Theoretical approach to two-dimensional traffic flow models

Abstract

Deterministic and stochastic cellular automata models available to study two-dimensional traffic fIow are compared in this paper. It is shown that a connection between them can be made only when the infinite time and infinite system limits are taken in the appropriate order. We also stress the crucial importance of the choice of boundary conditions in the deterministic model to obtain bulk properties. PACS number(s): 64.60.Cn, 05.20.Dd, 47.90.+a, 89.40.+k In the last few years there has been a growing interest in the study of cellular automata (CA) models which try to mimic, with simple rules, the features of traffic in highways [1]. Recently some work has also been focused on the behavior of traffic in cities. Along this line two CA models have been proposed [2,3]. Both describe two equal populations of cars moving in perpendicular directions from node to node of a square-lattice-like city with streets pointing only up and right and periodic boundary conditions (BC's). Movement occurs in discrete time steps and traffic lights rule it so that they allow horizontal and vertical movements alternately. The interaction between cars forbids a car to jump to a node if it is occupied by another car at the same time step.The model of Ref. [3] includes the ability of cars to turn with probability y e [0,1/2]. When y=0 this model reduces to that of Ref. [2]. In both models a phase transition from a freely moving phase to a jammed one occurs above a certain density of cars. Whereas the jammed phase is characterized in both models by a low value of the average velocity U (which approaches 0 as y goes to 0), there is a drastically different behavior of the average velocity as a function of the density of cars n in the freely moving phase. The deterministic model [2] shows that U remains constant and equal to its maximum value up to the transition density, while the stochastic model [3] exhibits an almost linear decrease with slope -1/2 which appears not to depend on y. A recent mean-field-like study of the latter model [4] confirms this fact. It is the purpose of this paper to make clear the connection between these two models and the origin of this discontinuity of the behavior of v(n) in the parameter 7.In Ref.[4] a microscopic description of these models was achieved by introducing a set of Boolean variables, namely (i) occupation numbers of site r and time step t for vertical and horizontal cars, p, ', and v'", respectively, (ii)

Freezing transition in a four-directional traffic model for facing and crossing pedestrian flow

“…(13) and (14) with (15)-(18) are consistent with the mean-field approximation of BML model derived by Molera et al [25].…”

“…The BML model is two-directional traffic. The BML model has been extended to various pedestrian traffics and have been studied extensively [24][25][26][27].…”

Statistical physics of vehicular traffic and some related systems

“…The analogs of the equations (64) and (65) (see Appendix I for details) have analogous physical interpretations. As usual, in the naive SOMF approximation one neglects the correlations between the occupations of different sites [255]. Since none of the sites is allowed to be occupied by more than one particle at a time, Pauli operators may be used to develop an analytical theory of vehicular traffic [256] but one should keep in mind that the particles representing the vehicles are purely classical and the system does not have any quantum mechanical characteristics.…”

“…A scheme for a truly microscopic analysis of the BML model begins [255] by introducing the corresponding generalized occupation variables n ↑ (x, y; t) and n → (x, y; t), which describe the state of occupation of the sites (x, y) by the north-bound and east-bound vehicles, respectively, on the two-dimensional street-network. The analogs of the equations (64) and (65) (see Appendix I for details) have analogous physical interpretations.…”

Derivation of the fundamental traffic diagram for two circular roads and a crossing using minplus algebra and Petri net modeling.

“…In the cellular automaton classification [4], the model used here is of Biham-Middleton-Levine [3] (BML) type (used to describe simplified regular towns). But contrary to BML, we accept the possibility of turning like in [11], [12], [13] but, here, with deterministic turn (if we give a number to each car entering in the crossing, the odd cars go south and the even cars go to west) and making a special attention to the case of only two roads and a crossing. This last case has been studied in detail in [8], [9], [10] without the possibility of turning with a stochastic modeling.…”