We investigate two models for traffic flow with modified acceleration (‘slow-to-start’) rules. Even in the simplest case vmax = 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 vmax = 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.