Scheduling algorithms for time shared computing facilities are considered in terms of a queueing theory model. The extremely useful limit of "processor sharing" is adopted, wherein the quantum of service shrinks to zero; this approach greatly simplifies the problem. A class of algorithms is studied for which the scheduling discipline may change for a given job as a function of the amount of service received by that job. These multilevel disciplines form a natural extension to many of the

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... ines previously considered. The average response time for jobs conditioned on their service requirement is solved for. Explicit solutions are given for the system M/G/1 in which levels may be first come first served (FCFS), feedback (FB), or round-robin (RR) in any order. The service time distribution is restricted to be a polynomial times an exponential for the case of RR. Examples are described for which the average response time is plotted. These examples display the great versatility of the results and demonstrate the flexibility available for the intelligent design of discriminatory treatment among jobs (in favor of short jobs and against long iobs) in time shared computer systems.