A method of operating a computer system is disclosed in which an instruction having an explicit prefetch request is issued directly from an instruction sequence unit to a prefetch unit of a processing unit. In a preferred embodiment, two prefetch units are used, the first prefetch unit being hardware independent and dynamically monitoring one or more active streams associated with operations carried out by a core of the processing unit, and the second prefetch unit being aware of the lower level storage subsystem and sending with the prefetch request an indication that a prefetch value is to be loaded into a lower level cache of the processing unit. The invention may advantageously associate each prefetch request with a stream ID of an associated processor stream, or a processor ID of the requesting processing unit (the latter feature is particularly useful for caches which are shared by a processing unit cluster). If another prefetch value is requested from the memory hierarchy, and it is determined that a prefetch limit of cache usage has been met by the cache, then a cache line in the cache containing one of the earlier prefetch values is allocated for receiving the other prefetch value.