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 (TT, pp)-policy or (pp, NN)-policy. For those two policies, the expected cost functions are established to determine the joint optimal threshold values of (TT, pp) and (pp, NN), 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 (pp, NN)-policy indeed outperforms the optimal (TT, pp)-policy. Numerical examples are also presented for illustrative purposes.