The purpose of this work is to analyze the energy saving schemes of user equipment in the first releases (Release 15) of the fifth generation (5G NR) cellular mobile communication systems, as well as to identify possible ways to improve the energy efficiency of these systems. The paper describes the identified drawbacks of the existing discontinuous reception scheme (DRX-scheme) used for user equipment energy saving in 5G NR. The effectiveness of discontinuous reception scheme in reducing user equipment energy consumption was analyzed in detail by simulation study for various traffic models and a key scenario for the deployment of 5G wireless cellular mobile communication systems. Analysis of the simulation results shows that the discontinuous reception scheme does not reach the upper limit of possible energy saving for all the traffic models studied for a number of reasons described in this paper.