Optimization on (N, p)-policy for an unreliable queue with second optional service and start-up

This article analyzes an M/G/1 queueing system with second optional service, server breakdowns, and general start-up times under (N, p)-policy. Customers arrive into the system according to a Poisson process. Service times, repair times, and start-up times are assumed to be generally distributed. The first service is essential for all arrived customers, and the second service is optional, which is demanded by some customers after their essential service is completed. For this queuing system, the expected cost function per unit time is constructed to search the joint optimum threshold values of the (N, p) at the minimum cost. We also obtain the explicit closed-form expression of the optimal solutions. A sensitivity analysis is performed to evaluate the impacts on the optimum threshold value. Finally, numerical results are presented for several cases.