Comparison of two randomized policy M/G/1 queues with second optional service, server breakdown and startup

The problem addressed in this paper is to compare the minimum cost of the two randomized control policies in the M/G/1 queueing system with an unreliable server, a second optional service, and general startup times. All arrived customers demand the first required service, and only some of the arrived customers demand a second optional service. The server needs a startup time before providing the first required service until the system becomes empty. After all customers are served in the queue, the server immediately takes a vacation and the system operates the ( T , p )-policy or ( p , N )-policy. For those two policies, the expected cost functions are established to determine the joint optimal threshold values of ( T , p ) and ( p , N ), respectively. In addition, we obtain the explicit closed form of the joint optimal solutions for those two policies. Based on the minimal cost, we show that the optimal ( p , N )-policy indeed outperforms the optimal ( T , p )-policy. Numerical examples are also presented for illustrative purposes.