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Border Monitoring Query (BMQ) has different query semantic from conventional continuous range query. It monitors the values of data streams and reports them only when data streams cross the borders of its range. In this paper, we first emphasize the importance and usefulness of BMQ through attractive service scenarios. Then, we propose BMQ-Index, which is specialized to BMQ evaluation. It efficiently processes a large number of BMQs in a shared and incremental manner. For shared processing, BMQ-Index adopts a query indexing approach, thereby achieving a high level of scalability. For incremental processing, BMQ-Index employs an incremental access method. Thus, successive BMQ evaluations are significantly accelerated. We present an index structure and a search algorithm to support onedimensional as well as multi-dimensional BMQ. Lastly, we demonstrate the performance benefits of BMQ-Index through analysis and experiments.