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While tasks may be preemptive or non-preemptive (due to their transactional operations), deadline guarantees in multi-core systems have been made only for those task sets in each of which all tasks are preemptive or non-preemptive, not a mixture thereof,i.e., fully preemptive or fully non-preemptive. In this paper, we first develop a schedulability analysis framework that guarantees the timing requirements of a given task set in which a task can be either preemptive or non-preemptive in multi-core systems. We then apply this framework to the prioritization polices of EDF (earliest deadline first) and FP (fixed priority), yielding schedulability tests of mpn-EDF (Mixed Preemptive/Non-preemptive EDF) and mpn-FP, which are generalizations of corresponding fully-preemptive and non-preemptive algorithms, i.e., fp-EDF and np-EDF, and fp-FP and np-FP. In addition to their timing guarantees for any task set that consists of a mixture of preemptive and non-preemptive tasks, the tests outperform the existing schedulability tests of np-EDF andnp-FP (i.e., special cases of mpn-EDF and mpn-FP). Using these tests, we also improve schedulability by developing an algorithm that optimally disallows preemption of a preemptive task under a certain assumption. We demonstrate via simulation that the algorithm finds up to 47.6 percent additional task sets that are schedulable with mpn-FP (likewise mpn-EDF), but not with fp-FP and np-FP (likewisefp-EDF and np-EDF).