Multi-Server Ticket Queue Incorporating Impatient Customers and a Single Synchronous Vacation

Ticket-based queueing systems are widely adopted in the service industry and government sectors for managing customer flow. In such systems, each arriving customer receives a numbered ticket that indicates the number currently being served. Unlike conventional classical queueing systems, these are governed by ticketing mechanisms that introduce unique operational dynamics. This paper investigates a multi-server ticket queue system incorporating impatient customers and a single synchronous vacation policy. To enhance analytical tractability and reduce computational complexity, we propose a simplified version of the model along with efficient approximation techniques. For this modified ticket queue, we recursively solve the steady-state probabilities using the matrix-geometric method. Additionally, we derive explicit expressions for key performance measures. Numerical sensitivity analysis and graphical illustrations are provided to examine the influence of various system parameters. Finally, we formulate a bi-objective optimization problem and solve it using a multi-objective genetic algorithm to identify the optimal operating conditions that minimize the operating costs.