Optimal inventory policies under stochastic production and demand rates

We consider a single stage production-inventory system whose production and demand rates are modulated by an environment process modeled as a finite state Continuous Time Markov Chain (CTMC). When the inventory level reaches zero, an order is placed from an external supplier, and it arrives instantaneously. We derive an Economic Order Quantity (EOQ) policy that minimizes the long-run average cost if one replaces the deterministic demand rate by the expected demand - production rate in steady state. Finally, we extend the model to allow backlogging.