Only two types of sleep cycles with two fixed sleep periods are defined in the standard of the discontinuous reception (DRX) mechanism in LTE/LTE-A. Explicitly, such a DRX mechanism is simple but lacks flexibility. To provide flexibility, two types of dynamic and adjustable DRX (DADRX) mechanisms are proposed and investigated in this paper. The first type of DADRX extends the number of types of sleep cycles through adding different pairs of sleep cycles with different sleep periods gradually. As for the second type of DADRX, it fixes the total number of types of sleep cycles like the first-type DADRX but further employs a set of level expanding probabilities to control probabilistically the possible pairs of sleep cycles that can be reached. To enable the analytical calculation, performance metrics for the proposed DADRX mechanisms are acquired, including average response delay and average long-term power consumption. Via both analytical and simulation approaches, we investigate the dynamics of performance for the proposed DADRX mechanisms extensively. We successfully show that these two proposed mechanisms can exhibit much more flexibility than the closely related mechanisms in the literature.