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The need for supporting dynamic real-time environments where changes in workloads occur frequently requires a scheduling framework that: (1) explicitly addresses overload conditions, (2) allows the system to achieve graceful degradation while guaranteeing the deadlines of the most critical tasks in the system, and (3) supports an efficient runtime selection mechanism capable of determining the load to be shed from the system to handle the overload. In this paper, we propose a novel scheduling framework for a real-time environment that experiences dynamic workload changes. This framework is capable of adjusting the system workload in incremental steps under overloaded conditions such that the most critical tasks in the system are always scheduled and the total value of the system is maximized. Each task has an assigned criticality value and consists of two parts, a mandatory part and an optional part. A timely answer is available after the mandatory part completes execution and its value may be improved by executing the entire optional part. The process of selecting tasks (mandatory or optional parts) to discard while maximizing the value of the system requires the exploration of a potentially large number of combinations. Since an optimal solution is too time-consuming to be computed online, an approximate algorithm is executed incrementally whenever the processor would otherwise be idle, progressively refining the quality of the solution. This scheme allows the scheduler to handle overloads with low cost while maximizing the use of the available resources and without jeopardizing the temporal constraints of the most critical tasks in the system. Simulation results show that few stages of the algorithm need to be executed for achieving a performance with near-optimal results.