An $M/M/1$ queueing-inventory system with working vacations, vacation interruptions and lost sales

We consider a single server queueing-inventory system under $(s,Q)$ replenishment strategy with working vacations, vacation interruptions and lost sales. The server provides service at a lower rate during working vacations than while in normal mode of service. If working vacation realizes while providing service in that mode, then the server continues in the present status until the current service is completed. Upon that he switches to normal mode of service, provided there is at least one customer waiting. If no customer is waiting at this point of time, he goes for vacation. Also we assume that if there are customers in the system at a service completion epoch during working vacation, the server will comeback to the normal working mode, else the server stays in the working vacation mode. With the system having infinite capacity, the condition for stability of the system is obtained, followed by computation of the steady-state probability vector is discussed. Various performance measures are evaluated. In addition, busy period analysis and the stationary waiting time distribution in the queue is derived. Numerical illustrations are provided to illustrate the system performance and an optimization problem is also discussed.