The IEEE 802.11p Enhanced Distributed Channel Access (EDCA) is a standardization for vehicle-to-vehicle and road-to-vehicle communications. The saturated throughputs of the IEEE 802.11p EDCA obtained from previous analytical expressions differ from those of simulations. The purpose of this paper is to explain the reason why the differences appear in the previous analytical model of the EDCA. It is clarified that there is a special state wherein the Backoff Timer (BT) is decremented in the first time slot of after a frame transmission, which cannot be expressed in the previous Markov model. In addition, this paper proposes modified Markov models, which allow the IEEE 802.11p EDCA to be correctly analyzed. The proposed models describe BT-decrement procedure in the first time slot accurately by adding new states to the previous model. As a result, the proposed models provide accurate transmission probabilities of network nodes. The validity of the proposed models is confirmed by the quantitative agreements between analytical predictions and simulation results.