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This paper studies the problem of minimizing the queueing delay for a time-varying channel with a single queue, subject to constraints on the average and peak power. First, by separating the time-scales of the arrival process, the channel process and the queueing dynamics it derives a heavy-traffic limit for the queue length in the form of a reflected diffusion process. Given a monotone function of the queue-length process that serves as a penalty, and constraints on the average and peak available power, it shows that the optimal power allocation policy is a channel-state based threshold policy. For each channel state j there corresponds a threshold value of the queue length, and it is optimal to transmit at peak power if the queue length exceeds this threshold, and not transmit otherwise. Numerical results compare the optimal policy for the original Markovian dynamics to the threshold policy which is optimal for the heavy-traffic approximation, to conclude that that latter performs very well even outside the heavy-traffic operating regime.