A queueing system with Erlang-distributed replenishment time, multiple vacations and continuous-type inventory

In this paper, we consider a continuous-type inventory system with positive service time, Erlang distributed leading time, and multiple vacations. The inventory policy is (r, Q). Once the inventory level is less than r, the manager sends a replenishment request to the external supplier. After a replenishment time obeying the Erlang distribution, a fixed replenishment quantity Q arrives in the system. If, during this period, the inventory is depleted, the server takes multiple vacations. We assume that the lost sale occurs while the server is on vacation. Normal service begins once the server returns from a vacation and sees that the system inventory is Q. We introduce a continuous-type inventory model with zero service time. We also prove that the steady-state probability distribution of the original model can be decomposed into the product of the distribution of the queue length of the classical M/M/1 queueing model and the inventory distribution of the continuous-type inventory model with zero service time. Numerica examples display the effects of parameters on the performance measures and the optimal system cost.