Energy-efficiency has been an important system issue in hardware and software designs to extend operation duration or cut power bills. This research explores systems with probabilistic distribution on the execution time of real-time tasks for systems with discrete frequencies. Most previous studies consider DVS systems with continuous frequencies for the minimization of expected energy consumption under timing constraints. However, these approaches cannot guarantee the minimization of expected energy consumption when only discrete frequencies are available. This paper presents new approaches to minimize the expected energy consumption. By applying intra-task frequency scheduling, we develop an efficient algorithm to derive optimal frequency scheduling for a single task. The algorithm is then extended to cope with periodic real-time tasks with different power characteristics. With inter-task and intra-task frequency scheduling, we present a linear-programming approach to derive optimal...