The control policy of an M[X]/G/1 queueing system with server startup and two vacation types

This paper studies the optimal control of an M^[X]/G/1 queue with two types of generally distributed random vacation: type 1 (long) and type 2 (short) vacations. The server is turned off and takes type 1 vacation whenever the system is empty. If the number of customers waiting in the system at the instant of a vacation completion is less than Q, the server will take a type 1 vacation. If the number of customers in the system is greater than or equal to Q and smaller than N, the server will take a type 2 vacation. If the server returns from a vacation and finds at least N customers in the system, he is immediately turned on and requires a startup time before providing the service until the system is again empty. We analyze the system characteristics for such a model. The total expected cost function per unit time is developed to determine the optimal thresholds of Q and N at a minimum cost.