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We utilize a multiprocessor server-based approach to schedule a general class of soft real-time systems with stochastic execution demands, when bounded average-case tardiness is sufficient for schedulability. A key feature of the task model considered here is that the stochastic execution-time demands can have arbitrary amounts of dependence within pre-specified time intervals of bounded length. This is an important practical step forward from requiring complete independence of execution times between successive jobs of the same task. Our main result does not require the scheduler to know the execution time of each job in advance, and requires only average-case utilization to be bounded by the number of processors. This constraint is mild compared to constraints on worst-case utilization because in multiprocessor systems, worst-case execution times may be orders of magnitude higher than average-case execution times.