One key usage of VANET is to support vehicle safety applications. This use case is characterized by the prominence of broadcasts in scaled settings. In this context, we try to answer the following questions: i) what is the probability of reception of a broadcast message by another car depending on its distance to the sender, ii) how to give priority access and an improved reception rate for important warnings, e.g., sent out in an emergency situation , and iii) how are the above two results affected by signal strength fluctuations caused by radio channel fading? We quantify via simulation the probability of reception for the two-ray-ground propagation model as well as for the Nakagami distribution in saturated environments. By making use of some IEEE 802.11e EDCA mechanisms for priority access, we do not only quantify how channel access times can be reduced but also demonstrate how improved reception rates can be achieved. Our results show that the mechanisms for priority access are ...