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State-of-the-art multi-core reconfigurable processors do not exploit the full potential of simultaneous multi-tasking with run-time adaptive reconfigurable fabric allocation. We propose a novel run-time system for simultaneous multi-tasking in a multi-core reconfigurable processor that adaptively allocates the mixed-grained reconfigurable fabric resource at run time among different tasks considering their performance constraints. Our scheme employs the novel concept of refined task-criticality (based on the functional-block-level performance constraints) considering the computational properties of dependent tasks and their inherent potential for acceleration. Our scheme dynamically compensates the deadline misses at the functional block level. It thereby reduces the potential task-level deadline misses under competing scenarios. With the help of a secure video conferencing application (with 4 dependent tasks of diverse computational properties), we demonstrate that our scheme reduces the deadline misses by (on average) 6× under given performance constraints, when compared to state-of-the-art reconfigurable processors.