Simple modeling techniques for base-stock inventory systems with state dependent demand rates

In this paper new modeling techniques of $$(S-1,S)$$ ( S - 1 , S ) inventory systems (continuous review base-stock inventory systems) with state dependent demand rates are proposed. Examples of single-location $$(S-1,S)$$ ( S - 1 , S ) inventory systems where the demand, experienced by the system, varies due to the state of the system are, e.g., inventory models with partial backorders, inventory models with lost sales, inventory models with perishable items, inventory models with emergency replenishments etc. Models of such inventory systems are in general hard to solve due to the fact that the Markov property is often lost, and the prevalent tool used in the literature for providing exact solutions of such models is the theory of partial differential equations. Instead of using partial differential equations with rather complicated analysis of boundary conditions, we suggest considerably simpler techniques which are based on elementary theory of queueing and renewal processes. First, we show that it is possible to use Markov theory in order to prove certain statistical properties of the limiting distribution of the ages of the items in the system. Secondly, we develop a corresponding procedure based on renewal theory, which forms a basis for more complicated models assuming non-Poisson customer demand processes.