  We consider a car-following model described by a delay difference equation
and give its exact solutions that present propagation of a traffic jam. This
model is a discrete-time version of the delayed optimal-velocity model; in the
continuum limit, we recover the delay differential equation for this model and
the exact solutions as well. We then work in the ultra-discrete limit,
obtaining a delay cellular-automaton model, which successfully inherits the
solutions. Also the dispersion relation for the present solutions suggests that
a quick response of drivers does not always result in fast dissolution of a
traffic jam.
