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In this paper, we present multiple access with lag time (MALT) for medium access control (MAC) with strong quality-of-service (QoS) supports and throughput-efficient power control in mobile ad hoc networks. On lop of conventional priority-based techniques such as different interframe spaces and backoff algorithms, MAPS supports effective differentiated service employing the distributed differentiated scheduling (DDS) to allow packets with higher priority or urgent deadline to schedule for a packet slot that is farther into the future. In this way, these higher-priority packets can schedule for their transmissions without competition from lower priority packets since the latter are not allowed to make reservation during slots with conflicting schedules yet. We demonstrate through simulations that MALT is considerably stronger than IEEE 802.11e and a power-controlled dual-channel variant of IEEE 802.11e in terms of its differentiation capability for delays, discarding/blocking ratios, and throughput. Our simulation results also show that MAPS can achieve higher throughput as compared to EDCF of IEEE 802.11e due to its supports for power-controlled variable-radius transmissions.