Two-threshold policy for M[x]/G/1 queueing system with two vacation types, startup time and an unreliable server

This paper studies the control policy of an M^[x]/G/1 queue with two types of generally distributed random vacation: type 1 (long) and type 2 (short) vacations, where arrivals in batch form a Poisson process and service times are generally distributed. The server is turned off and takes a 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 M, the server will take a type 1 vacation again. If the number of customers in the system is greater than or equal to M 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. Furthermore, it is assumed that the server breaks down according to a Poisson process and his repair time has a general distribution. The system characteristics of such a model are analyzed and the total expected cost function per unit time is developed to determine the optimal thresholds of M and N at a minimum cost.