Periodic broadcast of cooperative awareness messages (CAMs) is of great importance considering safety-related applications, for a network like VANET. This study proposes a vehicle-identifier (ID)-based MAC scheme for the periodic broadcast of CAMs in a vehicular network. Each CAM generated by a vehicle is associated with a unique ID. This study uses this attribute of CAM messages while participating in the channel assessment procedure. Each vehicle weighs the random back-off number with the unique ID incorporated in their respective CAM. This eventually leads to the selection of a vehicle-ID-based random back-off number, mitigating all possible probabilities of collision due to same back-off number selection. The simulation is performed considering saturated traffic conditions, to analyse its behaviour under most critical situations. The proposed model eliminates the chances of selecting zero as the initial back-off number, eliminating any chances of consecutive freeze process. The proposed one-dimensional Markovian model is validated by MATLAB. The proposed model provides a clear depiction that it performs better than the traditional IEEE 802.11OCB used for VANET. Also, it is worth noting that the achieved results outperform the existing work in terms of average packet delay since nearly one-fourth of the contention window size is utilised throughout the simulation.