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We develop a novel approach to energy-efficient transmissions with arbitrary packet arrival process and strict delay constraints over time-varying wireless channels. When the arrivals, deadlines, and channel realizations are known a priori, we formulate the problem as a convex program. Relying on the specific structure of the optimality conditions, we put forth an efficient algorithm with a linear computational complexity in the order of constraint number to find the (offline) optimal rate control strategy. It is revealed that the power usage under the optimal policy admits a multi-level water-filling form, where the determination of the multiple water-levels can be visualized by the trajectory of letting a string tie its two ends and then taut between what we call the "water" arrival and departure curves. Guided by the optimal strategy, development of energy-efficient online schedules in practical systems is discussed. Numerical results are provided to demonstrate the merits of the proposed novel scheme.