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, 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.