In an automated storage library (1), demount of data storage media may be permitted when all peripheral storage devices (4) in the library are occupied and one or more of such devices remains inactive for a predetermined time. Storage media may be demounted to carry out a mount request, provided that the peripheral storage devices in which the media are mounted are inactive. Even when no mount requests are pending, the status of the peripheral storage devices is continually monitored. If the peripheral storage devices are all occupied, and one or more of such devices remains inactive for the predetermined time, the least recently used storage medium is demounted and transferred to a storage cell in the library. Demount of storage media may also be permitted when one or more of the occupied peripheral storage devices remains inactive for a relatively long predetermined time. Storage media may be demounted to carry out a mount request, provided that the peripheral storage devices in which the media are mounted are inactive. A storage medium may also be demounted to create an unoccupied peripheral storage device, even though no mount request is pending, when all of the peripheral storage devices are occupied and one or more of such devices remains inactive for a relatively short predetermined time. Regardless of the existence of a pending mount request or of the occupancy status of the peripheral storage devices, the storage medium in each of the occupied peripheral storage devices remaining inactive for the relatively long predetermined time is demounted and transferred to a storage cell in the library. Thus, during long idle periods, the library will not continue to use the resources of inactive peripheral storage devices.