Priority decisions arise whenever limited facilities must be apportioned among competitive demands for service. Broadly viewed, even the familiar first-come-first-served discipline is a priority rule. It favors the longest-waiting user, and guards against excessive delays. Other priority rules, such as shortest-job-next, are keyed instead to considerations of operating efficiency. Urgency of request is still another common consideration. Since these considerations often conflict, the priority rule serves as mediator. Use of a common coat measure can help effect this mediation, as results from recent job-shop simulations illustrate. A priority operation of contemporary interest is scheduling a time-shared computer among its concurrent users. Service requirements are not known in advance of execution. To keep response times short for small requests, service intervals are partitioned and segments are served separately in round-robin fashion. A mathematical analysis pinpoints the tradeoff between overhead and discrimination implicit in this procedure, and allows alternate strategies to be costed. Extensions of the simple round-robin procedure are suggested, the objectives of time sharing are reviewed, and implications are drawn for the design of future priority and pricing systems.