We consider an online scheduling environment where decisions are made without knowledge of the data of jobs that may arrive later. However, additional jobs can only arrive at known future times. This environment interpolates between the classical offline and online scheduling environments, and approaches the classical online environment when there are many equally spaced potential job arrival times.<br/><br/>The objective is to minimize the sum of weighted completion times, a widely used measure of work-in-process inventory cost and customer service. For a nonpreemptive single machine environment, we show that a lower bound on the competitive ratio of any online algorithm is the solution of a mathematical program. This lower bound is between $(1+SQRT(5))/2 and 2, with the exact value depending on the potential job arrival times. We also provide a "best possible" online scheduling algorithm, and show that its competitive ratio matches this lower bound. <br/><br/>We analyze two practically motivated special cases where the potential job arrival times have a special structure. When there are many equally spaced potential job arrival times, the competitive ratio of our online algorithm approaches the best possible competitive ratio of 2 for the classical online problem
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