In this thesis, we analyze the optimal power saving with given delay constraints in the discontinuous reception (DRX) system in a Long-Term Evolution (LTE) network. To analyze the optimal power saving, we model the radio resource controller (RRC) connected system as a semi-Markov chain where packet arrivals follow a Poisson process. With our analytic model we investigate the impact of system parameters on the power and delay performance, and derive an optimal solution.