A sequence of tasks that must be performed on a sequential database can be scheduled in various ways. Schedules will differ with respect to the number of accesses made to peripheral storage devices and the amount of memory space consumed by buffers. Buffer requirements are discussed for task schedules that avoid accesses to peripherals storing the sequential database. The relationship between certain thrifty scheduling policies and loop jamming, a standard code optimization technique, is also identified. Application to UNIX pipelines and to file processing is discussed.