Optimization and sensitivity analysis of controlling arrivals in the queueing system with single working vacation

This paper analyzes the F -policy M/M/1/K queueing system with working vacation and an exponential startup time. The F -policy deals with the issue of controlling arrivals to a queueing system, and the server requires a startup time before allowing customers to enter the system. For the queueing systems with working vacation, the server can still provide service to customers rather than completely stop the service during a vacation period. The matrix-analytic method is applied to develop the steady-state probabilities, and then obtain several system characteristics. We construct the expected cost function and formulate an optimization problem to find the minimum cost. The direct search method and Quasi-Newton method are implemented to determine the optimal system capacity K , the optimal threshold F and the optimal service rates ( μ B , μ V ) at the minimum cost. A sensitivity analysis is conducted to investigate the effect of changes in the system parameters on the expected cost function. Finally, numerical examples are provided for illustration purpose.