CELLULAR AUTOMATA MODELS

Many of the arguments regarding the nature of a stochastic and possibly critical phase transition (32,35,33,31,34) have been made using so-called cellular automata (CA) models. CA models use coarse spatial, temporal, and state space resolution. For traffic, a standard way is to segment a 1-lane road into cells of length $\cellLen$, where $\cellLen$ is the length a vehicle occupies in the average in a jam, i.e.tex2html_verbatim_mark>#math437#.

As with the Krauss model, the time step for the CA models is best selected similar to the reaction time; a time step of 1 second works well in practice. Taking the time step together with $\cellLen$, one finds that a speed of 135 km/h corresponds to five cells per time step; this is often taken as maximum velocity .

Stochastic CA models contain a noise parameter that introduces randomness into the driving rules: With a probability , the deterministically calculated velocity gets reduced by one. This is the same idea as Eq. (9) in the model by Krauss. One can make dependent on the velocity (36); the resulting models are sometimes called models with ``velocity-dependent randomization (VDR)''. Often one uses just two probablilities: when the car is standing and when the car is driving. Standard values are and . This models that drivers, once stopped, are a bit sloppy in restarting again.

With this family of models, one can again plot the density variance (Fig. 2(c)). Instead of the noise amplitude , the parameter is used. means deterministic driving except when accelerating from zero; increasing means increasingly more randomness when moving. In this plot, one finds a behavior similar to Fig. 2(b): For small , the system displays three states (laminar, coexistence, and jammed). For , the system is a 1-phase system. Close to and still at , the system displays a lot of jam formation and structure which vanishes only when observed at very large scales (very large $\ell$). In consequence, it is now clear why there was so much discussion about possible fractals for the original model (37) where : It is indeed close to a critical point, and therefore fractal behavior up to a certain cut-off length scale should be expected.