Power saving mechanisms with sleep-mode operations have been used in IEEE 802.16e in order to extend the battery lifetime of mobile devices. The standard sleep-mode operation needs to meet a given delay constraint. However, it may incur unnecessary energy consumption due to frequent state transitions. In this paper, we propose a new energy-efficient sleep-mode operation while meeting a given delay requirement. We apply an M/G/1 queue model with multiple vacations for downlink traffic in an error-free environment in order to evaluate the energy efficiency performance of the proposed sleep-mode operation in terms of packets/Joule. We compare the proposed sleep-mode operation with the standard sleep-mode operation. The results show that our proposed sleep-mode operation improves the total energy efficiency consumption per unit frame time by 35% and energy efficiency by 21% while meeting a given delay requirement.