A Reflected (s,S) Markov Modulated Inventory Process with Order Cancellations  

We consider a stochastic fluid inventory model reflected at level 0 based on the (s; S) policy. The content level I = fI(t) : t > 0g increases or decreases according to a fluid-flow rate modulated by an n-state CTMC. I starts at I(0) = S; whenever I(t) drops to level s, an order is placed to take the inventory back to level S, which the supplier will carry out after an exponential leadtime. However, if during the leadtime the content level reaches S the order is suppressed. We obtain explicit formulas for the expected discounted costs and the long run average case. The derivations are based on OST of a multi-dimensional martingale and on fluid flow techniques.