On a discrete optimal velocity model and its continuous and   ultradiscrete relatives
  We propose a discrete traffic flow model with discrete time. Continuum limit of this model is equivalent to the optimal velocity model. It has also an ultradiscrete limit and a piecewise-linear type of traffic flow model is obtained. Both models show phase transition from free flow to jam in a fundamental diagram. Moreover, the ultradiscrete model includes the Fukui-Ishibashi model in a special case. 
Daisuke Takahashi and Junta Matsukidaira