In this paper, an analytical model for millimeter wave (mmWave) cellular systems that characterizes user equipment (UE) power-saving constraint is studied. On the basis of the realistic radio propagation, we model the channel quality by curve fitting and create a Markov chain based channel model. The reference signal for channel measurement in conventional cellular systems can be transmitted to the whole cell at a time, but the beam sweeping is needed in mmWave systems in order to overcome the high path loss of high-frequency electromagnetic wave. This constraint further affects the timing of channel measurement, channel quality feedback, and power consumption. A novel beam-aware dormant mechanism for power saving in mmWave systems is proposed. UEs may enter the dormant mode for power-saving when the base station (BS) is not transmitting on their beams. Therefore, the BS is suggested to consider the time gap between channel measurement and data transmission when scheduling UEs. Beam-aware scheduling framework is proposed and integrated with the beam-aware dormant mechanism. It is shown that the policy to schedule a UE after its channel measurement outperforms the one to schedule every time slot. The simulation results and mathematical expression both demonstrate that the beam-aware media access control (MAC) framework improves throughput under power constraint.