A make-to-stock production/inventory model with MAP arrivals and phase-type demands

We consider a make-to-stock production/inventory model in a random environment with finite storage capacity and restricted backlogging possibility. Our aim is to demonstrate that all cost quantities of interest can be derived in closed form under quite general assumptions on the demand arrival process and on the switches in the production rates. Specifically, the demands arrive according to a Markov additive process governed by a continuous-time Markov chain, and their sizes are independent and have phase-type distributions depending on the type of arrival. The production process switches between predetermined rates which depend on the state of the environment and on the presence or absence of backlogs. Four types of costs are considered: the holding cost for the stock, the cost of lost production due to the finite storage capacity, the shortage cost for the backlogged demand and the cost due to unsatisfied demand. We obtain explicit formulas for these cost functionals in the discounted case and under the long-run average criterion. The derivations are based on optional sampling of a multi-dimensional martingale and on fluid flow techniques.