  We investigate two models for traffic flow with modified acceleration
('slow-to-start') rules. Even in the simplest case $v_{max}=1$ these rules
break the 'particle-hole` symmetry of the model. We determine the fundamental
diagram (flow-density relationship) using the so-called car-oriented mean-field
approach (COMF) which yields the exact solution of the basic model with
$v_{max}=1$. Here we find that this is no longer true for the models with
modified acceleration rules, but the results are still in good agreement with
simulations. We also compare the effects of the two different slow-to-start
rules and discuss their relevance for real traffic. In addition, in one of
these models we find a new phase transition to a completely jammed state.
