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