Discontinuous reception (DRX) is a way for user equipment (UE) to save energy. DRX forces a UE to turn off its transceivers for a DRX cycle when it does not have a packet to receive from a base station, called an eNB. However, if a packet arrives at an eNB when the UE is performing a DRX cycle, the transmission of the packet is delayed until the UE finishes the DRX cycle. Therefore, as the length of the DRX cycle increases, not only the amount of UE energy saved by the DRX but also the transmission delay of a packet increase. Different applications have different traffic arrival patterns and require different optimal balances between energy efficiency and transmission delay. Thus, understanding the tradeoff between these two performance metrics is important for achieving the optimal use of DRX in a wide range of use cases. In this paper, we mathematically analyze DRX to understand this tradeoff. We note that previous studies were limited in that their analysis models only partially reflect the DRX operation, and they make assumptions to simplify the analysis, which creates a gap between the analysis results and the actual performance of the DRX. To fill this gap, in this paper, we present an analysis model that fully reflects the DRX operation. To quantify the energy efficiency of the DRX, we also propose a new metric called a real power-saving (RPS) factor by considering all the states and state transitions in the DRX specification. In addition, we improve the accuracy of the analysis result for the average packet transmission delay by removing unrealistic assumptions. Through extensive simulation studies, we validate our analysis results. We also show that compared with the other analysis results, our analysis model improves the accuracy of the performance metrics.