This paper considers packet scheduling with delay constraint that minimizes the average transmission energy expenditure in a wireless environment. There were recent studies that modeled transmission rate as a linear combination of input packet arrivals in the derivation of optimal scheduling policies for independent input arrivals. In this paper, we extend the scheduling policies in two aspects. We first consider the impact of input time-correlation and derive an optimal scheduling policy for energy minimization. It is observed that existing energy optimized scheduling policies save energy by transmitting packets slowly. They may result in unexpected deadline misses. We establish a relationship between the maximum reliable transmission rate and a QoS constraint, characterized by a deadline miss rate. The relationship facilitates the trade-off study of power consumption and delay constraint. Simulation results verify the relationship and show the superiority of the scheduler in comparison with the existing approaches