The ultra-dense edge networks with mmWave and beamforming are envisioned as a potential solution to satisfy the high rate and capacity requirements in 5G networks. In IEEE 802.11 ad, which is the first beamforming-enabled WLAN standard, all stations (STs) contend for beamforming (BF) training opportunities in associated beamforming training (A-BFT) slots. However, due to limited number of A-BFT slots, BF training suffers from a severe collision issue, especially in dense networks, which results in a low channel utilization in the A-BFT stage. To achieve the maximum channel utilization, it is of significance to allocate A-BFT slots efficiently. Therefore, in this paper, we propose an analytical model to analyze IEEE 802.11 ad medium access control (MAC) protocol in BF training stage. In particular, we analyze the successful transmission probability and channel utilization of IEEE 802.11 ad MAC protocol in the dense network. Based on theoretical analysis, we provide the optimal number of A-BFT slots. In addition, theoretical analysis indicates that the maximum channel utilization in the A-BFT stage is barely e^-1 which is the same as that of slotted ALOHA protocol. Simulation results are provided to validate the accuracy of the analytical model and theoretical analysis.