In this paper, the directional carrier sense multiple access/collision avoidance (CSMA/CA) protocol in the immediate acknowledgement mode for IEEE 802.15.3c is analyzed under saturation environments. For the analysis, a sensing region and an exclusive region with a directional antenna are computed probabilistically and a Markov chain model in which the features of IEEE 802.15.3c and the effects of using directional antennas are incorporated is analyzed. An algorithm to find the maximal number of concurrently transmittable frames is proposed. The system throughput and the average transmission delay are obtained in closed forms. The numerical results show the impact of directional antennas on the CSMA/CA media access control (MAC) protocol. For instance, the throughput with a small beamwidth of antenna is more than ten times larger than that for an omnidirectional antenna. The overall analysis is verified by a simulation. The obtained results will be helpful in developing an MAC protocol for enhancing the performance of mmWave wireless personal area networks.