Production-inventory systems in stochastic environment and stochastic lead times

We consider a production-inventory system where the production and demand rates are modulated by a finite state Continuous Time Markov Chain (CTMC). When the inventory position (inventory on hand - backorders + inventory on order) falls to a reorder point r, we place an order of size q from an external supplier. We consider the case of stochastic leadtimes, where the leadtimes are i.i.d. exponential(mu) random variables, and orders may or may not be allowed to cross. We derive the distribution of the inventory level, and analyze the long run holding, backlogging, and ordering cost rate per unit time. We use simulation to study the sensitivity of the system to the distribution of the lead times.