An optimizing unit for use with an interleaved memory and suitable for use in a computer graphics system is described. The unit utilizes knowledge of the repetitive and predictable nature of texture buffer accesses to potentially reduce the number of memory fetches. The unit maintains a queue of pending requests for tiles of data from the memory, and predicts the retrieval of redundant data within short sequences of requests. The redundant data is retrieved from the memory once, and repeated as necessary from local temporary storage registers.