In this paper, we aim to evaluate the system performance of the energy saving mode for Discontinuous Reception (DRX) in Long Term Evolution (LTE) systems. We consider the digital nature of modern communications, and build a discrete-time queueing model with a two-stage vacation. By using the method of an embedded Markov chain, we analyze the system model in the steady-state, and derive the formulas for the handover ratio, the energy saving ratio and the response time of data frames. Moreover, we present numerical results to show how the system performance measures change with the system parameters. Finally, we demonstrate the influence of system parameters on the system performance, and investigate the trade-off between different performance measures.