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Even though middleware support for grid computing has been the subject of extensive research, scheduling policies for the grid context have not been much studied. In addition to processor utilization, it is important to consider the response times of jobs in evaluating the performance of grid scheduling strategies. In this paper we propose distributed scheduling algorithms that use multiple simultaneous requests at different sites. Trace-based simulations show that the use of multiple simultaneous requests provides significant performance benefits. We also show how this scheme can be adapted to provide priority to local jobs, without much loss of performance.