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With the increasing number of mobile location services, the ability to answer future queries, based on the current mobile data motion status (reference position and velocity vector), becomes a desirable feature in spatiotemporal databases. A predictive continuous k nearest neighbor (k-CNN) query retrieves the k-nearest neighbors among a mobile date set that satisfy the query in a given time interval (from current to near future). The difficulty in such a case is that both the query and the objects change their position continuously, and then we use time-parameterized objects' position and queries to provide the answer. In this paper we investigate mechanisms to perform k-CNN search on time-parameterized R-tree structure, and develop an efficient method in order to process predictive k-CNN queries in moving object databases. We propose novel metrics to support our branch-and-bound algorithms. An extensive performance evaluation shows that the proposed method achieves significant improvements compared to existing techniques.