We mathematically analyze the sleep mode operation of IEEE 802.16m. The sleep mode operation for downlink traffic is modeled as a 3-dimensional discrete time Markov chain. We obtain the average power consumption of a mobile station and the average delay of a message. Numerical results match simulations very well. Numerical results show that there is a tradeoff between power consumption and message delay. We find the optimal lengths of sleep cycle and close-down time that minimize the power consumption while satisfying the quality of service (QoS) constraint on message delay. The power consumption of the sleep mode in IEEE 802.16m is better than that of sleep modes in legacy IEEE 802.16e standard under the same delay bound.