 4ρ(1 − ρ) and the initial conditions for the Riemann problem (6) are given by 1 ρ1,0 = , 4 We easily deduce that   3 , Ω1 = 0, 4

1 ρ3,0 = , 2

ρ2,0 = 1,

Ω2 = [0, 1] ,

Ω3 = [0, 1] ,

3 ρ4,0 = . 4 

3 Ω4 = 0, 4



and so     1 2 3 1 3 3 × [0, 1] : 0 ≤ γ1 + γ2 ≤ 1, 0 ≤ γ1 + γ2 ≤ ; Ω = (γ1 , γ2) ∈ 0, 4 4 3 4 3 4 see Figure 1. Therefore the point of maximum in Ω for the function E, defined

in (11), is given by (¯ γ1 , γ¯2 ) = 95 , 1 and consequently (¯ γ3 , γ¯4 ) = 29 ,3 . 36 4 Finally we have r r 1 1 1 5 1 1 29 3 ρ¯1 = + − , ρ¯2 = , ρ¯3 = + − , ρ¯4 = . 2 4 36 2 2 4 144 4 8

I4 I1

I2

I5 111 000 000 111 000 111 000 111 0000 1111 000 111 0000 1111 00000 11111 000 111 0000 1111 00000 11111 000 111 000 111 0000 1111 00000 11111 000 111 00000 11111 000 111 00000 11111 000 111 000 111 000 111

I6 I3

I7 Figure 2: A junction with multibuffer.

4

Riemann solver with multibuffer

In this section we introduce a new way to solve the Riemann problem at the junction J. We imagine that the junction J is composed by n incoming roads, m outgoing roads and m different buffers in front of each outgoing road; see Figure 2. Let us index buffers by j ∈ {n + 1, . . . , n + m}. Then a function rj (t), which represents the load of the j-th buffer at time t > 0, is associated to each for every j ∈ {n + 1, . . . , n + m}, the P buffer. Moreover, max numbers µj > i αjif (σ) and rj > 0 denote, respectively, the maximum number of cars, which can enter or exit the j-th buffer per unit of time, and the maximum number of cars which can be stored in the j-th buffer; hence we deduce the constraints 0 ≤ rj (t) ≤ rjmax for every j ∈ {n + 1, . . . , n + m}. P Remark 3 Notice that the assumption µj > i αji f (σ) means that buffers have large capacities w.r.t. maximal road flux. This is a necessary assumption to obtain a consistent solution to Riemann problems at junctions. Such situation is illustrated in Example 2. If one of the buffers is full, then we impose that no car passes through the junction. If instead all the buffers are not full, then every car enters in the buffer associated to the desired destination and, finally, it enter in the outgoing road by FIFO policy. Remark 4 Note that if one of the buffers is full, then it is necessary not allow cars to cross J, otherwise the preferences of the drivers may not be satisfied. Indeed every buffer should receive a non zero percentage of the flux entering the junction J. However, if some buffer is full, then it can not receive any car; therefore the only way to respect the constraints, imposed by the matrix A, is to block the intersection. 9

Let A ∈ N, P ∈ P be the matrices of the preferences of drivers and of priority coefficients among incoming roads. We define a Riemann solver with multibuffer Qn+m  max  0, rj RS : [0, 1]n+m × j=n+1 −→ [0, 1]n+m × [0, n f (σ)]m
in in (ρ1,0 , . . . , ρn+m,0 , rn+1 , . . . , rn+m ) 7−→ ρ¯1 , . . . , ρ¯n+m , fn+1 , . . . , fn+m (13) as follows. 1. For every i ∈ {1, . . . , n} and j ∈ {n + 1, . . . , n + m}, define max γi,j = αji γimax ,

i.e. the maximum flux, which can exit from Ii and enter in Ij . 2. For every j ∈ {n + 1, . . . , n + m}, define ( n ) X max fˆjin = min γi,j , µj and i=1

 fˆjout = min γjmax , µj ,

respectively the maximum flux which can enter in the j-th buffer and which can enter in the road Ij .  max n+m 3. If minj=n+1 rj − rj > 0 (i.e. no buffer is full), then define the convex set: ) ( n X max kji = fˆjin ∀j , Rn×m ⊃ K = {kji} i=1,...,n : 0 ≤ kji ≤ γi,j , j=n+1,...,n+m

i=1

˜ ∈ Rn×m by setting q˜ji = pjifˆjin . Let Q = projK (Q) ˜ ∈ and the point Q K, where projK is the orthogonal projection over the convex set K. Finally, for every i ∈ {1, . . . , n}, we set γ¯i =

n+m X

qji ,

j=n+1

where qji are the components of Q and represent the number of cars going from Ii to the j-th buffer.  max n+m 4. If minj=n+1 rj − rj = 0 (i.e. at least one buffer is full), we set γ¯i = 0 for every i ∈ {1, . . . , n}.

10

5. For every j ∈ {n + 1, . . . , n + m}, we set ( fˆjout , if rj > 0, γ¯j = in ˆout ˆ min{fj , fj }, if rj = 0. 6. For every j ∈ {n + 1, . . . , n + m}, define ( Pn  max n+m if minj=n+1 rj − rj > 0, i=1 qji in fj =  max n+m 0 if minj=n+1 rj − rj = 0.

7. For every i ∈ {1, . . . , n}, define ρ¯i either by ρi,0 if f (ρi,0 ) = γ¯i , or by the solution to f (ρ) = γ¯i such that ρ¯i ≥ σ. 8. For every j ∈ {n + 1, . . . , n + m}, define ρ¯j either by ρj,0 if f (ρj,0 ) = γ¯j , or by the solution to f (ρ) = γ¯j such that ρ¯j ≤ σ. 9. Define in in RS(ρ1,0 , . . . , ρn+m,0 , rn+1 , . . . , rn+m ) = (¯ ρ1 , . . . , ρ¯n+m , fn+1 , . . . , fn+m ).

Remark 5 Let us comment on the various steps to define the Riemann solver with multibuffer. Steps 1. and 2. define maximal fluxes from incoming roads to buffers and from buffers to outgoing roads. In Step 3., if all buffers are not full, we define fluxes from incoming roads by projecting a vector representing priorities over the set of admissible fluxes. If, on the contrary, at least one buffer is full, then in Step 4. we simply set all fluxes from incoming roads to vanish. Then Step 5. determine fluxes from buffers to outgoing roads, which values depend on the status of the buffer: empty or non empty. Step 6. defines fluxes entering buffers. Finally, Steps 7. and 8. describe how to compute the boundary values both for incoming and outgoing roads. Example 2 Consider a junction with one incoming road I1 and two outgoing roads I2 and I3 and distribution matrix (α21 , α31 ). For initial conditions we assume 0 < ρ1,0 = ρ2,0 = ρ3,0 < σ and empty buffers, while capacities satisfy µ2 < α21 f (ρ1,0 ) while µ3 > f (σ). Applying the Riemann solver with multibuffer we get: fˆ2in = µ2 ,

fˆ3in = α31 f (ρ1,0 ),

thus RS(ρ1,0 , ρ2,0 , ρ3,0 , 0, 0) = (¯ ρ1 , ρ¯2 , ρ¯3 , µ2 , α31 f (ρ1,0 )), 11

where f (¯ ρ1 ) = µ2 + α31 f (ρ1,0 ), f (¯ ρ2 ) = µ2 , f (¯ ρ3 ) = α31 f (ρ1,0 ). Now if we apply RS to the initial conditions (¯ ρ1 , ρ¯2 , ρ¯3 , 0, 0) we get RS(¯ ρ1 , ρ¯2 , ρ¯3 , 0, 0) = (ˆ ρ1 , ρ¯2 , ρˆ3 , µ2, α31 f (σ)), where f (ˆ ρ1 ) = µ2 + α31 f (σ), f (ˆ ρ3 ) = α31 f (σ). This proves that if µj is below P i αji f (σ), then the Riemann solver lacks of consistency. We give the following definition of solution to Riemann problems with multibuffer.

Definition 4.1 Let rn+1,0 , . . . , rn+m,0 be the initial loads of the buffers. A solution to the Riemann problem (6) with multibuffer is given by
Qn+m 1 (ρ1 , . . . , ρn+m ) ∈ C [0, +∞[; l=1 Lloc (Il ) (rn+1 , . . . , rn+m ) ∈ W 1,∞ ([0, +∞[; Rm )

such that 1. for every l ∈ {1, . . . , n + m}, ρl is an entropy-admissible solution to (1) on Ij and, for a.e. t > 0, ρl (t, ·) has a version with finite total variation; 2. for every l ∈ {1, . . . , n + m}, ρl (0, x) = ρ0,l for a.e. x ∈ Il ; 3. for a.e. t > 0

and

RS (ρ1 (t, 0), . . . , ρn+m (t, 0), rn+1 (t), . . . , rn+m (t))
in in = ρ1 (t, 0), . . . , ρn+m (t, 0), fn+1 (t), . . . , fn+m (t) rj (t) = rj,0 +

Z

0

t


fjin (s) − f (¯ ρj (t, 0)) ds

for every j ∈ {n + 1, . . . , n + m}.

4.1

Model justification

In this Subsection, we propose an example in order to underline the different behavior of the following Riemann solvers: the one introduced in [3], the Riemann solver with buffer proposed by Herty, Lebacque and Moutari [16] and the Riemann solver with multibuffer considered in this paper. Let us fix a junction J with a single incoming road I1 and two outgoing roads I2 and I3 . Consider the Riemann problem (6) with ρ1,0 = σ, ρ2,0 = 1 and σ < ρ3,0 < 1. We also assume that buffers are empty (if present). 12

t

I1

I2

t

1

¯1t x=λ 1

I3

t σ

0

x

x

¯2t x=λ ρ3,0 x

Figure 3: The solutions to the Riemann solver RS CGP in I1 , I2 and I3 . The Riemann solver RS CGP introduced in [3]. It is simple to see that RS CGP (ρ1,0 , ρ2,0 , ρ3,0 ) = (1, 1, 0) independently from the distribution matrix at the junction. So the solution to (6), with respect to the Riemann solver RS CGP , is given by the triple (ρ1 , ρ2 , ρ3 ) defined by ( ¯ 1 t, σ, if x < λ ρ1 (t, x) = ¯ 1 t < x < 0, 1, if λ ρ2 (t, x) = 1 ( ρ3 (t, x) =

0,

if

ρ3,0 ,

if

¯2 t 0λ

¯ 2 = 3,0 ; see Figure 3. This example shows that, ¯ 1 = − f (σ) and λ where λ 1−σ ρ3,0 if an outgoing road is full, then no car crosses the junction. This implies that the road I3 empties and in the incoming road I1 it appears a shock with negative speed, connecting σ with the maximum possible density. It is questionable that the model does not allow any car going to the empty road I3 . f (ρ

)

The Riemann solver with buffer of [16]. 13

t

I1

t

I2

ρ3,0

¯ 1 (t − t¯) x=λ

1

σ

x r

t

I3

x

ρ3,0 t 0

x

t¯

Figure 4: The solutions to the Riemann solver of [16] in I1 , I2 and I3 and the load of the buffer. In this part we describe the solution to the same Riemann problem for a junction with a buffer, as introduced in [16]. For the complete description of this Riemann solver see [16]. Assume that the capacity of the buffer µ is greater than or equal to f (σ). rmax For 0 < t < t¯ (t¯ = f (σ)−f ) the solution is given by (ρ3,0 ) ρ1 (t, x) = σ ρ2 (t, x) = 1 ρ3 (t, x) = ρ3,0 r(t) = [f (σ) − f (ρ3,0 )] t while, if t > t¯, ρ1 (t, x) =

(

σ,

if

ρ3,0 ,

if

¯ 1 (t − t¯), x<λ ¯ 1 (t − t¯) < x < 0, λ

ρ2 (t, x) = 1 ρ3 (t, x) = ρ3,0 r(t) = rmax ; see Figure 4. Note that in this case the queue in the buffer increases and when it reaches the maximum value, then a shock with negative speed appears in I1 and connects the states σ and ρ3,0 . 14

Notice that all cars will finally reach road ρ3,0 for every time. Therefore, the presence of a unique buffer erases the original will of drivers. This phenomenon is a drawback from modelling point of view.

The Riemann solver with multibuffer. Assume that µ2 = µ3 ≥ f (σ). Call (α21 , α31 ) the traffic distribution matrix. r max For simplicity, we further assume that α31 f (σ) < f (ρ3,0 ). Define t¯ = α212 f (σ) . Then, for 0 < t < t¯ the solution is given by ρ1 (t, x) = σ ρ2 (t, x) = 1 ( ρ3 (t, x) =

ρ3,0 ,

if

¯3t x>λ

ρ¯3 ,

if

¯3t 0
r2 (t) = α1,2 f (σ) t r3 (t) = 0 ¯ 3 = f (ρ3,0 )−f (¯ρ3 ) . For t > t¯ the solution where ρ¯3 < σ, f (¯ ρ3 ) = α31 f (σ) and λ ρ3,0 −¯ ρ3 is given by ( ¯ 1 (t − t¯) σ, if x ≤ λ ρ1 (t, x) = ¯ 1 (t − t¯) < x ≤ 0 1, if λ ρ2 (t, x) = 1   0,   ρ3 (t, x) = ρ¯3 ,    ρ3,0 ,

if

˜ 3 (t − t¯) 0≤x<λ

if

˜ 3 (t − t¯) ≤ x < λ ¯3t λ ¯3t < x λ

if

r2 (t) = r2max r3 (t) = 0

¯ 1 = − f (σ) and λ ˜ 3 = f (¯ρ3 ) ; see Figures 5 and 6. where λ 1−σ ρ¯3 Notice that cars flow to road I3 meanwhile the buffer of road I2 is not yet full, then they stop. Finally, the situation is intermediate between the two above solvers. More precisely, the flow is neither stopped immediately nor continue for all time. Moreover, cars going through the junction are not redirected, but travel towards the outgoing roads or buffers accordingly to drivers’ preferences, expressed by the traffic distribution matrix.

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t

I1 ¯ 1 (t − t¯) x=λ

t

I2 1

1 t σ

I3

x

0

x

ρ¯3

ρ3,0 x

Figure 5: The solutions to the Riemann solver with multibuffer in I1 , I2 and I3 .

r2

r3

t

t t¯

Figure 6: The solutions to the Riemann solver with multibuffer for the buffers r2 and r3 .

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5

Comparison with other models with buffers

The idea of using roads, or in general arcs, with buffers in front was used by various authors. Here we compare our model with those used in recent literature. In [12], a coupled ODE-PDE model for supply chains (and networks) was proposed. It consists of arcs with dynamics described by a conservation law for part density and ODEs for buffers in front of each arc. More precisely the conservation law is of the type: ρt + (min{vρ, µ})x = 0, where ρ is the part density, v the constant velocity and µ the processing rate. Notice that such model produces waves having only positive speed. Thus, the evolution of part density affects the network only in the forward direction. The ODEs for buffers are of the type: r˙ = f in − f out

(14)

where f in is the flux entering the buffer from the previous arc and f out is the flux exiting the buffer to the next arc. f in depends only on the density of the previous arc and is independent from the buffer status, thus the buffers is necessarily with infinite size. f out is defined similarly to our case, namely it is equal to f in , if the buffer is empty (and f in is below the processing capacity of next arc), otherwise it is equal to the processing capacity of next arc. Summarizing the main differences with our model are the following: • The conservation laws admits only waves with positive velocity, so no backward effect is possible. • Consequently fluxes entering buffers can not depend on buffer status, therefore buffers are necessarily of infinite size. In [16], authors propose a model for vehicular traffic, which considers junctions with an arbitrary number of incoming and outgoing roads and a buffer in between. Conservation laws are of the same type we considered here. On the other side, the equation for the buffer is of the type (14). The buffer has limited size, thus fluxes from incoming roads will stop when the buffer is full. If the buffer is not full, then fluxes from incoming roads enter the buffers, possibly limited by a maximal processing rate. Finally, the flux exiting the buffer distribute over outgoing roads according to traffic distribution coefficients. The main differences with our model are the following: 17

• There is a unique buffer for all outgoing roads, opposed to our multibuffer. • The traffic from incoming road is stopped only when the common buffer is full, while in our case a single full buffer will stop the traffic. • The traffic distribution coefficients do not depend on incoming roads. Indeed a traffic distribution matrix can not be respected as shown by Example of Section 4.1.

6

Conclusions

We have proposed a new way for describing dynamics at intersections in road networks, when car density evolution is governed by the Lighthill-WhithamRichards model. Due to finite speed of waves in the LWR model, we can focus on a single junction. We supposed that a buffer is attached in front of each outgoing road of the junction, and we completely described the dynamics inside the buffers and between roads and buffers by means of a Riemann solver with multibuffer. Moreover we provided examples and analytical comparisons between our approach and some previously introduced in the literature.

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