Event-based optimization of service rate control in retrial queues

This paper deals with the problem of service rate control for a retrial queue where the controller can observe only the number of total customers in the system. Service rates are adjustable based on the partial information (the arrival or departure event) instead of the perfect system state (customer distribution among orbit and server). The goal is to find the optimal service rates that minimize the long-run average cost. This problem is formulated as an event-based optimization instead of a state-based optimization. Using the sensitivity-based optimization theory, we obtain interesting structures of the optimal service rate control policy, which show that the optimal policy is a bang-bang control and even has a threshold form under some mild conditions. The necessary and sufficient condition of optimal policies is also derived. Furthermore, by the difference formula of the system performances under any two policies, we develop a policy iteration-type algorithm to find the optimal policy for the case with general cost functions. With different initial values of service rates, our algorithm is demonstrated to efficiently find the optimal service rates. Moreover, we study the difference of system performances from the corresponding state-based optimization by numerical experiments, which indicates the managerial insights about the value of information for decision-makers.