Car-oriented mean-field theory for traffic flow models

We present a new analytical description of the cellular automaton model for singlelane traffic. In contrast to previous approaches we do not use the occupation number of sites as dynamical variable but rather the distance between consecutive cars. Therefore certain longerranged correlations are taken into account and even a mean-field approach yields non-trivial results. In fact for the model with vmax = 1 the exact solution is reproduced. For vmax = 2 the fundamental diagram shows a good agreement with results from simulations. Despite a large number of publications about the cellular automaton approach to traffic flow (see e.g. [1] and references therein) only a few of those deal with a systematic analytical description. Most works make use of large-scale computer simulations which can be carried out very efficiently for this class of models. Nevertheless, analytical results—exact or approximate—may give important information relevant for a complete understanding of those models. The most important exact result is certainly the solution of the model for vmax = 1 [2]. This result has been obtained using n-cluster approximation [2, 3], i.e. an improved meanfield theory taking into account correlations between n neighbouring sites. For vmax = 1 the 2-cluster approximation is exact [2]. For higher velocities the n-cluster approximation for small n already yields very good results for the so-called fundamental diagram (flow-density relationship) [2, 3]. In computer simulation studies there are in principle two different approaches [3] called site-oriented and car-oriented¶. In the site-oriented approach the state of the system is specified by storing the state of each cell which can either be empty or occupied by a single car with velocity v = 0, 1, . . . , vmax. In the car-oriented approach, on the other hand, one stores the velocity of each car and the distance to the next car ahead. Since the cluster approximation corresponds to a site-oriented approach, this analogy inspired us to investigate an analytical description based on the car-oriented approach, the socalled car-oriented mean-field theory (COMF)+. The COMF already takes into account some longer-ranged correlations so that one can hope that it yields at least a good approximation. For completeness we briefly repeat the definition of the CA model for single-lane traffic flow [5] in the following. The street is divided into L cells of a certain length (for realistic § E-mail address: [email protected] ‖ E-mail address: [email protected] ¶ In [3] this approach has been called particle-oriented. + A brief account of some preliminary results has already been given in [4]. 0305-4470/97/040069+07$19.50 c © 1997 IOP Publishing Ltd L69 L70 Letter to the Editor applications 7.5 m) which can be occupied by at most one car or be empty. The cars have an internal parameter (‘velocity’) which can take on only integer values v = 0, 1, 2, . . . , vmax. The dynamics of the model are described by the following update rules for the velocities and the motion of cars [5]. In the first step all cars with velocities vi < vmax are accelerated by one velocity unit, vi → v′ i = vi + 1. The following step describes the slowing down due to other cars and prevents accidents. All cars with velocities v′ i > di (where di is the number of free cells in front of car i) decelerate to velocity v′′ i = di . The last step in the velocity update is a randomization effect taking into accout several aspects of the driver’s behaviour, e.g. fluctuations in driving style, overreaction at breaking, and retarded acceleration: every car with velocity v′′ i > 0 will slow down one unit with probability p, i.e. v′′ i p → v′′′ i = v′′ i − 1. In the final step the car then moves v′′′ i sites. These four rules, referred to as step 1 to step 4 in the following, are applied to all cars at the same time (parallel dynamics). COMF for vmax = 1 We denote the probability to find at time t (exactly) n empty sites in front of a vehicle by Pn(t). As in [2, 3] we change the order of the update steps to 2–3–4–1. This change has to be taken into account when calculating the flux f (c, p). It has the advantage that after step 1 there are no cars with velocity 0, i.e. all cars have velocity 1. The time evolution of the probabilities Pn(t) can conveniently be expressed through the probability g(t) (ḡ(t) = 1 − g(t)) that a car moves (does not move) in the next timestep. In order to find the time evolution of the Pn(t) we first determine from which configurations at time t a given state at time t + 1 could have been evolved under the rules 2–3–4–1. Take for instance a car—called second car in the following—which has n > 1 free sites in front, i.e. its distance to the next car ahead (called first car in the following) is n+ 1 sites. This configuration might have evolved from four different configurations at time t , depending on whether (i) both cars moved in the timestep t → t+1 (which happens with probability qg(t)), (ii) both cars did not move (with probability pḡ(t)), (iii) only the first car moved (with probability pg(t)), or (iv) only the second car moved (with probability qḡ(t)). This means that the second car at time t had either n free site in front (cases (i) and (ii)), or n− 1 free sites (case (iii)), or n+ 1 free sites (case (iv)). The special cases n = 0, 1 can be treated in a analogous fashion. In this way one obtains the time evolution of the probabilities as P0(t + 1) = ḡ(t)[P0(t)+ qP1(t)] (1) P1(t + 1) = g(t)P0(t)+ [qg(t)+ pḡ(t)]P1(t)+ qḡ(t)P2(t) (2) Pn(t + 1) = pg(t)Pn−1(t)+ [qg(t)+ pḡ(t)]Pn(t)+ qḡ(t)Pn+1(t). (3) A car will move in the next timestep if there is at least one empty cell in front of it (probability ∑ n>1 Pn(t)) and if it does not decelerate in the randomization step 3 (probability q = 1 − p). Therefore, the probabilities g(t) and ḡ(t) are given by g(t) = q ∑ n>1 Pn(t) = q[1 − P0(t)] ḡ(t) = P0(t)+ p ∑ n>1 Pn(t) = p + qP0(t) (4) where we have used the normalization