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)

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