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A non·preemptive M/GI/l queue with several job classes is considered. At the completion of the service time the multiple feedback occurs. The objective is to maximize the expected discounted reward with the infInite horizon. Using the Harrison's method, the model is formulated as a bandit problem and its optimal policy is characterized by the index rule. Next it is considered that the service time is decomposed into quantum of the unit span. This model is a queueing system of a single feedback. Using the index rule, three type of optimal job schedulings are discussed. served at a time and the served job is not interrupted by the other job, that is, nonpreemptive. When the service is completed, the multiple feedback will occur. The cost structure is a pure expected reward received at a decision epoch. Our problem is to obtain the optimal job scheduling which maximizes the expected discounted reward with the infinite horizon. Our model is formu­ lated as a bandit problem with Poisson inputs and the multiple feedback. In a single server queueing system the optimal job scheduling which mini­ mi~es the expected holding cost has been studied. The optimal policy is characterized by a priority service discipline. When the service time is known in a nonpreemptive queue "the shortest-processing-time-first" discipline (SPTF) is optimal. When the expectation of the .service time is known in a nonpreemptive queue "the shortest-expected-processing-time-first" discipli.ne (SEPTF) is optimal. When the service time is known in a preemptive queue,

OPTIMAL JOB SCHEDULING OF M/GI/1 QUEUE WITH FEEDBACK: THE DISCOUNTED CASE