Threshold Control for a Removable and Un-Reliable Server with Different Type Vacations and Startup

SYNOPTIC ABSTRACT

This paper studies the optimal control of an M/G/l queue with two types of generally distributed random vacation: type 1 (long) and type 2 (short) vacations, where arrivals 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.

Key Words and Phrases:

• breakdowns
• cost model
• queue
• startup
• different type vacations