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Dynamic Control of an M/M/1 Service System with Adjustable Arrival and Service Rates

We study a service facility in which the system manager dynamically controls the arrival and service rates to maximize the long-run average value generated. We initially consider a rate-setting problem where the service facility is modeled as an M/M/1 queue with adjustable arrival and service rates and solve this problem explicitly. Next, we use this solution to study a price-setting problem, where customers are utility maximizing and price- and delay-sensitive, and the system manager chooses state-dependent service rates and prices. We find that the optimal arrival rate is decreasing and the optimal service rate is increasing in the number of customers in the system; however, the optimal price need not be monotone. We also show that under the optimal policy, the service facility operates as one with a finite buffer. Finally, we study a numerical example to compare the social welfare achieved using a dynamic policy to that achieved using static policies and show the dynamic policy offers significant welfare gains.