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This paper derives the form of a minimal cost rate stationary operating policy for an M/M/I queueing system with K possible service rates. The two costs of the system are a general cost rate dependent on the state of the system and a cost rate associated with each of the possible service rates. The optimal policy is completely characterized by if K − 1 numbers. The optimal service rate is shown to be nondecreasing in the state of the system.
Thomas B. Crabill (1972) studied this question.