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Reduced models are simplified versions of a given domain, designed to accelerate the planning process. Interest in reduced models has grown since the surprising success of determinization in the first international probabilistic planning competition, leading to the development of several enhanced determinization techniques. To address the drawbacks of previous determinization methods, we introduce a family of reduced models in which probabilistic outcomes are classified as one of two types: primary and exceptional. In each model that belongs to this family of reductions, primary outcomes can occur an unbounded number of times per trajectory, while exceptions can occur at most a finite number of times, specified by a parameter. Distinct reduced models are characterized by two parameters: the maximum number of primary outcomes per action, and the maximum number of occurrences of exceptions per trajectory. This family of reductions generalizes the well-known most-likely-outcome determinization approach, which includes one primary outcome per action and zero exceptional outcomes per plan. We present a framework to determine the benefits of planning with reduced models, and develop a continual planning approach that handles situations where the number of exceptions exceeds the specified bound during plan execution. Using this framework, we compare the performance of various reduced models and consider the challenge of generating good ones automatically. We show that each one of the dimensions---allowing more than one primary outcome or planning for some limited number of exceptions---could improve performance relative to standard determinization. The results place previous work on determinization in a broader context and lay the foundation for a systematic exploration of the space of model reductions.