A make-to-stock production system with an emergency production rate

We consider a make-to-stock production system in which items can be produced under two production rates: normal and emergency. Under Poisson demands and exponential production lead-times, the optimal control policy is characterized by two critical inventory levels. For each production rate, there exists a critical inventory level below which it is optimal to produce under this rate. An M/M/1/S queueing model with state-dependent arrival rates is analyzed to compute the average total cost per unit time under the optimal control policy. Through numerical study, we show that the production system with two production rates has a significant benefit over the one with single production rate.