Power management is one of the most important issues in mobile communications. Much research has been done in reducing wireless station's power consumptions. IEEE 802.11 addresses this issue by adopting a MAC layer active-doze Power Saving Mechanism. In an 802.11 ad hoc network, this Power Saving Mechanism works as follows. Any wireless station with data to send must first announce its traffic and then contends for the channel with other stations for data transmissions, all based on the DCF protocol. Stations not involved in any data transmissions can go to the doze mode to conserve energy. In this paper, we first show that this mechanism has the problem of power management inefficiency when used in a multiple rate ad hoc network. We then propose a novel scheduling mechanism, STFS, to reduce the total power consumptions of the wireless stations in the network. Simulation results show that the proposed scheduling mechanism does have better performance than that of 802.11 PSM.