Notes of M/G/1 system under the ⟨p, T⟩ policy with second optional service

We optimize the operating cost of the < p, T > policy for an M/G/1 queueing system with second optional service, where the customer may depart from the system either after the first essential service with probability 1 - r or at the end of the first service may immediately go for a second service with probability r. Moreover, the server takes a vacation of fixed length T if the system becomes empty. If customers are found in the queue after T time units have elapsed since the end of the busy period, the server reactivates with probability p or leaves for a vacation of the same length T with probability 1 - p. Alternatively, if no customers present in the queue upon returning from the vacation, the server leaves for another a vacation of the same length. We call this < p, T > policy. The total expected cost function per unit time is developed to determine the optimal thresholds of p and T at a minimum cost. Based on the optimal cost the explicit form for joint optimum values of p and T are obtained.