A Queueing-Inventory System with Modified Delayed Vacation under Bernoulli Schedule

In this paper, we consider a queueing-inventory system with modified delayed vacation under Bernoulli schedule. If the inventory is empty upon completion of the service, the server will take the modified delayed vacation. During the modified delayed vacation period, if the replenishment is completed, the customer will still be served as normal. After this period, the server will take Bernoulli schedule, where the server reverts to normal work with probability \(q(0\le q\le 1)\) $q(0\le q\le 1)$ or takes multiple vacations with probability \(1-q\) $1-q$ . The customers arrive according to a Poisson process. Upon customer arrival, if the inventory is not empty, the customer accepts the service and leaves the system carrying a product. The service time, lead time, modified delayed vacation time and multiple vacations time are all assumed to be exponentially distributed. We derive the stability condition of the system and the matrix geometric solution of steady-state probabilities using an algorithm. Then performance measures of the system are derived. Finally, numerical results are presented to demonstrate the impact of system parameters on performance measures and the expected cost function.