Average cost minimization in a multi-server retrial queueing system with a controllable reserve group of servers

The paper deals with a dynamic optimal activation and deactivation of a reserve group of servers in a multi-server retrial queueing system. The repeated attempts to occupy the server from the orbit occur due to a classical retrial discipline, and the activation of reserves requires a random set-up time. The optimal policy minimizes the average cost per unit of time which consists of the holding and usage cost components. The optimal control policy belongs to a class of hysteretic control policies defined by two threshold levels for the orbit size and prescribes the necessity to activate and deactivate a standby reserve group of servers. A dynamic-programming approach for the Markov decision process is used to show that the optimal thresholds depend on the number of busy servers as well. In order to provide a less computationally expensive method for calculating optimal thresholds, explicit heuristic expressions are proposed to guarantee at least a quasi-optimal solution to the optimization problem. Numerical examples confirm the efficiency of the proposed approach.