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In this paper, we study the problem of packet scheduling in a wireless environment with the objective of minimizing the average transmission energy expenditure under individual packet delay constraints. Most past studies assumed that the input arrivals followed a Poisson process or were statistically independent. However, traffic from a real source typically has strong time correlation. We model a packet scheduling and queuing system for a general input process in linear time-invariant systems. We propose an energy-efficient packet scheduling policy that takes the correlation into account. Meanwhile, a slower transmission rate implies that packets stay in the transmitter for a longer time, which may result in unexpected transmitter overload and buffer overflow. We derive the upper bounds of the maximum transmission rate under an overload probability and the upper bounds of the required buffer size under a packet drop rate. Simulation results show that the proposed scheduler improves up to 15 percent in energy savings compared with the policies that assume statistically independent input. Evaluation of the bounds in providing QoS control shows that both deadline misses and packet drops can be effectively bounded by a predefined constraint.