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We consider the problem of providing QoS guarantees to Grid users through advance reservation of resources. Advance reservation mechanisms provide the ability to allocate resources to users based on agreed-upon QoS requirements and increase the predictability of a Grid system, yet incorporating such mechanisms into current Grid environments has proven to be a challenging task due to the resulting resource fragmentation. We use concepts from computational geometry to present a framework for tackling the resource fragmentation, and for formulating a suite of scheduling strategies. We also develop efficient implementations of the scheduling algorithms that scale to large Grids. We conduct a comprehensive performance evaluation study using simulation, and we present numerical results to demonstrate that our strategies perform well across several metrics that reflect both user-and system-specific goals. Our main contribution is a timely, practical, and efficient solution to the problem of scheduling resources in emerging on-demand computing environments.