A data storage system is constructed to rapidly respond to a backup request by streaming backup data from primary storage to tape. It is desirable to permit the data to be removed from the primary storage at a faster rate than it can be written to tape. The backup data is buffered in a memory buffer, and when the memory buffer becomes substantially full, a portion of the backup data is buffered in disk storage. When the memory buffer becomes substantially empty, the portion of the backup data in the disk storage is written to tape. In a preferred embodiment, the memory buffer is in random access memory of a data mover computer that transfers the backup data from primary storage to a tape library unit. When the memory buffer becomes full, the data mover stores the overflow in a cached disk storage subsystem. When the memory buffer becomes empty, the data mover retrieves the overflow from the cached disk storage subsystem and transmits the overflow to the tape library unit. In this fashion, the sequence of data tracks as stored on the tape can be different from the sequence of data tracks as stored in the primary storage. To facilitate restoration of the primary storage with a backup version, the data tracks as stored in records on the tape are identified by a track number included in each record on the tape.