Randomized policy of a poisson input queue with J vacations

This paper studies the operating characteristics of an M/G/1 queuing system with a randomized control policy and at most J vacations. After all the customers are served in the queue exhaustively, the server immediately takes at most J vacations repeatedly until at least N customers are waiting for service in the queue upon returning from a vacation. If the number of arrivals does not reach N by the end of the Jth vacation, the server remains idle in the system until the number of arrivals in the queue reaches N. If the number of customers in the queue is exactly accumulated N since the server remains idle or returns from vacation, the server is activated for services with probability p and deactivated with probability (1 − p). For such variant vacation model, other important system characteristics are derived, such as the expected number of customers, the expected length of the busy and idle period, and etc. Following the construction of the expected cost function per unit time, an efficient and fast procedure is developed for searching the joint optimum thresholds (N*, J*) that minimize the cost function. Some numerical examples are also presented.