Bi‐level threshold policy of MX/(G1,G2/1 queue with early setup and single vacation

In this paper, we analyse a queueing system with a second optional service channel under bi‐level control policy and server's single vacation. There is a single server who provides the first essential service to all arriving customers. As soon as the first service of a customer is complete, then with probability r, the customer may opt for the second service in which case the second service will immediately commence, or else with probability (1 − r) he may opt to leave the system, so that another customer at the head of the queue (if any) is taken up for his first essential service. The server operates an (m, N) policy with an early setup and takes a single vacation whenever the system becomes empty. For this model, the stationary probability generating function of the queue length distribution is obtained through supplementary variable technique and a decomposition property is discussed. The expected length of the cycle, various performance measures and the optimal values of thresholds m and N which minimise the total expected operating cost are also calculated.