A multi-server retrial queueing-inventory system with asynchronous multiple vacations

This study deals with asynchronous server vacation and customer retrial facilities in a multi-server queueing-inventory system. Customers arrive according to a Poisson process. The system comprises identical servers, a finite-size waiting area, and a storage area. The service time is distributed exponentially. If each server finds insufficient customers and items in the system after the busy period, they start a vacation. Once the server's vacation is over and it recognizes there is no chance of getting busy, it goes into an idle state; otherwise, it will take another vacation. Each server's vacation period occurs independently of the other servers. The system accepts a control policy for inventory replenishment. For the steady-state analysis, Neuts and Rao's matrix geometric approximation approach is used owing to the structure of an infinitesimal generator matrix. The necessary stability condition is computed. After calculating the sufficient system performance measures, an expected total cost of the system will be constructed and numerically incorporated with the parameters. Additional numerical analyses are conducted to examine customers' waiting time in the queue and orbit and the expected customer loss rate.