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Discontinuous reception (DRX) saves battery power of user equipment (UE) usually at the expense of potential increase in latency in the Long Term Evolution (LTE) networks. Therefore, an optimization is needed to find the best tradeoff between latency and power saving. In this paper, we first develop an analytical model to estimate power saving achieved and latency incurred by DRX mechanism for active and background mobile traffic. A tradeoff scheme is then formulated to maintain a balance between these two performance parameters based on operator's preference for power saving and latency requirement of traffic. The analytical model is validated using system level simulation results obtained from OPNET Modeler. The results show that the proposed tradeoff scheme is efficient in keeping a balance between power saving and latency. The results also indicate that DRX short cycles are very effective in reducing latency for active traffic, while shorter inactivity timer is desirable for background traffic to enhance power saving. We also proposed a mechanism to switch DRX configuration based on traffic running at UE, using UE assistance procedure recently adopted by 3GPP in Release 11. DRX configuration switching increases the power saving significantly without any noticeable increase in latency of active traffic.