The traditional vehicular ad hoc network (VANET), which is evolving into the internet of vehicles (IoV), has drawn great attention for its enormous potential in road safety improvement, traffic management, infotainment service support, and even autonomous driving. IEEE 802.11p, as the vital standard for wireless access in vehicular environments, has been released for more than one decade and its evolution, IEEE 802.11bd, has also been released for a few months. Since the analytical models for the IEEE 802.11p/bd medium access control (MAC) play important roles in terms of performance evaluation and MAC protocol optimization, a lot of analytical models have been proposed. However, the existing analytical models are still not accurate as a result of ignoring some important factors of the MAC itself and real communication scenarios. Motivated by this, a novel analytical model is proposed, based on a novel two-dimensional (2-D) Markov chain model. In contrast to the existing studies, all the important factors are considered in this proposed model, such as the backoff freezing mechanism, retry limit, post-backoff states, differentiated packet arrival probabilities for empty buffer queue, and queue model of packets in the buffer. In addition, the influence of the capture effect under a Nakagami-m fading channel has also been considered. Then, the expressions of successful transmission, collided transmission, normalized unsaturated throughput, and average packet delay are all meticulously derived, respectively. At last, the accuracy of the proposed analytical model is verified via the simulation results, which show that it is more accurate than the existing analytical models.