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Piecewise linear car-following modeling
Transportation Research Part C 25 (2012) 100–112
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Transportation Research Part C journal homepage: www.elsevier.com/locate/trc
Piecewise linear car-following modeling Nadir Farhi Université Paris-Est, IFSTTAR, GRETTIA, F-93166 Noisy-le-Grand, France
a r t i c l e
i n f o
Article history: Received 18 August 2011 Received in revised form 15 May 2012 Accepted 16 May 2012
Keywords: Car-following modeling Optimal control Variational formulations
a b s t r a c t We present a traffic model that extends the linear car-following model as well as the minplus traffic model (a model based on the min-plus algebra). A discrete-time car-dynamics describing the traffic on a 1-lane road without passing is interpreted as a dynamic programming equation of a stochastic optimal control problem of a Markov chain. This variational formulation permits to characterize the stability of the car-dynamics and to determine the stationary regimes when they exist. The model is based on a piecewise linear approximation of the fundamental traffic diagram. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Car-following models are microscopic traffic models that describe the car-dynamics with stimulus-response equations expressing the drivers’ behavior. Each driver responds, by choosing its speed or acceleration, to a given stimulus that can be composed of many factors such as inter-vehicular distances, relative velocities, instantaneous velocities, etc. We present in this article a car-following model that extends the linear car-following model (Chandler et al., 1958; Herman et al., 1959; Gazis et al., 1959, 1961), as well as the min-plus traffic model (Lotito et al., 2005). The vehicular traffic on a 1-lane road without passing is described by first order discrete-time dynamics. The latter guarantee a conservation law of cars and express a given behavior law of drivers. The model presented here is very flexible in such a way that a large class of driver behavior law curves can be included. Moreover, the traffic dynamics are interpreted in term of optimal control. More precisely, the dynamics are seen as dynamic programming equations (DPEs) of stochastic optimal control problems of Markov chains, where the car position is interpreted as the value function of the DPE. This interpretation is important because it permits to understand the effect of the individual driver behavior on the collective traffic. The results we obtain here can be summarized as follows. We distinguished the case of traffic on a ring road from that of traffic on an open road. For the former case, where a given number of cars, having the same behavior (using the same behavioral law) move on a ring road, we first characterize the stability conditions of the car dynamics. We then solve analytically the algebraic system describing the stationary regime. From that, we obtain the mean speed, as well as the stationary distribution of cars on the road. Moreover, the mean speed is derived as a function of the mean car density on the road, giving thus the fundamental traffic diagram. For the case of traffic on an open road, we follow the trajectories of a given number of cars on an open road. The cars are supposed to have the same behavior. We follow the same steps as in the first case, but here, in order to study the stationary regime, we assume that the velocity of the first car tends to a fixed value. We solve analytically the algebraic system describing the stationary regime, and derive the local car density as a function of the stationary first car velocity. The latter relationship is nothing but the inverse of the fundamental diagram obtained in the ring road case.
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N. Farhi / Transportation Research Part C 25 (2012) 100–112
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We are concerned in this article by microscopic traffic modeling with a Lagrangian description of the traffic dynamics, where the function xðn; tÞ, giving the position of car n at time t (or the cumulated distance traveled by a car n up to time t), is used. The discrete-time variational formulation made here is similar to the time-continuous one used by Daganzo and Geroliminis (2008) to show the existence of a concave macroscopic fundamental diagram. In the macroscopic kinematic traffic modeling, Eulerian descriptions of the traffic dynamics are usually used, with the function nðt; xÞ giving the cumulated number of cars passing through position x up to time t (which coincides with the Moscowitz function (Daganzo, 2006) in the case of traffic without passing). The combination of a conservation law with an equilibrium law gives the well known first order traffic model of Lighthill and Whitham (1955) and Richards (1956). In this introduction, we first notice that the same approach used in macroscopic traffic modeling, combining a conservation law with an equilibrium law, can be used to derive microscopic traffic models. By this, we introduce car-following models and give a review on basic and most important existing ones. In particular, we recall the linear and the min-plus car-following models, which are particular cases of the model we present here. Finally, we recall a theoretical result on non-expansive and connected dynamic systems which we need in our developments, and present the outline of this article. _ The first order partial derivative of the function xðn; tÞ in time, denoted xðn; tÞ expresses the velocity v ðn; tÞ of car n at time t. The first order differentiation of xðn; tÞ in the car numbers Dxðn; tÞ expresses the inverse of the inter-vehicular distance _ yðn; tÞ ¼ xðn 1; tÞ xðn; tÞ between cars n and n 1 at time t. The equality of the second derivatives Dxðn; tÞ and D_ xðn; tÞ gives then the following conservation law of distance.
_ yðn; tÞ ¼ Dv ðn; tÞ:
ð1Þ
If we assume that a fundamental diagram V e , giving the velocity v as a function of the inter-vehicular distance yðv ¼ V e ðyÞÞ at the stationary traffic, exists, and that the diagram V e holds also on the transient traffic, then we have
_ yðn; tÞ ¼ v_ ðn; tÞ=V 0e ðyÞ;
ð2Þ
V 0e ðyÞ
where denotes the derivative of V e with respect to y. By combining (1) and (2) we obtain the model:
v_ ðn; tÞ ¼ V 0e ðyðn; tÞÞDv ðn; tÞ:
ð3Þ
(3) is a car-following model that gives the acceleration of car n at time t as a response to a stimulus composed of the relative speed Dv ðn; tÞ and the term V 0e ðyðn; tÞÞ. For example, if V e ðyÞ ¼ v 0 exp a=y, where v 0 denotes the free (or desired) velocity, and a is a parameter, then V 0 ðyÞ ¼ aVðyÞ=y2 and (3) gives a particular case of the Gazis, Herman, and Rothery model (Gazis et al., 1961). Almost all car following models are based on the assumption of the existence of a behavioral law V e . The latter has been taken linear in (Greenshields, 1935; Herman et al., 1959), logarithmic in (Greenber, 1959), exponential in (Newell, 1961), and with more complicated forms in other works. Bando et al. (1995) used the sigmiudal function
V e ðyÞ ¼ tanhðy hÞ þ tanhðhÞ;
ð4Þ
where tanh denotes the hyperbolic tangent function, and h is a constant. Kerner and Konhäuser (1993), Herrmann and Kerner (1998), and then Lenz et al. (1999), and Hoogendoorn et al. (2006) have used the function
( ) 1 p1 V e ðyÞ ¼ v 0 1 þ exp p3 ; p cy 2
ð5Þ
where p1 ; p2 ; p3 ; v 0 and c are parameters estimated from data. p1 ; p2 and p3 have been fixed to p1 ¼ 1000; p2 ¼ 10=2:1 and p3 ¼ 5:34 109 . In (Lenz et al., 1999), c is taken equal to 7.5. See also (Helly, 1959, 1961; Treiber et al., 2000). The simplest car-following model is the linear one, where the car dynamics are written
x_ n ðt þ TÞ ¼ aðxn1 ðtÞ xn ðtÞÞ þ b;
ð6Þ
where T is a reaction time, a is a sensitivity parameter, and b is a constant. That model can be derived from (3) with a linear fundamental diagram V e . The stability of the linear car-following model (6) and the existence of a stationary regime have been treated in (Herman et al., 1959). The min-plus traffic model (Lotito et al., 2005; Farhi, 2008) is a microscopic discrete-time car-following model based on the min-plus algebra (Baccelli et al., 1992). It consists in the following dynamics.1
xn ðt þ 1Þ ¼ minfxn ðtÞ þ v 0 ; xn1 ðtÞ rg;
ð7Þ
where v 0 is the free velocity and r is a safety distance. The idea of the model (7) is that the car dynamics is linear in the min-plus algebra (Baccelli et al., 1992), where the addition is the operation ‘‘min’’ and the product is the standard addition ‘‘þ’’. The average growth rate vector per time unit of the 1 We accept here, and in the sequel of the article, to add velocities to distances, since we take the time step dt as the time unit ðdt ¼ 1Þ. We would write xn ðt þ dtÞ ¼ minfxn ðtÞ þ v 0 dt; xn1 ðtÞ rg.
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; v ; . . . ; v Þ, where v is the stationary system, defined by v ¼ limt!1 xðtÞ=t, is then obtained (see (Lotito et al., 2005)) v ¼ ðv car-speed, derived as the unique min-plus eigenvalue of the min-plus linear map of the dynamics. The fundamental diagram is then obtained
v ¼ minðv 0 ; y rÞ q ¼ minðv 0 q; 1 rqÞ;
ð8Þ ð9Þ
(respectively q) is the stationary average car-speed (respectively car-flow), and y (respectively q) is the average where v . and q ¼ 1=y inter-vehicular distance (respectively car-density) on the ring, with q ¼ qv In (7), for t P 0, if xn1 ðtÞ xn ðtÞ > v 0 þ r, then the dynamics is xn ðt þ 1Þ xn ðtÞ ¼ v 0 . If xn1 ðtÞ xn ðtÞ 6 v 0 þ r then the dynamics is xn ðt þ 1Þ xn ðtÞ ¼ xn1 ðtÞ xn ðtÞ r. Therefore, (7) is linear in both phases of free and congested traffic. The min-plus model (7) permits to distinguish two phases in which the traffic dynamics is linear, but with a sensitivity parameter (a in (6)) in {0,1}. The linear model (6) is not constrained in the sensitivity parameter value, but it permits the modeling of only one traffic phase. The model we present here extends (6) and (7) in a way that an arbitrary number of traffic phases can be modeled, with flexibility in the sensitivity parameter value on each phase. Dual Eulerian modeling approaches with extensions to the 2D-traffic have been proposed in (Farhi, 2008; Farhi et al., 2011). For our model, we apply a similar but more general approach than the min-plus one used to analyze the dynamic system (7). Indeed, the dynamics (7) is additive homogeneous of degree one2 and is monotone.3 It is then non-expansive4 (Candrall and Tartar, 1980). The stability of the dynamic system (7) is thus guaranteed from its non-expansiveness. Moreover, (7) is connected (or communicating)5 (Gaubert and Gunawardena, 1998, 1999). An important result from (Gunawardena and Keane, 1995; Gaubert and Gunawardena, 1998) (Theorem 1 below) permits the analysis of non-expansive and connected dynamic systems. Theorem 1 (Gunawardena and Keane, 1995; Gaubert and Gunawardena, 1998). If a dynamic system xðtÞ ¼ f ðxðt 1ÞÞ is non þ x ¼ f ðxÞ admits a solution ðv ; xÞ, where x is defined up to an expansive and connected, then the additive eigenvalue problem v 2 R is unique. Moreover, the dynamic system admits an average growth additive constant, not necessarily in a unique way, and v ; v ; . . . ; v Þ. rate vector v, which is unique (independent of the initial condition) and given by vðf Þ ¼ t ðv The model treated in this article can be seen as an extension of the min-plus model (7). In Section 2, we present the model. It is called piecewise linear car-following model because it is based on a piecewise linear fundamental diagram V e . The model describes the traffic of cars on a road of one lane without passing. First (sub-Section 2.1), the cars move on a ring road. The stability conditions of the dynamic system describing the traffic are determined. Under those conditions, the cardynamics are interpreted as a dynamic programming equation (DPE) associated to a stochastic optimal control problem of a Markov chain. The DPE is solved analytically. We show that the individual driver-behavior law V e supposed in the model is realized on the collective driver-behavior at the stationary regime. The effect of the stability condition on the shape of the fundamental diagram is shown. Second (sub-Section 2.2), the traffic on an ‘‘open’’ road (a highway stretch for example) is treated. We give equivalent results of those given in the ring road case. In Section 3, we give a result on the transient traffic and present an example, where we simulate the traffic basing on a piecewise linear approximation of the diagram (5). The results obtained in this article are important because, first they permit to model (first-order) individual driver behaviors with an arbitrary precision. Second, the fundamental traffic diagram at the stationary regime is analytically derived, giving thus the collective drivers behaviors at the equilibrium. Third, the variational formulation permits to interpret the individual driver behavior in term of optimal control. Finally, many extensions of the model are possible. First, anticipation in driving is recently introduced; see (Farhi et al., in press), where we are still able to solve the dynamics analytically, and where other qualitative results are obtained. Second, the heterogeneity in driving (drivers with different behaviors) can be easily modeled, and we think that the same analytic approach can be adapted to that case. Third, the same modeling approach can be used to model lane-change in one-dimensional traffic. Finally, it may also be possible to extend this approach, at least in modeling, for the two dimensional traffic. 2. Piecewise linear car following model The behavioral law V e is an increasing curve bounded by the free speed v 0 . Moreover, V e ðyÞ ¼ 0 for y 2 ½0; yj where yj denotes the jam inter-vehicular distance. We propose here to approximate the curve V e with a piecewise-linear curve
V e ðyÞ ¼ min maxfauw y þ buw g; u2U
2
w2W
ð10Þ
A dynamic system xðtÞ ¼ f ðxðt 1ÞÞ is additive homogeneous of degree 1 if f is so, that is if 8x 2 Rn ; 8k 2 R; f ðk þ xÞ ¼ k þ f ðxÞ. A dynamic system xðtÞ ¼ f ðxðt 1ÞÞ is monotone if f is so, that is if 8x1 ; x2 2 Rn ; x1 6 x2 ) f ðx1 Þ 6 f ðx2 Þ, where the order 6 is pointwise in Rn . 4 A dynamic system xðtÞ ¼ f ðxðt 1ÞÞ is non-expansive if f is so, that is if there exists a norm k k in Rn such that 8x1 ; x2 2 Rn ; kf ðx2 Þ f ðx1 Þk 6 kx2 x1 k. 5 An additive homogeneous of degree 1 and monotone dynamic system xðtÞ ¼ f ðxðt 1ÞÞ with x 2 Rn is connected if its associated graph is strongly connected. The graph associated to that dynamic system is the graph with n nodes and whose arcs are determined as follows. There exists an arc from a node i th to a node j if limm!1 fj ðmei Þ ¼ 1, where ei denotes the i vector of the canonic basis of Rn . 3
N. Farhi / Transportation Research Part C 25 (2012) 100–112
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where U and W are two finite sets of indices. We explain hereafter how the indices u 2 U and w 2 W are used and interpreted in the traffic dynamics. In (10), those indices just permit (at that stage) to have a sufficient number of linear segments for the approximation of the curve V e . Since V e is increasing, we have auw P 0; 8ðu; wÞ 2 U W. We are interested here on the discrete-time first-order dynamics
xn ðt þ 1Þ ¼ xn ðtÞ þ minfau ðxn1 ðtÞ xn ðtÞÞ þ bu g;
ð11Þ
xn ðt þ 1Þ ¼ xn ðtÞ þ min maxfauw ðxn1 ðtÞ xn ðtÞÞ þ buw g:
ð12Þ
u2U
and u2U
w2W
It is clear that (12) extends (11). The model (12) is also an extension of both linear model (6) and min-plus model (7). In this article, we characterize the stability of the dynamics (12), calculate the stationary regimes, show that the fundamental diagrams are effectively realized at the stationary regime, and analyze the transient traffic. We will distinguish two cases: Traffic on a ring road and traffic on an ‘‘open’’ road. 2.1. Traffic on a ring road We follow here the modeling of Lotito et al. (2005). Let us consider m cars moving a one-lane ring road in one direction without passing. We assume that the cars have the same length that we take here as the unity of distance. The road is . assumed to be of size l; that is, it can contain at most l cars. The car density on the road is thus q ¼ m=l ¼ 1=y 2.1.1. Stochastic optimal control model We consider here the car dynamics (11). That is to say that each car n maximizes its velocity at time t under the constraints
xn ðt þ 1Þ 6 xn ðtÞ þ au ðxn1 ðtÞ xn ðtÞÞ þ bu ;
8u 2 U:
ð13Þ
Each constraint of (13) bounds the velocity xn ðt þ 1Þ xn ðtÞ by a sum of a fixed term bu and a term depending linearly on the inter-vehicular distance. Let us first notice that (11), on the ring road, is written
xn ðt þ 1Þ ¼ xn ðtÞ þ minfau ðxn1 ðtÞ xn ðtÞÞ þ bu g;
for n P 2;
u2U
x1 ðt þ 1Þ ¼ x1 ðtÞ þ minfau ðxn ðtÞ þ l x1 ðtÞÞ þ bu g; u2U
which can also be written
xn ðt þ 1Þ ¼ minfð1 au Þxn ðtÞ þ au xn1 ðtÞ þ bu g; u2U
for n P 2;
x1 ðt þ 1Þ ¼ minfð1 au Þx1 ðtÞ þ au xn ðtÞ þ au m=q þ bu g: u2U
Let us denote by M u ; u 2 U the family of matrices defined by
2 6 6 Mu ¼ 6 6 4
1 au
0
au .. .
1 au .. .
..
0
0
3
au
7 7 7; 7 5
0 .
au 1 au
and by cu ; u 2 U, the family of vectors defined by
cu ¼ t ½au m=q þ bu ; bu ; . . . ; bu : The dynamics (11) are then written:
xn ðt þ 1Þ ¼ min ½M u xðtÞn þ cun ; u2U
1 6 n 6 m:
ð14Þ
The system (14) is additive homogeneous of degree 1 by the definition of the matrices M u ; u 2 U. It is monotone under the condition that all the components of M u ; u 2 U are non-negative, which is equivalent to au 2 ½0; 1; 8u 2 U. Hence, under that condition, the system (14) is non-expansive. Moreover, the matrices Mu ; u 2 U are stochastic.6 Those matrices can then be seen as transition matrices of a controlled Markov chain, where the set of controls is U. The connectedness of the system (14), as defined in the previous section, is related to the irreducibility of the Markov chain with transition matrices M u ; u 2 U. It is easy to check that (14) is connected if and only 6
We mean here Muij P 0; 8i; j and
P
u j M ij
¼ 1; 8i.
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N. Farhi / Transportation Research Part C 25 (2012) 100–112
if 9u 2 U; au 2 ð0; 1; see Appendix A for the proof. That condition is interpreted in term of traffic by saying that every car moves by taking into account the position of the car ahead. Consequently, under the condition 8u 2 U; au 2 ½0; 1 and 9u 2 U; au 2 ð0; 1, the dynamic system (14) is non-expansive and is connected. Therefore, by Theorem 1, we conclude that the additive eigenvalue problem
v þ xn ¼ min f½M u xn þ cun g; u2U
1 6 n 6 m;
ð15Þ
; xÞ, where x is defined up to an additive describing the stationary regime of the dynamic system (14), admits a solution ðv 2 R is unique. Moreover, the dynamic system admits a unique average constant, not necessarily in a unique way, and v . growth rate per time unit v, whose components are all equal and coincide with v The average growth rate per time unit v of the system (14) is interpreted in term of traffic as the stationary car-velocity. The additive eigenvector x gives the asymptotic distribution of cars on the ring. x is given up to an additive constant, since the ; xÞ is a solution of (15) then ðv ; e þ xÞ is also a car-dynamics (14) is additive homogeneous of degree 1. That is to say that if ðv solution for (15), for every constant e 2 R. Let us now give an interpretation of the model in term of ergodic stochastic optimal control. Indeed, (15) can be seen as a dynamic programming equation of an ergodic stochastic optimal control problem of a Markov chain with transition matrices Mu ; u 2 U and costs cu ; u 2 U, and with a set of states N ¼ f1; 2; . . . ; mg. The stochastic optimal control problem of the chain is written
( minE s2S
) T 1X cuntt ; T!þ1 T t¼0 lim
ð16Þ
where S is a set of feedback strategies on N . A strategy s 2 S associates to every state n 2 N a control u 2 U (that is ut ¼ sðnt Þ). ; xÞ for the dynamic programming Eq. (15). The following result gives one solution ðv ; xÞ given by: Proposition 1. The system (15) admits a solution ðv
þ bu g; v ¼ min fau y u2U ; ðm 2Þy ; x ¼ t ½ðm 1Þy
...;
; 0: y
Proof. First, because of the symmetry of the system (15), it is natural that the asymptotic car-positions xn ; 1 6 n 6 m are uniformly distributed on the ring, and that the optimal strategy is independent of the state x. Let us prove it. 2 U be defined by au y . Let x be the vector given in Proposition 1. Then 8n 2 f1; 2; . . . ; mg þ bu ¼ minu2U fau y þ bu g ¼ v Let u we have
þ bu Þ þ xn ¼ minðau y þ bu Þ þ xn ¼ min ½M u xn þ cun ¼ v þ xn : ½M u xn þ cun ¼ ðau y u2U
u2U
In term of traffic, Proposition 1 shows that the car-dynamics is stable under the condition au 2 ½0; 1, and the average car speed is given by the additive eigenvalue of the equilibrium equation in the case where the system is connected. Moreover, it affirms that the fundamental diagram supposed in the model is realized at the stationary regime.
þ bu g; v ¼ min fau y u2U
ð17Þ
q ¼ minfau þ bu qg:
ð18Þ
u2U
It is important to note here that, up to the assumption au 2 ½0; 1; 8u 2 U, every concave curve V e or Q e can be approximated with (17) or (18). Indeed, approximating fundamental diagrams using those formulas is nothing but computing Fenchel transforms; see (Daganzo, 2006; Aubin et al., 2009). More precisely, if we denote by V the set V ¼ fbu ; u 2 Ug and define the function g by:
g: V ! R
v ¼ bu
# au ;
then
q ¼ Q e ðqÞ ¼ minðqv gðv ÞÞ ¼ g ðqÞ; v 2V
where g denotes the Fenchel transform of g. Finally, we note that the min-plus linear model is a particular case of the model presented in this section, where U ¼ fu1 ; u2 g with ða1 ; b1 Þ ¼ ð0; v Þ and ða2 ; b2 Þ ¼ ð1; rÞ. In this case, the fundamental traffic diagram is approximated with a piecewise linear curve of two segments.
N. Farhi / Transportation Research Part C 25 (2012) 100–112
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2.1.2. Stochastic game model We consider in this section the car dynamics (12), again with the assumption 8ðu; wÞ 2 U W; auw 2 ½0; 1 and 9ðu; wÞ 2 U W; auw 2 ð0; 1. The dynamic system (12) is interpreted as a stochastic dynamic programming equation associated to a stochastic game problem on a controlled Markov chain. As above, a generalized eigenvalue problem is solved. The extension we make here approximates non-concave fundamental diagrams. In term of traffic, we take into account the drivers’ behavior changing from low densities to high ones. The difference between these two situations is that in low densities, drivers, moving, or being able to move with high velocities, they try to leave large safety distances between each other, so the safety distances are maximized; whilst in high densities, drivers, moving, or having to move with low velocities, they try to leave small safety distances between each other in order to avoid jams; so they minimize safety distances. To illustrate this idea, let us consider the following two dynamics of a given car n.
xn ðt þ 1Þ ¼ minfxn ðtÞ þ v ; xn1 ðtÞ rg;
ð19Þ
xn ðt þ 1Þ ¼ minfxn ðtÞ þ v ; maxfxn1 ðtÞ r; ðxn ðtÞ þ xn1 ðtÞÞ=2gg:
ð20Þ
The dynamics (19) is a min-plus dynamics which grossly tell that cars move with their desired velocity v at the fluid regime and they keep a safety distance r at the congested regime. The dynamics (20) distinguishes two situations at the congested regime: In a relatively low density situation where the cars are separated by a distance that equals at least to 2r we have
maxfxn1 ðtÞ r; ðxn ðtÞ þ xn1 ðtÞÞ=2g ¼ xn1 ðtÞ r: In a high density situation, where the cars are separated by distances less than 2r we have
maxfxn1 ðtÞ r; ðxn ðtÞ þ xn1 ðtÞÞ=2g ¼ ðxn ðtÞ þ xn1 ðtÞÞ=2: In the latter case, we accept the cars moving closer but by reducing the approach speed in order to avoid collisions. We think that the additional assumption that drivers leave small safety distances between each other in the congested traffic phase, introduced in (20), is acceptable in term of driving behavior. However, that assumption cannot be obtained without introducing a maximum operator in the dynamics (i.e. with only minimum operators). Indeed, with only minimum operators the approach is mechanically reduced with the increasing of the car-density (in fact this is the concaveness of the fundamental diagram). Because of the realness of such scenarios, we think that the fundamental traffic diagram should be composed of two parts, a concave part at the fluid regime, and a convex part at the congested regime. The dynamics (12) generalizes this idea. The dynamics (12) can be written
xn ðt þ 1Þ ¼ min max ½Muw xðtÞn þ cuw ; n u2U
1 6 n 6 m;
w2W
ð21Þ
where
2 M uw
6 6 ¼6 6 4
1 auw
0
auw .. .
1 auw .. .
0
0
auw
0 ..
.
3 7 7 7; 7 5
auw 1 auw
and
cuw ¼ t ½auw m=q þ buw ; buw ; . . . ; buw : The stationary traffic is then described by the eigenvalue problem
uw ½M xn þ cuw ; n
v þ xn ¼ min max u2U w2W
1 6 n 6 m:
The following result generalizes Proposition 1. ; xÞ given by: Theorem 2. The system (22) admits a solution ðv
þ buw g; v ¼ min maxfauw y u2U w2W ; ðm 2Þy ; x ¼ t ½ðm 1Þy
...;
; 0: y
Proof. Similar to the proof of Proposition 1. h
ð22Þ
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N. Farhi / Transportation Research Part C 25 (2012) 100–112
We can easily check that the dynamic system (21) is non-expansive under the condition auw 2 ½0; 1; 8ðu; wÞ 2 U W (by the same way as in the case of the dynamic system (14)). Its stability is thus guaranteed under that condition. If, in addition, 9ðu; wÞ 2 U W; auw 2 ð0; 1, then the system is connected. Therefore, by Theorems 1 and 2, we conclude that the dynamic (given by Theroem 2 and system (21) admits a unique average growth rate vector v, whose components are all equal to v interpreted as the average car speed). The behavior law supposed in the model is realized at the stationary regime.
þ buw g; v ¼ min maxfauw y u2U w2W
ð23Þ
q ¼ min maxfauw þ buw qg:
ð24Þ
u2U
w2W
In term of stochastic optimal control, the system (22) can be seen as a dynamic programming equation associated to a stochastic game, with two players, on a Markov chain. The set of states of the chain is again N ¼ f1; 2; . . . ; mg. The chain is controlled by two players, a minimizer one with a finite set U of controls, and a maximizer one with a finite set W of controls. The transitions and the costs of the chain are given by the matrices Muw and the vectors cuw ; ðu; wÞ 2 U W defined above. The stochastic optimal control problem is
(
) T 1X ut wt min max js2S E lim cnðtÞ ; T!þ1 T t¼0
ð25Þ
where S is the set of strategies assoicating to every state n 2 N a couple of commands ðu; wÞ 2 U W. It is assumed here that the maximizer knows at each step the decision of the minimizer. We now give a consequence of the stability condition auw 2 ½0; 1; 8ðu; wÞ 2 U W, on the shape of the fundamental diagrams (23) and (24). As shown in Fig. 1, where we have drawn the fundamental diagram (5) (with v 0 ¼ 14 meter by half second, and c ¼ 7:5), the stability condition puts the curves (23) and (24) in specific respective regions in the plan. Indeed, for the diagram (23), if we assume that V e is bounded by v 0 , V e ðyÞ ¼ 0; 8y 2 ½0; yj , and that V e is continuous (and increasing), then starting by the point ðyj ; 0Þ, one cannot join any point above the line passing by ðyj ; 0Þ and having the slope 1, with any sequence of segments of slopes auw 2 ½0; 1. We can write
V e ðyÞ 6 maxð0; minðv 0 ; y yj ÞÞ: Similarly, on the diagram (24), if we assume that Q e is continuous and Q e ðqÞ ¼ 0; 8q 2 ½qj ; 1, then going back from the point ðqj ; 0Þ, one cannot attain any point above the line passing by ðqj ; 0Þ and ð0; 1Þ, with a sequence of segments having their ordinates at the origin (auw Þ in ½0; 1. We can write
Q e ðqÞ 6 maxð0; minðv 0 q; 1 q=qj ÞÞ:
2.2. Traffic on an open road We study in this section the traffic on an open road with one lane and without passing. The fact that the traffic description is lagrangian is important here. Indeed, we follow the movement of a given number of cars on an open road, and we are interested on what happen on only the stretch where all the cars move. The average car-density is then defined as the total number of cars divided by the stretch occupied by the cars. Therefore, the average car-density changes over time.
Fig. 1. The effect of the stability condition auw 2 ½0; 1 on the shape of the fundamental diagram.
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We are interested, in particular, in the stationary regime of the traffic dynamics. We will assume that the velocity of the first car becomes stationary from a given time horizon, and look for the stationary car-density in that case. Note that this is the inverse question of the one posed in the ring road case, where we have the average car-density, and look for the average car-speed. We are interested in the following dynamics.
x1 ðt þ 1Þ ¼ x1 ðtÞ þ v 1 ðtÞ;
ð26Þ
xn ðt þ 1Þ ¼ min maxfxn ðtÞ þ auw ½xn1 ðtÞ xn ðtÞ þ buw g: u2U
w2W
If we denote by M uw the matrices
2
1
0
6a 6 uw M uw ¼ 6 6 .. 4 .
1 auw .. .
0
3
0
7 7 7; 7 5
0 ..
.
auw 1 auw
0
and by cuw ðtÞ the vectors
cuw ðtÞ ¼ t ½v 1 ðtÞ; buw ; . . . ; buw ; buw ; for ðu; wÞ 2 U W and t 2 N, then the dynamic system (26) is written
xn ðt þ 1Þ ¼ min max ½Muw xðtÞn þ cuw n ðtÞ ; u2U
w2W
1 6 n 6 m:
ð27Þ
It is easy to check that the dynamic system (27) is additive homogeneous of degree 1, and is monotone under the condition 8ðu; wÞ 2 U W; auw 2 ½0; 1. Therefore, under that condition, (27) is non-expansive. However, (27) is not connected for every ðu; wÞ 2 U W. We will be interested here, in particular, in the stationary regime, where v 1 ðtÞ reaches a fixed value v 1 . For this case, the eigenvalue problem associated to (26) is
v þ x1 ¼ x1 þ v 1 ; v þ xn ¼ min maxfxn þ auw ðxn1 xn Þ þ buw g: u2U w2W
ð28Þ
The system (28) is also written
uw ½M xn þ cuw ; n
v þ xn ¼ min max u2U w2W
1 6 n 6 m;
ð29Þ
where
cuw ¼ t ½v 1 ; buw ; . . . ; buw ; buw : Then the following result is a corollary of Theorem 2. ; xÞ is a solution for the system (28), where Corollary 1. For all y 2 R satisfying minu2U maxw2W ðauw y þ buw Þ ¼ v 1 , the couple ðv v ¼ v 1 and x is given up to an additive constant by
x ¼ t ½ðm 1Þy; ðm 2Þy; . . . ; y; 0:
ð30Þ
; wÞ 2U W Proof. The proof is similar to that of Proposition 1. Let y 2 R satisfying minu2U maxw2W ðauw y þ buw Þ ¼ v 1 . Let ðu such that auw y þ buw ¼ v 1 . Let x be given by (30). Then 8n 2 f1; 2; . . . ; mg we have
½Muw xn þ cunw ¼ ðauw y þ buw Þ þ xn ¼ min maxðauw y þ buw Þ þ xn ¼ min max½Muw xn þ cuw n ¼ v 1 þ xn : u2U
w2W
u2U
w2W
We can easily check that for ðu; wÞ 2 U W such that auw ¼ 0 and buw ¼ v 1 , every inter-vehicular distance y 2 R satisfies the condition minu2U maxw2W ðauw y þ buw Þ ¼ v 1 . Thus, such couples ðu; wÞ do not count for that condition. Corollary 1 can then be announced differently. Let us denote by W u for u 2 U the family of index sets
W u ¼ fw 2 W; ðauw ; buw Þ–ð0; v 1 Þg; the asymptotic average inter-vehicular distance: and by y
¼ max min y u2U
w2W u
v 1 buw auw
;
ð31Þ
2 R (i.e. where we use the convention a=0 ¼ þ1 if a > 0 and a=0 ¼ 1 if a < 0. Then Corollary 1 tells simply that if y < þ1), then the dynamic system (28) admits a solution ðv ; xÞ where v ¼ v 1 is unique, and where x is not necessarily 1 < y unique and is given up to an additive constant by
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; ðm 2Þy ; . . . ; y ; 0: x ¼ t ½ðm 1Þy Non-uniform asymptotic car-distributions can also be obtained. Let us clarify the following three cases. ¼ þ1. In this case, the distance between the first car and the other (a1) If 9u 2 U; 8w 2 W u ; auw ¼ 0 and buw < v 1 , then y cars increases over time and goes to þ1. The asymptotic car-distribution on the road is not uniform. ¼ 1. In this case, the first car is passed by all other cars, and the (a2) If 8u 2 U; 9w 2 W u ; auw ¼ 0 and buw > v 1 , then y distance between the first car and the other cars increases over time and goes to þ1. The asymptotic car-distribution on the road is not uniform. (a3) If 8u 2 U; 8w 2 W u ; auw ¼ 0 and if minu2U maxw2W u buw ¼ v 1 , then for all x 2 Rm , ðv 1 ; xÞ is a solution for the system (28). In this case, every distribution of the cars moving all with the constant velocity v 1 is stationary. The formula (31) is the fundamental traffic diagram expressing the average inter-vehicular distance as a function of the car speed at the stationary regime. In the case where only a minimum operator is used in (26), the formula (31) is reduced to the convex fundamental diagram
¼ max y u2U
v 1 bu au
ð32Þ
:
If we define the average car-density as the number of vehicles divided by the length of the stretch occupied by all the vehicles (that is the distance between the first car and the last one), then the stationary average car-density q is the inverse of the stationary average inter-vehicular distance. Therefore, from (31), we have
auw
q ¼ min max u2U
v 1 buw
w2W
ð33Þ
:
In the case where a stationary flow q is known (which is in fact not sufficient to determine uniquely the stationary cardensity), rather than the stationary velocity of the first car, then from (33), the possible stationary average car-densities satisfy
1 ¼ min max u2U
w2W
auw q buw q
ð34Þ
;
which is simply reversing the formula (24), up to the particular cases clarified above (for auw ¼ 0). Note that we usually have fundamental diagrams where two possible stationary car-densities are possible for a positive non-maximal stationary carflow, only one possible stationary car-density (called critical) for maximum stationary car-flows, and an infinity of stationary car-densities (zero and all the densities bigger than the jam density) for the null stationary car-flow. 3. Transient traffic In this section, we first give a result on the uniqueness of the additive eigenvector associated to the car-dynamics.7 The result characterizes also the convergence of the car trajectories to the uniform distribution of cars on the road. We then show, on an example, the simulation of the model. Proposition 2. If auw 2 ð0; 1Þ; 8ðu; wÞ 2 U W (respectively au 2 ð0; 1Þ; 8u 2 U), then the eigenvector x given in Theorem 2 (respectively Proposition 1) and Corollary 1 is unique, and the vector xðtÞ; t 2 N whose dynamics is defined by (21) (respectively (14) and (27) converges (uniquely) to the eigenvector x.
Proof. Let us do the proof for the dynamics (21) and Theorem 2. We define a new vector z as follows.
for 1 6 n 6 m 1;
zn ¼ xn xnþ1 ; z m ¼ xm x 1 þ l :
The vector z gives the inter-vehicular distances on the ring road. It is then easy to check that the dynamics of z is simply
zn ðt þ 1Þ ¼ min max½M uw zðtÞn ; u2U
w2W
1 6 n 6 m:
ð35Þ
Now, the key of the proof is that if auw 2 ð0; 1Þ; 8ðu; wÞ 2 U W, then all the (multiplicative) eigenvalues of any matrix M uw , for ðu; wÞ 2 U W, have their modules less than 1, except the eigenvalue 1 which is simple.8 Moreover, the eigenvector asso ð1; 1; . . . ; 1Þ, where ciated to the eigenvalue 1 is9 (1, 1, . . . , 1). Therefore, the sequence of vectors fzðtÞgt2N converges to a vector y 7 8 9
Note that Theorem 1 guarantees the uniqueness of the additive eigenvalue but not that of the associated eigenvector. It is easy to determine the eigenvalues of M uw since it is a sum of two circulant matrices. Indeed the matrices M uw are stochastic, so all the lines sum to 1.
N. Farhi / Transportation Research Part C 25 (2012) 100–112
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gives the common (or the average) inter-vehicular distance at the stationary regime. The proof is similar for the dynamics (14) y and (27). h Proposition 2 tells that under the condition auw 2 ð0; 1Þ; 8ðu; wÞ 2 U W, the transient traffic consists to closer the car positions on the road to the stationary car position. We give below an example where we approximated a fundamental traffic diagram (collective behavioral law), and simulated the corresponding car dynamics, according to the model presented in this article. Nevertheless, we notice that the heterogeneity in driving should be modeled in order to be able to perform better simulations. Indeed, a behavioral law should be determined and approximated for every driver, and the dynamics (26) has to be adapted for this. This can be done easily in term of medeling (it consists to index the parameters auw and buw with respect to every driver (car), or at least to a given number of classes of drivers.) In the case of traffic on an open road, the dynamics taking into account the driving heterogeneity can be written as follows.
x1 ðt þ 1Þ ¼ x1 ðtÞ þ v 1 ðtÞ; xn ðt þ 1Þ ¼ min maxfxn ðtÞ þ anuw ½xn1 ðtÞ xn ðtÞ þ bnuw g: u2U
ð36Þ
w2W
The stability condition in the heterogeneous driving case is similar to that of the homogeneous driving case. That is anuw 2 ½0; 1; 8ðn; u; wÞ 2 f1; 2; . . . ; mg U W. We project to study this extension in a forthcoming article, where we need to model vehicle passing and/or lane change in order to allow faster vehicles to pass or to change lane when bothered by slower vehicles. The stationary regimes (average car-speed and car-distribution on the road or on each lane) need also to be determined. Another important factor that we would like to take into account is the anticipation in driving. That is that drivers respond to a stimulus that takes into account headway spacings with respect to a number of leaders (instead of only the headway spacing with respect to the first leader). The multi-anticipation in driving, in piecewise-linear car-following modeling, have been treated recently in (Farhi et al., in press), where the stability is again characterized and the stationary car distributions are determined. Example 1 is a short simulation test of the model (without heterogeneity nor anticipation). In (Farhi et al., in press), further numerical tests are done, and a parameter identification approach is proposed to piecewise-linear car following models. Heterogeneity in driving has to be included in order to perform reliable parameter identifications. Example 1. Let us simulate the car-dynamics (26). We take as the time unit half a second (1/2 s), and as the distance unit 1 meter (m). The parameters of the model are determined by approximating the behavior law (5), with a free velocity v 0 ¼ 14 m=ð1=2Þ s (which is about 100 km/h) and c ¼ 7:5 as in (Lenz et al., 1999); see Fig. 2. The behavior law is approximated by the following piecewise linear curve of six segments.
e ðyÞ ¼ maxfa1 y þ b ; minfa2 y þ b ; a3 y þ b ; a4 y þ b ; a5 y þ b ; a6 y þ b gg; V 1 2 3 4 5 6
ð37Þ
where the parameters ai and bi for i ¼ 1; 2; . . . ; 6 are given in Table 1. We simulate the piecewise linear car-following model associated to the approximation above.
x1 ðtÞ ¼ x1 ðt 1Þ þ v 1 ðtÞ; e ðxn1 ðt 1Þ xn ðt 1ÞÞ: xn ðtÞ ¼ xn ðt 1Þ þ V
ð38Þ
The velocity of the first car v 1 ðtÞ; t P 0 is varied in the time interval ½0; 1000, then fixed to the free velocity v 0 ¼ 14 m=ð1=2Þ s in the time interval ½1000; 3000, and finally fixed on a velocity that exceeds v 0 in the remaining time ½3000; 7200. The average inter-vehicular distance is then computed at every time t, and the results are shown in Fig. 3.
Fig. 2. Approximation of the behavioral law (5) with a piecewise linear curve.
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Table 1 Parameters ai and bi ; i ¼ 1; 2; . . . ; 6 corresponding to the approximation (37). Segments
1
2
3
4
5
6
ai
0 0
0.54 8.1
0.32 1.47
0.13 6.11
0.34 10.6
0 14
bi
Fig. 3. Simulation results. On the left-side: the first car velocity (solid line), and the average inter-vehicular distance (dash line) functions of time. On the right side: the approximation of the behavior law (5) (dash line), and the average inter-vehicular distance obtained by simulation in function of the velocity of the first car (solid line).
The simple simulation we made here permits to have an idea of the traffic in the transient regime. Fig. 3 shows how the average of the inter-vehicular distance is changed due to a changing in the velocity of the first car. In the right side of Fig. 3, we compare the fundamental law assumed in the model with the diagram giving the average inter-vehicular distance (with respect to the number of cars) function of the velocity of the first car. We observe that loops are obtained on that diagram in the transient traffic. The loops are interpreted by the fact that once the velocity of the first car is temporarily stationary, the velocities of the following cars, and thus also the average velocity of the cars, make some time to attain the first car velocity. It can also be interpreted by saying that even though the cars have, individually, the same response to a changing in intervehicular distance; their collective response depends on whether the inter-vehicular distance is increasing or decreasing. The apparition of such loops is due to the reaction time of drivers. It can also be related to the number of anticipation cars in case of multi-anticipative modeling. However, one may measure on a given section, different car-flows for the same car density (or occupancy rate) depending on the traffic acceleration or deceleration, and interpret it as the hysteresis phenomenon (Edie, 1961; Treiterer and Myers, 1974; Zhang, 1999; Geroliminis and Sun, 2011). Finally let us point out the infinite sate-space case, where an infinite number of cars is considered. First, we notice that in that case, we need to assume that the length of the road is infinite. Therefore, only the open road case can be considered. Moreover, the set of cars must be countable, in order to be able to write the dynamics. It is well known (see for example (Hernandez-Lerma and Lasserre, 1999; Whittle, 1986)) that in average-cost programming, the results of Theorem 1 do not hold for the infinite state-space case, without additional assumptions.10 Nevertheless, the dynamics we study here is a particular case for which we can tell more. First, we can easily check that, except the non-uniform car-distribution cases (a1), (a2) and (a3) discussed above (in Section 2), the additive eigenvalue problem (stationary regime equation) admits a solution (the same one as that of the finite state-space case). Then we know that for every state x0
such that ðx0 xÞ is bounded jx0n xn j < þ1; 8n 2 N , the average growth rate per time-unit of the dynamics (27), starting from x0 , coincides with the additive eigenvalue. Therefore, under that condition, we are in the same situation as in the finite statespace case, i.e. the average car-speed on the road is v 1 . 4. Conclusion We proposed in this article a car-following model that extends the linear car-following model as well as the min-plus model. The stability and the stationary regimes of the model are characterized thanks to a variational formulation of the car-dynamics. This formulation, although already made with continuous-time models, it permits to clarify the stimulus10 In the infinite state-space case of the average-cost programming, even the existence of an average-cost optimal policy or a solution for the additive eigenvalue problem are in question. Moreover, the existence of a solution to the additive eigenvalue problem is no more sufficient for optimality. That is to say that even when an additive eigenvalue exists, it may not coincide with the average growth rate per time-unit of the car dynamics (the average car-speed in our case).
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response process in microscopic discrete-time traffic models, and to interpret it in term of stochastic optimal control. Among the important questions to be treated in the future, the impacts of heterogeneity and anticipation in driving, on the transient and stationary traffic regimes, based on the model proposed in this article. Anticipation in driving has been introduced recently for our model (see (Farhi et al., in press)). Other works on multilane traffic and driver heterogeneity are underway. Appendix A. Connectedness of system (14) Let x 2 Rm . We denote by h : Rm ! Rm the operator defined by hðxðtÞÞ ¼ xðt þ 1Þ, where xn ðt þ 1Þ, for 1 6 n 6 m are given by the definition of the system (14). That is
xn ðt þ 1Þ ¼ min ½M u xðtÞn þ cun ; u2U
1 6 n 6 m:
If 9u 2 U, such that au 2 ð0; 1, then for all n 2 f1; 2; . . . ; mg, there exists an arc, on the graph associated to h, going from n 1 to n (n being cyclic in f1; 2; . . . ; mg). Indeed,
xn ðt þ 1Þ ¼ ð1 au Þxn ðtÞ þ au xn1 ðtÞ þ bu : Then since au > 0, we get:
lim hn ðmen1 Þ ¼ lim ½au m þ bu ¼ 1:
m!1
m!1
where en1 denotes the ðn 1Þth vector of the canonic basis of Rm . Therefore the graph associated to h is strongly connected. If 8u 2 U; au ¼ 0, then we can easily check that all arcs of the graph associated to h are loops. Hence that graph is not strongly connected.
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Contents lists available at SciVerse ScienceDirect
Transportation Research Part C journal homepage: www.elsevier.com/locate/trc
Piecewise linear car-following modeling Nadir Farhi Université Paris-Est, IFSTTAR, GRETTIA, F-93166 Noisy-le-Grand, France
a r t i c l e
i n f o
Article history: Received 18 August 2011 Received in revised form 15 May 2012 Accepted 16 May 2012
Keywords: Car-following modeling Optimal control Variational formulations
a b s t r a c t We present a traffic model that extends the linear car-following model as well as the minplus traffic model (a model based on the min-plus algebra). A discrete-time car-dynamics describing the traffic on a 1-lane road without passing is interpreted as a dynamic programming equation of a stochastic optimal control problem of a Markov chain. This variational formulation permits to characterize the stability of the car-dynamics and to determine the stationary regimes when they exist. The model is based on a piecewise linear approximation of the fundamental traffic diagram. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Car-following models are microscopic traffic models that describe the car-dynamics with stimulus-response equations expressing the drivers’ behavior. Each driver responds, by choosing its speed or acceleration, to a given stimulus that can be composed of many factors such as inter-vehicular distances, relative velocities, instantaneous velocities, etc. We present in this article a car-following model that extends the linear car-following model (Chandler et al., 1958; Herman et al., 1959; Gazis et al., 1959, 1961), as well as the min-plus traffic model (Lotito et al., 2005). The vehicular traffic on a 1-lane road without passing is described by first order discrete-time dynamics. The latter guarantee a conservation law of cars and express a given behavior law of drivers. The model presented here is very flexible in such a way that a large class of driver behavior law curves can be included. Moreover, the traffic dynamics are interpreted in term of optimal control. More precisely, the dynamics are seen as dynamic programming equations (DPEs) of stochastic optimal control problems of Markov chains, where the car position is interpreted as the value function of the DPE. This interpretation is important because it permits to understand the effect of the individual driver behavior on the collective traffic. The results we obtain here can be summarized as follows. We distinguished the case of traffic on a ring road from that of traffic on an open road. For the former case, where a given number of cars, having the same behavior (using the same behavioral law) move on a ring road, we first characterize the stability conditions of the car dynamics. We then solve analytically the algebraic system describing the stationary regime. From that, we obtain the mean speed, as well as the stationary distribution of cars on the road. Moreover, the mean speed is derived as a function of the mean car density on the road, giving thus the fundamental traffic diagram. For the case of traffic on an open road, we follow the trajectories of a given number of cars on an open road. The cars are supposed to have the same behavior. We follow the same steps as in the first case, but here, in order to study the stationary regime, we assume that the velocity of the first car tends to a fixed value. We solve analytically the algebraic system describing the stationary regime, and derive the local car density as a function of the stationary first car velocity. The latter relationship is nothing but the inverse of the fundamental diagram obtained in the ring road case.
E-mail address: [email protected] 0968-090X/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.trc.2012.05.005
N. Farhi / Transportation Research Part C 25 (2012) 100–112
101
We are concerned in this article by microscopic traffic modeling with a Lagrangian description of the traffic dynamics, where the function xðn; tÞ, giving the position of car n at time t (or the cumulated distance traveled by a car n up to time t), is used. The discrete-time variational formulation made here is similar to the time-continuous one used by Daganzo and Geroliminis (2008) to show the existence of a concave macroscopic fundamental diagram. In the macroscopic kinematic traffic modeling, Eulerian descriptions of the traffic dynamics are usually used, with the function nðt; xÞ giving the cumulated number of cars passing through position x up to time t (which coincides with the Moscowitz function (Daganzo, 2006) in the case of traffic without passing). The combination of a conservation law with an equilibrium law gives the well known first order traffic model of Lighthill and Whitham (1955) and Richards (1956). In this introduction, we first notice that the same approach used in macroscopic traffic modeling, combining a conservation law with an equilibrium law, can be used to derive microscopic traffic models. By this, we introduce car-following models and give a review on basic and most important existing ones. In particular, we recall the linear and the min-plus car-following models, which are particular cases of the model we present here. Finally, we recall a theoretical result on non-expansive and connected dynamic systems which we need in our developments, and present the outline of this article. _ The first order partial derivative of the function xðn; tÞ in time, denoted xðn; tÞ expresses the velocity v ðn; tÞ of car n at time t. The first order differentiation of xðn; tÞ in the car numbers Dxðn; tÞ expresses the inverse of the inter-vehicular distance _ yðn; tÞ ¼ xðn 1; tÞ xðn; tÞ between cars n and n 1 at time t. The equality of the second derivatives Dxðn; tÞ and D_ xðn; tÞ gives then the following conservation law of distance.
_ yðn; tÞ ¼ Dv ðn; tÞ:
ð1Þ
If we assume that a fundamental diagram V e , giving the velocity v as a function of the inter-vehicular distance yðv ¼ V e ðyÞÞ at the stationary traffic, exists, and that the diagram V e holds also on the transient traffic, then we have
_ yðn; tÞ ¼ v_ ðn; tÞ=V 0e ðyÞ;
ð2Þ
V 0e ðyÞ
where denotes the derivative of V e with respect to y. By combining (1) and (2) we obtain the model:
v_ ðn; tÞ ¼ V 0e ðyðn; tÞÞDv ðn; tÞ:
ð3Þ
(3) is a car-following model that gives the acceleration of car n at time t as a response to a stimulus composed of the relative speed Dv ðn; tÞ and the term V 0e ðyðn; tÞÞ. For example, if V e ðyÞ ¼ v 0 exp a=y, where v 0 denotes the free (or desired) velocity, and a is a parameter, then V 0 ðyÞ ¼ aVðyÞ=y2 and (3) gives a particular case of the Gazis, Herman, and Rothery model (Gazis et al., 1961). Almost all car following models are based on the assumption of the existence of a behavioral law V e . The latter has been taken linear in (Greenshields, 1935; Herman et al., 1959), logarithmic in (Greenber, 1959), exponential in (Newell, 1961), and with more complicated forms in other works. Bando et al. (1995) used the sigmiudal function
V e ðyÞ ¼ tanhðy hÞ þ tanhðhÞ;
ð4Þ
where tanh denotes the hyperbolic tangent function, and h is a constant. Kerner and Konhäuser (1993), Herrmann and Kerner (1998), and then Lenz et al. (1999), and Hoogendoorn et al. (2006) have used the function
( ) 1 p1 V e ðyÞ ¼ v 0 1 þ exp p3 ; p cy 2
ð5Þ
where p1 ; p2 ; p3 ; v 0 and c are parameters estimated from data. p1 ; p2 and p3 have been fixed to p1 ¼ 1000; p2 ¼ 10=2:1 and p3 ¼ 5:34 109 . In (Lenz et al., 1999), c is taken equal to 7.5. See also (Helly, 1959, 1961; Treiber et al., 2000). The simplest car-following model is the linear one, where the car dynamics are written
x_ n ðt þ TÞ ¼ aðxn1 ðtÞ xn ðtÞÞ þ b;
ð6Þ
where T is a reaction time, a is a sensitivity parameter, and b is a constant. That model can be derived from (3) with a linear fundamental diagram V e . The stability of the linear car-following model (6) and the existence of a stationary regime have been treated in (Herman et al., 1959). The min-plus traffic model (Lotito et al., 2005; Farhi, 2008) is a microscopic discrete-time car-following model based on the min-plus algebra (Baccelli et al., 1992). It consists in the following dynamics.1
xn ðt þ 1Þ ¼ minfxn ðtÞ þ v 0 ; xn1 ðtÞ rg;
ð7Þ
where v 0 is the free velocity and r is a safety distance. The idea of the model (7) is that the car dynamics is linear in the min-plus algebra (Baccelli et al., 1992), where the addition is the operation ‘‘min’’ and the product is the standard addition ‘‘þ’’. The average growth rate vector per time unit of the 1 We accept here, and in the sequel of the article, to add velocities to distances, since we take the time step dt as the time unit ðdt ¼ 1Þ. We would write xn ðt þ dtÞ ¼ minfxn ðtÞ þ v 0 dt; xn1 ðtÞ rg.
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; v ; . . . ; v Þ, where v is the stationary system, defined by v ¼ limt!1 xðtÞ=t, is then obtained (see (Lotito et al., 2005)) v ¼ ðv car-speed, derived as the unique min-plus eigenvalue of the min-plus linear map of the dynamics. The fundamental diagram is then obtained
v ¼ minðv 0 ; y rÞ q ¼ minðv 0 q; 1 rqÞ;
ð8Þ ð9Þ
(respectively q) is the stationary average car-speed (respectively car-flow), and y (respectively q) is the average where v . and q ¼ 1=y inter-vehicular distance (respectively car-density) on the ring, with q ¼ qv In (7), for t P 0, if xn1 ðtÞ xn ðtÞ > v 0 þ r, then the dynamics is xn ðt þ 1Þ xn ðtÞ ¼ v 0 . If xn1 ðtÞ xn ðtÞ 6 v 0 þ r then the dynamics is xn ðt þ 1Þ xn ðtÞ ¼ xn1 ðtÞ xn ðtÞ r. Therefore, (7) is linear in both phases of free and congested traffic. The min-plus model (7) permits to distinguish two phases in which the traffic dynamics is linear, but with a sensitivity parameter (a in (6)) in {0,1}. The linear model (6) is not constrained in the sensitivity parameter value, but it permits the modeling of only one traffic phase. The model we present here extends (6) and (7) in a way that an arbitrary number of traffic phases can be modeled, with flexibility in the sensitivity parameter value on each phase. Dual Eulerian modeling approaches with extensions to the 2D-traffic have been proposed in (Farhi, 2008; Farhi et al., 2011). For our model, we apply a similar but more general approach than the min-plus one used to analyze the dynamic system (7). Indeed, the dynamics (7) is additive homogeneous of degree one2 and is monotone.3 It is then non-expansive4 (Candrall and Tartar, 1980). The stability of the dynamic system (7) is thus guaranteed from its non-expansiveness. Moreover, (7) is connected (or communicating)5 (Gaubert and Gunawardena, 1998, 1999). An important result from (Gunawardena and Keane, 1995; Gaubert and Gunawardena, 1998) (Theorem 1 below) permits the analysis of non-expansive and connected dynamic systems. Theorem 1 (Gunawardena and Keane, 1995; Gaubert and Gunawardena, 1998). If a dynamic system xðtÞ ¼ f ðxðt 1ÞÞ is non þ x ¼ f ðxÞ admits a solution ðv ; xÞ, where x is defined up to an expansive and connected, then the additive eigenvalue problem v 2 R is unique. Moreover, the dynamic system admits an average growth additive constant, not necessarily in a unique way, and v ; v ; . . . ; v Þ. rate vector v, which is unique (independent of the initial condition) and given by vðf Þ ¼ t ðv The model treated in this article can be seen as an extension of the min-plus model (7). In Section 2, we present the model. It is called piecewise linear car-following model because it is based on a piecewise linear fundamental diagram V e . The model describes the traffic of cars on a road of one lane without passing. First (sub-Section 2.1), the cars move on a ring road. The stability conditions of the dynamic system describing the traffic are determined. Under those conditions, the cardynamics are interpreted as a dynamic programming equation (DPE) associated to a stochastic optimal control problem of a Markov chain. The DPE is solved analytically. We show that the individual driver-behavior law V e supposed in the model is realized on the collective driver-behavior at the stationary regime. The effect of the stability condition on the shape of the fundamental diagram is shown. Second (sub-Section 2.2), the traffic on an ‘‘open’’ road (a highway stretch for example) is treated. We give equivalent results of those given in the ring road case. In Section 3, we give a result on the transient traffic and present an example, where we simulate the traffic basing on a piecewise linear approximation of the diagram (5). The results obtained in this article are important because, first they permit to model (first-order) individual driver behaviors with an arbitrary precision. Second, the fundamental traffic diagram at the stationary regime is analytically derived, giving thus the collective drivers behaviors at the equilibrium. Third, the variational formulation permits to interpret the individual driver behavior in term of optimal control. Finally, many extensions of the model are possible. First, anticipation in driving is recently introduced; see (Farhi et al., in press), where we are still able to solve the dynamics analytically, and where other qualitative results are obtained. Second, the heterogeneity in driving (drivers with different behaviors) can be easily modeled, and we think that the same analytic approach can be adapted to that case. Third, the same modeling approach can be used to model lane-change in one-dimensional traffic. Finally, it may also be possible to extend this approach, at least in modeling, for the two dimensional traffic. 2. Piecewise linear car following model The behavioral law V e is an increasing curve bounded by the free speed v 0 . Moreover, V e ðyÞ ¼ 0 for y 2 ½0; yj where yj denotes the jam inter-vehicular distance. We propose here to approximate the curve V e with a piecewise-linear curve
V e ðyÞ ¼ min maxfauw y þ buw g; u2U
2
w2W
ð10Þ
A dynamic system xðtÞ ¼ f ðxðt 1ÞÞ is additive homogeneous of degree 1 if f is so, that is if 8x 2 Rn ; 8k 2 R; f ðk þ xÞ ¼ k þ f ðxÞ. A dynamic system xðtÞ ¼ f ðxðt 1ÞÞ is monotone if f is so, that is if 8x1 ; x2 2 Rn ; x1 6 x2 ) f ðx1 Þ 6 f ðx2 Þ, where the order 6 is pointwise in Rn . 4 A dynamic system xðtÞ ¼ f ðxðt 1ÞÞ is non-expansive if f is so, that is if there exists a norm k k in Rn such that 8x1 ; x2 2 Rn ; kf ðx2 Þ f ðx1 Þk 6 kx2 x1 k. 5 An additive homogeneous of degree 1 and monotone dynamic system xðtÞ ¼ f ðxðt 1ÞÞ with x 2 Rn is connected if its associated graph is strongly connected. The graph associated to that dynamic system is the graph with n nodes and whose arcs are determined as follows. There exists an arc from a node i th to a node j if limm!1 fj ðmei Þ ¼ 1, where ei denotes the i vector of the canonic basis of Rn . 3
N. Farhi / Transportation Research Part C 25 (2012) 100–112
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where U and W are two finite sets of indices. We explain hereafter how the indices u 2 U and w 2 W are used and interpreted in the traffic dynamics. In (10), those indices just permit (at that stage) to have a sufficient number of linear segments for the approximation of the curve V e . Since V e is increasing, we have auw P 0; 8ðu; wÞ 2 U W. We are interested here on the discrete-time first-order dynamics
xn ðt þ 1Þ ¼ xn ðtÞ þ minfau ðxn1 ðtÞ xn ðtÞÞ þ bu g;
ð11Þ
xn ðt þ 1Þ ¼ xn ðtÞ þ min maxfauw ðxn1 ðtÞ xn ðtÞÞ þ buw g:
ð12Þ
u2U
and u2U
w2W
It is clear that (12) extends (11). The model (12) is also an extension of both linear model (6) and min-plus model (7). In this article, we characterize the stability of the dynamics (12), calculate the stationary regimes, show that the fundamental diagrams are effectively realized at the stationary regime, and analyze the transient traffic. We will distinguish two cases: Traffic on a ring road and traffic on an ‘‘open’’ road. 2.1. Traffic on a ring road We follow here the modeling of Lotito et al. (2005). Let us consider m cars moving a one-lane ring road in one direction without passing. We assume that the cars have the same length that we take here as the unity of distance. The road is . assumed to be of size l; that is, it can contain at most l cars. The car density on the road is thus q ¼ m=l ¼ 1=y 2.1.1. Stochastic optimal control model We consider here the car dynamics (11). That is to say that each car n maximizes its velocity at time t under the constraints
xn ðt þ 1Þ 6 xn ðtÞ þ au ðxn1 ðtÞ xn ðtÞÞ þ bu ;
8u 2 U:
ð13Þ
Each constraint of (13) bounds the velocity xn ðt þ 1Þ xn ðtÞ by a sum of a fixed term bu and a term depending linearly on the inter-vehicular distance. Let us first notice that (11), on the ring road, is written
xn ðt þ 1Þ ¼ xn ðtÞ þ minfau ðxn1 ðtÞ xn ðtÞÞ þ bu g;
for n P 2;
u2U
x1 ðt þ 1Þ ¼ x1 ðtÞ þ minfau ðxn ðtÞ þ l x1 ðtÞÞ þ bu g; u2U
which can also be written
xn ðt þ 1Þ ¼ minfð1 au Þxn ðtÞ þ au xn1 ðtÞ þ bu g; u2U
for n P 2;
x1 ðt þ 1Þ ¼ minfð1 au Þx1 ðtÞ þ au xn ðtÞ þ au m=q þ bu g: u2U
Let us denote by M u ; u 2 U the family of matrices defined by
2 6 6 Mu ¼ 6 6 4
1 au
0
au .. .
1 au .. .
..
0
0
3
au
7 7 7; 7 5
0 .
au 1 au
and by cu ; u 2 U, the family of vectors defined by
cu ¼ t ½au m=q þ bu ; bu ; . . . ; bu : The dynamics (11) are then written:
xn ðt þ 1Þ ¼ min ½M u xðtÞn þ cun ; u2U
1 6 n 6 m:
ð14Þ
The system (14) is additive homogeneous of degree 1 by the definition of the matrices M u ; u 2 U. It is monotone under the condition that all the components of M u ; u 2 U are non-negative, which is equivalent to au 2 ½0; 1; 8u 2 U. Hence, under that condition, the system (14) is non-expansive. Moreover, the matrices Mu ; u 2 U are stochastic.6 Those matrices can then be seen as transition matrices of a controlled Markov chain, where the set of controls is U. The connectedness of the system (14), as defined in the previous section, is related to the irreducibility of the Markov chain with transition matrices M u ; u 2 U. It is easy to check that (14) is connected if and only 6
We mean here Muij P 0; 8i; j and
P
u j M ij
¼ 1; 8i.
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if 9u 2 U; au 2 ð0; 1; see Appendix A for the proof. That condition is interpreted in term of traffic by saying that every car moves by taking into account the position of the car ahead. Consequently, under the condition 8u 2 U; au 2 ½0; 1 and 9u 2 U; au 2 ð0; 1, the dynamic system (14) is non-expansive and is connected. Therefore, by Theorem 1, we conclude that the additive eigenvalue problem
v þ xn ¼ min f½M u xn þ cun g; u2U
1 6 n 6 m;
ð15Þ
; xÞ, where x is defined up to an additive describing the stationary regime of the dynamic system (14), admits a solution ðv 2 R is unique. Moreover, the dynamic system admits a unique average constant, not necessarily in a unique way, and v . growth rate per time unit v, whose components are all equal and coincide with v The average growth rate per time unit v of the system (14) is interpreted in term of traffic as the stationary car-velocity. The additive eigenvector x gives the asymptotic distribution of cars on the ring. x is given up to an additive constant, since the ; xÞ is a solution of (15) then ðv ; e þ xÞ is also a car-dynamics (14) is additive homogeneous of degree 1. That is to say that if ðv solution for (15), for every constant e 2 R. Let us now give an interpretation of the model in term of ergodic stochastic optimal control. Indeed, (15) can be seen as a dynamic programming equation of an ergodic stochastic optimal control problem of a Markov chain with transition matrices Mu ; u 2 U and costs cu ; u 2 U, and with a set of states N ¼ f1; 2; . . . ; mg. The stochastic optimal control problem of the chain is written
( minE s2S
) T 1X cuntt ; T!þ1 T t¼0 lim
ð16Þ
where S is a set of feedback strategies on N . A strategy s 2 S associates to every state n 2 N a control u 2 U (that is ut ¼ sðnt Þ). ; xÞ for the dynamic programming Eq. (15). The following result gives one solution ðv ; xÞ given by: Proposition 1. The system (15) admits a solution ðv
þ bu g; v ¼ min fau y u2U ; ðm 2Þy ; x ¼ t ½ðm 1Þy
...;
; 0: y
Proof. First, because of the symmetry of the system (15), it is natural that the asymptotic car-positions xn ; 1 6 n 6 m are uniformly distributed on the ring, and that the optimal strategy is independent of the state x. Let us prove it. 2 U be defined by au y . Let x be the vector given in Proposition 1. Then 8n 2 f1; 2; . . . ; mg þ bu ¼ minu2U fau y þ bu g ¼ v Let u we have
þ bu Þ þ xn ¼ minðau y þ bu Þ þ xn ¼ min ½M u xn þ cun ¼ v þ xn : ½M u xn þ cun ¼ ðau y u2U
u2U
In term of traffic, Proposition 1 shows that the car-dynamics is stable under the condition au 2 ½0; 1, and the average car speed is given by the additive eigenvalue of the equilibrium equation in the case where the system is connected. Moreover, it affirms that the fundamental diagram supposed in the model is realized at the stationary regime.
þ bu g; v ¼ min fau y u2U
ð17Þ
q ¼ minfau þ bu qg:
ð18Þ
u2U
It is important to note here that, up to the assumption au 2 ½0; 1; 8u 2 U, every concave curve V e or Q e can be approximated with (17) or (18). Indeed, approximating fundamental diagrams using those formulas is nothing but computing Fenchel transforms; see (Daganzo, 2006; Aubin et al., 2009). More precisely, if we denote by V the set V ¼ fbu ; u 2 Ug and define the function g by:
g: V ! R
v ¼ bu
# au ;
then
q ¼ Q e ðqÞ ¼ minðqv gðv ÞÞ ¼ g ðqÞ; v 2V
where g denotes the Fenchel transform of g. Finally, we note that the min-plus linear model is a particular case of the model presented in this section, where U ¼ fu1 ; u2 g with ða1 ; b1 Þ ¼ ð0; v Þ and ða2 ; b2 Þ ¼ ð1; rÞ. In this case, the fundamental traffic diagram is approximated with a piecewise linear curve of two segments.
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2.1.2. Stochastic game model We consider in this section the car dynamics (12), again with the assumption 8ðu; wÞ 2 U W; auw 2 ½0; 1 and 9ðu; wÞ 2 U W; auw 2 ð0; 1. The dynamic system (12) is interpreted as a stochastic dynamic programming equation associated to a stochastic game problem on a controlled Markov chain. As above, a generalized eigenvalue problem is solved. The extension we make here approximates non-concave fundamental diagrams. In term of traffic, we take into account the drivers’ behavior changing from low densities to high ones. The difference between these two situations is that in low densities, drivers, moving, or being able to move with high velocities, they try to leave large safety distances between each other, so the safety distances are maximized; whilst in high densities, drivers, moving, or having to move with low velocities, they try to leave small safety distances between each other in order to avoid jams; so they minimize safety distances. To illustrate this idea, let us consider the following two dynamics of a given car n.
xn ðt þ 1Þ ¼ minfxn ðtÞ þ v ; xn1 ðtÞ rg;
ð19Þ
xn ðt þ 1Þ ¼ minfxn ðtÞ þ v ; maxfxn1 ðtÞ r; ðxn ðtÞ þ xn1 ðtÞÞ=2gg:
ð20Þ
The dynamics (19) is a min-plus dynamics which grossly tell that cars move with their desired velocity v at the fluid regime and they keep a safety distance r at the congested regime. The dynamics (20) distinguishes two situations at the congested regime: In a relatively low density situation where the cars are separated by a distance that equals at least to 2r we have
maxfxn1 ðtÞ r; ðxn ðtÞ þ xn1 ðtÞÞ=2g ¼ xn1 ðtÞ r: In a high density situation, where the cars are separated by distances less than 2r we have
maxfxn1 ðtÞ r; ðxn ðtÞ þ xn1 ðtÞÞ=2g ¼ ðxn ðtÞ þ xn1 ðtÞÞ=2: In the latter case, we accept the cars moving closer but by reducing the approach speed in order to avoid collisions. We think that the additional assumption that drivers leave small safety distances between each other in the congested traffic phase, introduced in (20), is acceptable in term of driving behavior. However, that assumption cannot be obtained without introducing a maximum operator in the dynamics (i.e. with only minimum operators). Indeed, with only minimum operators the approach is mechanically reduced with the increasing of the car-density (in fact this is the concaveness of the fundamental diagram). Because of the realness of such scenarios, we think that the fundamental traffic diagram should be composed of two parts, a concave part at the fluid regime, and a convex part at the congested regime. The dynamics (12) generalizes this idea. The dynamics (12) can be written
xn ðt þ 1Þ ¼ min max ½Muw xðtÞn þ cuw ; n u2U
1 6 n 6 m;
w2W
ð21Þ
where
2 M uw
6 6 ¼6 6 4
1 auw
0
auw .. .
1 auw .. .
0
0
auw
0 ..
.
3 7 7 7; 7 5
auw 1 auw
and
cuw ¼ t ½auw m=q þ buw ; buw ; . . . ; buw : The stationary traffic is then described by the eigenvalue problem
uw ½M xn þ cuw ; n
v þ xn ¼ min max u2U w2W
1 6 n 6 m:
The following result generalizes Proposition 1. ; xÞ given by: Theorem 2. The system (22) admits a solution ðv
þ buw g; v ¼ min maxfauw y u2U w2W ; ðm 2Þy ; x ¼ t ½ðm 1Þy
...;
; 0: y
Proof. Similar to the proof of Proposition 1. h
ð22Þ
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We can easily check that the dynamic system (21) is non-expansive under the condition auw 2 ½0; 1; 8ðu; wÞ 2 U W (by the same way as in the case of the dynamic system (14)). Its stability is thus guaranteed under that condition. If, in addition, 9ðu; wÞ 2 U W; auw 2 ð0; 1, then the system is connected. Therefore, by Theorems 1 and 2, we conclude that the dynamic (given by Theroem 2 and system (21) admits a unique average growth rate vector v, whose components are all equal to v interpreted as the average car speed). The behavior law supposed in the model is realized at the stationary regime.
þ buw g; v ¼ min maxfauw y u2U w2W
ð23Þ
q ¼ min maxfauw þ buw qg:
ð24Þ
u2U
w2W
In term of stochastic optimal control, the system (22) can be seen as a dynamic programming equation associated to a stochastic game, with two players, on a Markov chain. The set of states of the chain is again N ¼ f1; 2; . . . ; mg. The chain is controlled by two players, a minimizer one with a finite set U of controls, and a maximizer one with a finite set W of controls. The transitions and the costs of the chain are given by the matrices Muw and the vectors cuw ; ðu; wÞ 2 U W defined above. The stochastic optimal control problem is
(
) T 1X ut wt min max js2S E lim cnðtÞ ; T!þ1 T t¼0
ð25Þ
where S is the set of strategies assoicating to every state n 2 N a couple of commands ðu; wÞ 2 U W. It is assumed here that the maximizer knows at each step the decision of the minimizer. We now give a consequence of the stability condition auw 2 ½0; 1; 8ðu; wÞ 2 U W, on the shape of the fundamental diagrams (23) and (24). As shown in Fig. 1, where we have drawn the fundamental diagram (5) (with v 0 ¼ 14 meter by half second, and c ¼ 7:5), the stability condition puts the curves (23) and (24) in specific respective regions in the plan. Indeed, for the diagram (23), if we assume that V e is bounded by v 0 , V e ðyÞ ¼ 0; 8y 2 ½0; yj , and that V e is continuous (and increasing), then starting by the point ðyj ; 0Þ, one cannot join any point above the line passing by ðyj ; 0Þ and having the slope 1, with any sequence of segments of slopes auw 2 ½0; 1. We can write
V e ðyÞ 6 maxð0; minðv 0 ; y yj ÞÞ: Similarly, on the diagram (24), if we assume that Q e is continuous and Q e ðqÞ ¼ 0; 8q 2 ½qj ; 1, then going back from the point ðqj ; 0Þ, one cannot attain any point above the line passing by ðqj ; 0Þ and ð0; 1Þ, with a sequence of segments having their ordinates at the origin (auw Þ in ½0; 1. We can write
Q e ðqÞ 6 maxð0; minðv 0 q; 1 q=qj ÞÞ:
2.2. Traffic on an open road We study in this section the traffic on an open road with one lane and without passing. The fact that the traffic description is lagrangian is important here. Indeed, we follow the movement of a given number of cars on an open road, and we are interested on what happen on only the stretch where all the cars move. The average car-density is then defined as the total number of cars divided by the stretch occupied by the cars. Therefore, the average car-density changes over time.
Fig. 1. The effect of the stability condition auw 2 ½0; 1 on the shape of the fundamental diagram.
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We are interested, in particular, in the stationary regime of the traffic dynamics. We will assume that the velocity of the first car becomes stationary from a given time horizon, and look for the stationary car-density in that case. Note that this is the inverse question of the one posed in the ring road case, where we have the average car-density, and look for the average car-speed. We are interested in the following dynamics.
x1 ðt þ 1Þ ¼ x1 ðtÞ þ v 1 ðtÞ;
ð26Þ
xn ðt þ 1Þ ¼ min maxfxn ðtÞ þ auw ½xn1 ðtÞ xn ðtÞ þ buw g: u2U
w2W
If we denote by M uw the matrices
2
1
0
6a 6 uw M uw ¼ 6 6 .. 4 .
1 auw .. .
0
3
0
7 7 7; 7 5
0 ..
.
auw 1 auw
0
and by cuw ðtÞ the vectors
cuw ðtÞ ¼ t ½v 1 ðtÞ; buw ; . . . ; buw ; buw ; for ðu; wÞ 2 U W and t 2 N, then the dynamic system (26) is written
xn ðt þ 1Þ ¼ min max ½Muw xðtÞn þ cuw n ðtÞ ; u2U
w2W
1 6 n 6 m:
ð27Þ
It is easy to check that the dynamic system (27) is additive homogeneous of degree 1, and is monotone under the condition 8ðu; wÞ 2 U W; auw 2 ½0; 1. Therefore, under that condition, (27) is non-expansive. However, (27) is not connected for every ðu; wÞ 2 U W. We will be interested here, in particular, in the stationary regime, where v 1 ðtÞ reaches a fixed value v 1 . For this case, the eigenvalue problem associated to (26) is
v þ x1 ¼ x1 þ v 1 ; v þ xn ¼ min maxfxn þ auw ðxn1 xn Þ þ buw g: u2U w2W
ð28Þ
The system (28) is also written
uw ½M xn þ cuw ; n
v þ xn ¼ min max u2U w2W
1 6 n 6 m;
ð29Þ
where
cuw ¼ t ½v 1 ; buw ; . . . ; buw ; buw : Then the following result is a corollary of Theorem 2. ; xÞ is a solution for the system (28), where Corollary 1. For all y 2 R satisfying minu2U maxw2W ðauw y þ buw Þ ¼ v 1 , the couple ðv v ¼ v 1 and x is given up to an additive constant by
x ¼ t ½ðm 1Þy; ðm 2Þy; . . . ; y; 0:
ð30Þ
; wÞ 2U W Proof. The proof is similar to that of Proposition 1. Let y 2 R satisfying minu2U maxw2W ðauw y þ buw Þ ¼ v 1 . Let ðu such that auw y þ buw ¼ v 1 . Let x be given by (30). Then 8n 2 f1; 2; . . . ; mg we have
½Muw xn þ cunw ¼ ðauw y þ buw Þ þ xn ¼ min maxðauw y þ buw Þ þ xn ¼ min max½Muw xn þ cuw n ¼ v 1 þ xn : u2U
w2W
u2U
w2W
We can easily check that for ðu; wÞ 2 U W such that auw ¼ 0 and buw ¼ v 1 , every inter-vehicular distance y 2 R satisfies the condition minu2U maxw2W ðauw y þ buw Þ ¼ v 1 . Thus, such couples ðu; wÞ do not count for that condition. Corollary 1 can then be announced differently. Let us denote by W u for u 2 U the family of index sets
W u ¼ fw 2 W; ðauw ; buw Þ–ð0; v 1 Þg; the asymptotic average inter-vehicular distance: and by y
¼ max min y u2U
w2W u
v 1 buw auw
;
ð31Þ
2 R (i.e. where we use the convention a=0 ¼ þ1 if a > 0 and a=0 ¼ 1 if a < 0. Then Corollary 1 tells simply that if y < þ1), then the dynamic system (28) admits a solution ðv ; xÞ where v ¼ v 1 is unique, and where x is not necessarily 1 < y unique and is given up to an additive constant by
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; ðm 2Þy ; . . . ; y ; 0: x ¼ t ½ðm 1Þy Non-uniform asymptotic car-distributions can also be obtained. Let us clarify the following three cases. ¼ þ1. In this case, the distance between the first car and the other (a1) If 9u 2 U; 8w 2 W u ; auw ¼ 0 and buw < v 1 , then y cars increases over time and goes to þ1. The asymptotic car-distribution on the road is not uniform. ¼ 1. In this case, the first car is passed by all other cars, and the (a2) If 8u 2 U; 9w 2 W u ; auw ¼ 0 and buw > v 1 , then y distance between the first car and the other cars increases over time and goes to þ1. The asymptotic car-distribution on the road is not uniform. (a3) If 8u 2 U; 8w 2 W u ; auw ¼ 0 and if minu2U maxw2W u buw ¼ v 1 , then for all x 2 Rm , ðv 1 ; xÞ is a solution for the system (28). In this case, every distribution of the cars moving all with the constant velocity v 1 is stationary. The formula (31) is the fundamental traffic diagram expressing the average inter-vehicular distance as a function of the car speed at the stationary regime. In the case where only a minimum operator is used in (26), the formula (31) is reduced to the convex fundamental diagram
¼ max y u2U
v 1 bu au
ð32Þ
:
If we define the average car-density as the number of vehicles divided by the length of the stretch occupied by all the vehicles (that is the distance between the first car and the last one), then the stationary average car-density q is the inverse of the stationary average inter-vehicular distance. Therefore, from (31), we have
auw
q ¼ min max u2U
v 1 buw
w2W
ð33Þ
:
In the case where a stationary flow q is known (which is in fact not sufficient to determine uniquely the stationary cardensity), rather than the stationary velocity of the first car, then from (33), the possible stationary average car-densities satisfy
1 ¼ min max u2U
w2W
auw q buw q
ð34Þ
;
which is simply reversing the formula (24), up to the particular cases clarified above (for auw ¼ 0). Note that we usually have fundamental diagrams where two possible stationary car-densities are possible for a positive non-maximal stationary carflow, only one possible stationary car-density (called critical) for maximum stationary car-flows, and an infinity of stationary car-densities (zero and all the densities bigger than the jam density) for the null stationary car-flow. 3. Transient traffic In this section, we first give a result on the uniqueness of the additive eigenvector associated to the car-dynamics.7 The result characterizes also the convergence of the car trajectories to the uniform distribution of cars on the road. We then show, on an example, the simulation of the model. Proposition 2. If auw 2 ð0; 1Þ; 8ðu; wÞ 2 U W (respectively au 2 ð0; 1Þ; 8u 2 U), then the eigenvector x given in Theorem 2 (respectively Proposition 1) and Corollary 1 is unique, and the vector xðtÞ; t 2 N whose dynamics is defined by (21) (respectively (14) and (27) converges (uniquely) to the eigenvector x.
Proof. Let us do the proof for the dynamics (21) and Theorem 2. We define a new vector z as follows.
for 1 6 n 6 m 1;
zn ¼ xn xnþ1 ; z m ¼ xm x 1 þ l :
The vector z gives the inter-vehicular distances on the ring road. It is then easy to check that the dynamics of z is simply
zn ðt þ 1Þ ¼ min max½M uw zðtÞn ; u2U
w2W
1 6 n 6 m:
ð35Þ
Now, the key of the proof is that if auw 2 ð0; 1Þ; 8ðu; wÞ 2 U W, then all the (multiplicative) eigenvalues of any matrix M uw , for ðu; wÞ 2 U W, have their modules less than 1, except the eigenvalue 1 which is simple.8 Moreover, the eigenvector asso ð1; 1; . . . ; 1Þ, where ciated to the eigenvalue 1 is9 (1, 1, . . . , 1). Therefore, the sequence of vectors fzðtÞgt2N converges to a vector y 7 8 9
Note that Theorem 1 guarantees the uniqueness of the additive eigenvalue but not that of the associated eigenvector. It is easy to determine the eigenvalues of M uw since it is a sum of two circulant matrices. Indeed the matrices M uw are stochastic, so all the lines sum to 1.
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gives the common (or the average) inter-vehicular distance at the stationary regime. The proof is similar for the dynamics (14) y and (27). h Proposition 2 tells that under the condition auw 2 ð0; 1Þ; 8ðu; wÞ 2 U W, the transient traffic consists to closer the car positions on the road to the stationary car position. We give below an example where we approximated a fundamental traffic diagram (collective behavioral law), and simulated the corresponding car dynamics, according to the model presented in this article. Nevertheless, we notice that the heterogeneity in driving should be modeled in order to be able to perform better simulations. Indeed, a behavioral law should be determined and approximated for every driver, and the dynamics (26) has to be adapted for this. This can be done easily in term of medeling (it consists to index the parameters auw and buw with respect to every driver (car), or at least to a given number of classes of drivers.) In the case of traffic on an open road, the dynamics taking into account the driving heterogeneity can be written as follows.
x1 ðt þ 1Þ ¼ x1 ðtÞ þ v 1 ðtÞ; xn ðt þ 1Þ ¼ min maxfxn ðtÞ þ anuw ½xn1 ðtÞ xn ðtÞ þ bnuw g: u2U
ð36Þ
w2W
The stability condition in the heterogeneous driving case is similar to that of the homogeneous driving case. That is anuw 2 ½0; 1; 8ðn; u; wÞ 2 f1; 2; . . . ; mg U W. We project to study this extension in a forthcoming article, where we need to model vehicle passing and/or lane change in order to allow faster vehicles to pass or to change lane when bothered by slower vehicles. The stationary regimes (average car-speed and car-distribution on the road or on each lane) need also to be determined. Another important factor that we would like to take into account is the anticipation in driving. That is that drivers respond to a stimulus that takes into account headway spacings with respect to a number of leaders (instead of only the headway spacing with respect to the first leader). The multi-anticipation in driving, in piecewise-linear car-following modeling, have been treated recently in (Farhi et al., in press), where the stability is again characterized and the stationary car distributions are determined. Example 1 is a short simulation test of the model (without heterogeneity nor anticipation). In (Farhi et al., in press), further numerical tests are done, and a parameter identification approach is proposed to piecewise-linear car following models. Heterogeneity in driving has to be included in order to perform reliable parameter identifications. Example 1. Let us simulate the car-dynamics (26). We take as the time unit half a second (1/2 s), and as the distance unit 1 meter (m). The parameters of the model are determined by approximating the behavior law (5), with a free velocity v 0 ¼ 14 m=ð1=2Þ s (which is about 100 km/h) and c ¼ 7:5 as in (Lenz et al., 1999); see Fig. 2. The behavior law is approximated by the following piecewise linear curve of six segments.
e ðyÞ ¼ maxfa1 y þ b ; minfa2 y þ b ; a3 y þ b ; a4 y þ b ; a5 y þ b ; a6 y þ b gg; V 1 2 3 4 5 6
ð37Þ
where the parameters ai and bi for i ¼ 1; 2; . . . ; 6 are given in Table 1. We simulate the piecewise linear car-following model associated to the approximation above.
x1 ðtÞ ¼ x1 ðt 1Þ þ v 1 ðtÞ; e ðxn1 ðt 1Þ xn ðt 1ÞÞ: xn ðtÞ ¼ xn ðt 1Þ þ V
ð38Þ
The velocity of the first car v 1 ðtÞ; t P 0 is varied in the time interval ½0; 1000, then fixed to the free velocity v 0 ¼ 14 m=ð1=2Þ s in the time interval ½1000; 3000, and finally fixed on a velocity that exceeds v 0 in the remaining time ½3000; 7200. The average inter-vehicular distance is then computed at every time t, and the results are shown in Fig. 3.
Fig. 2. Approximation of the behavioral law (5) with a piecewise linear curve.
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Table 1 Parameters ai and bi ; i ¼ 1; 2; . . . ; 6 corresponding to the approximation (37). Segments
1
2
3
4
5
6
ai
0 0
0.54 8.1
0.32 1.47
0.13 6.11
0.34 10.6
0 14
bi
Fig. 3. Simulation results. On the left-side: the first car velocity (solid line), and the average inter-vehicular distance (dash line) functions of time. On the right side: the approximation of the behavior law (5) (dash line), and the average inter-vehicular distance obtained by simulation in function of the velocity of the first car (solid line).
The simple simulation we made here permits to have an idea of the traffic in the transient regime. Fig. 3 shows how the average of the inter-vehicular distance is changed due to a changing in the velocity of the first car. In the right side of Fig. 3, we compare the fundamental law assumed in the model with the diagram giving the average inter-vehicular distance (with respect to the number of cars) function of the velocity of the first car. We observe that loops are obtained on that diagram in the transient traffic. The loops are interpreted by the fact that once the velocity of the first car is temporarily stationary, the velocities of the following cars, and thus also the average velocity of the cars, make some time to attain the first car velocity. It can also be interpreted by saying that even though the cars have, individually, the same response to a changing in intervehicular distance; their collective response depends on whether the inter-vehicular distance is increasing or decreasing. The apparition of such loops is due to the reaction time of drivers. It can also be related to the number of anticipation cars in case of multi-anticipative modeling. However, one may measure on a given section, different car-flows for the same car density (or occupancy rate) depending on the traffic acceleration or deceleration, and interpret it as the hysteresis phenomenon (Edie, 1961; Treiterer and Myers, 1974; Zhang, 1999; Geroliminis and Sun, 2011). Finally let us point out the infinite sate-space case, where an infinite number of cars is considered. First, we notice that in that case, we need to assume that the length of the road is infinite. Therefore, only the open road case can be considered. Moreover, the set of cars must be countable, in order to be able to write the dynamics. It is well known (see for example (Hernandez-Lerma and Lasserre, 1999; Whittle, 1986)) that in average-cost programming, the results of Theorem 1 do not hold for the infinite state-space case, without additional assumptions.10 Nevertheless, the dynamics we study here is a particular case for which we can tell more. First, we can easily check that, except the non-uniform car-distribution cases (a1), (a2) and (a3) discussed above (in Section 2), the additive eigenvalue problem (stationary regime equation) admits a solution (the same one as that of the finite state-space case). Then we know that for every state x0
such that ðx0 xÞ is bounded jx0n xn j < þ1; 8n 2 N , the average growth rate per time-unit of the dynamics (27), starting from x0 , coincides with the additive eigenvalue. Therefore, under that condition, we are in the same situation as in the finite statespace case, i.e. the average car-speed on the road is v 1 . 4. Conclusion We proposed in this article a car-following model that extends the linear car-following model as well as the min-plus model. The stability and the stationary regimes of the model are characterized thanks to a variational formulation of the car-dynamics. This formulation, although already made with continuous-time models, it permits to clarify the stimulus10 In the infinite state-space case of the average-cost programming, even the existence of an average-cost optimal policy or a solution for the additive eigenvalue problem are in question. Moreover, the existence of a solution to the additive eigenvalue problem is no more sufficient for optimality. That is to say that even when an additive eigenvalue exists, it may not coincide with the average growth rate per time-unit of the car dynamics (the average car-speed in our case).
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response process in microscopic discrete-time traffic models, and to interpret it in term of stochastic optimal control. Among the important questions to be treated in the future, the impacts of heterogeneity and anticipation in driving, on the transient and stationary traffic regimes, based on the model proposed in this article. Anticipation in driving has been introduced recently for our model (see (Farhi et al., in press)). Other works on multilane traffic and driver heterogeneity are underway. Appendix A. Connectedness of system (14) Let x 2 Rm . We denote by h : Rm ! Rm the operator defined by hðxðtÞÞ ¼ xðt þ 1Þ, where xn ðt þ 1Þ, for 1 6 n 6 m are given by the definition of the system (14). That is
xn ðt þ 1Þ ¼ min ½M u xðtÞn þ cun ; u2U
1 6 n 6 m:
If 9u 2 U, such that au 2 ð0; 1, then for all n 2 f1; 2; . . . ; mg, there exists an arc, on the graph associated to h, going from n 1 to n (n being cyclic in f1; 2; . . . ; mg). Indeed,
xn ðt þ 1Þ ¼ ð1 au Þxn ðtÞ þ au xn1 ðtÞ þ bu : Then since au > 0, we get:
lim hn ðmen1 Þ ¼ lim ½au m þ bu ¼ 1:
m!1
m!1
where en1 denotes the ðn 1Þth vector of the canonic basis of Rm . Therefore the graph associated to h is strongly connected. If 8u 2 U; au ¼ 0, then we can easily check that all arcs of the graph associated to h are loops. Hence that graph is not strongly connected.
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Treiterer, J., Myers, J., 1974. The hysteresis phenomena in traffic flow. In: Proceedings of the Sixth Symposium on Transportation and Traffic Theory, pp. 13– 38. Whittle, P., 1986. Optimization Over Time, vol. 2. Wiley. Zhang, H.M., 1999. A mathematical theory of traffic hysteresis. Transportation Research Part B: Methodological 33 (1), 1–23,