  In this paper, we show that a variant of the discrete Burgers equation can be
obtained through the Cole--Hopf transformation to a generalized discrete
diffusion equation corresponding to the correlated random walk, which is also
known as a generalization of the well known random walk. By applying the
technique called ultradiscretization, we obtain the generalized ultradiscrete
diffusion equation, the ultradiscrete Cole--Hopf transformation and a variant
of the ultradiscrete Burgers equation. Moreover, we show that the resulting
ultradiscrete Burgers equation yields cellular automata which can be
interpreted as a traffic flow model.
