Discrete product inventory control system with positive service time and two operation modes

Consideration is given to the Markov inventory control system of a discrete product of maximum volume $S$ under the strategies $(s,Q)$ and $(s, S)$ with a positive service time. Upon arrival, the order is queued if the inventory level is positive or, otherwise, leaves the system unserviced. One server handles the queued orders one-by-one in the sequence of their arrival. If the inventory level exceeds $s$, then the service time has the exponential distribution of intensity $\mu$; otherwise, of intensity $\alpha\mu$, $0<\alpha\leqslant 1$. The product in the inventory is consumed only at the instant when the service (of the order) ends. Inventory deficit is not allowed. When the inventory is empty new orders are not admitted into the system, and the service process of the queued orders (if any) is stopped. The lead time is assumed to be exponentially distributed. Analytical relations are established for the basic stationary performance characteristics of the system.