The optimal control of an M/G/1 queueing system with server startup and two vacation types

Abstract

This paper considers the optimal control of an M/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, the server is immediately turned on and requires a startup time before providing the service until the system is again empty. We analyse the system characteristics for this system by means of two important probability functions.

The total expected cost function per unit time is developed to determine the suitable thresholds of Q and N at a minimum cost.

Keywords: Cost model; Queue; Startup; Vacation