Load balancing in a network of queueing-inventory systems

We study a supply chain consisting of production-inventory systems at several locations which are coupled by common supplier. Demand customers arrives each production system according to Poisson process and is lost if the local inventory depleted ("lost sales"). To satisfy customer's demand server needs raw material from associated inventory. The supplier manufactures replenish inventories, controlled continuous review base stock policy. routing items depends on on-hand with aim obtain "load balancing". show that stationary distribution has product form marginal distributions subsystem inventory-replenishment subsystem. For we derive an explicit solution for resp. recursive algorithm some special cases.