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This work presents a performance evaluation of a scheduling algorithm on a computational grid where each site employs a different local scheduling policy. It is demonstrated that when some sites apply a priority policy in favor of local jobs, other sites suffer from much longer response time. We propose an adaptive site selection algorithm for a grid scheduler based on priority policies of local schedulers in order to reduce the severity of such effect without interfering the autonomy of local schedulers. The results show that the proposed algorithm can lower the difference in average wait times among the sites with different priority-based scheduling policies. The algorithm can perform effectively under various levels of workload and fractions of sites with different policies.