LTE/LTE-A systems have specified a discontinuous reception (DRX) mechanism for battery energy saving of user equipment (UE). Until now, most analytical works have been on the DRX mechanism based on a single packet server. However, a single packet server typically yields low performance in energy saving under high arrival rates. On the other hand, practical LTE/LTE-A systems can allocate a variable size of resource for UE at one service time. To reflect the variable size of resource allocation and analyze the exact DRX mechanism under high arrival rates, we first model the DRX mechanism based on a batch server with varying buffer thresholds and server capacities. We also investigate the effect of system parameters such as buffer threshold, server capacity, DRX cycle, and inactivity timer in terms of delay and sleep ratio. Then, we find the optimal system parameter values of the batch packet server based DRX mechanism while satisfying the given delay requirements and maximizing the sleep ratio simultaneously. Finally, we show that the batch packet server yields the longer sleep ratio from 0.6 to 0.9 at high arrival rates, compared to a single packet server.