Queueing Inventory System in a Two-level Supply Chain with One-for-One Ordering Policy

Consider a two-level inventory system consisting of one supplier and one retailer. the retailer faces a poisson demand with a known rate and applies base stock (one-for-one ordering) policy. that is, his inventory position is set to a pre-determined level, so the demand pattern is transferred exactly to the supplier. the supplier has an inventory system and a service unit with exponentially distributed service time to process the orders received from the retailer. supplier also follows a base stock policy, and its lead time is exponentially distributed. when the supplier has some on-hand inventory, an arriving order from retailer joins the queue. but when the supplier has no on-hand inventory, the retailer does not accept any demand, i.e. the demand is lost. when the retailer has no on-hand inventory but the supplier has on-hand inventory, the arriving demand to the retailer will be backordered. for this system, we derive the steady state joint distributions of the ‘number of retailers order in service unit and the ‘on-hand inventory of the supplier’ and show that it has a product form. furthermore, we derive the total expected system cost per unit time. after convexity analysis of the cost function, we derive the optimal inventory policy of supplier and retailer. finally a numerical example is provided.