Control strategy and optimisation of the single-server queue with secondary optional service and working vacations

With the continuous evolution of business models, service mechanisms of the system have undergone significant innovation. By incorporating a lottery mechanism into the service process, this paper establishes and analyses a single-server queueing model with secondary optional service and working vacations. Using a difference-equation approach, the steady-state probabilities of the system and the individual equilibrium balking control strategies of customers are derived. From the perspectives of both customers and merchants, a social benefit function and a merchant benefit function are constructed, respectively. The socially optimal control strategy and the merchant optimal control strategy are derived, and a sensitivity analysis is conducted. In the optimisation analysis, particular attention is paid to the impact of the winning probability and the discount rate on profit when the lottery mechanism is introduced after the basic service. The results provide control decision support for businesses considering such service mechanisms in optimal management.