Stochastic Analysis of a System of Two Interconnected Inventories

This paper considers a continuous review inventory system for two interconnected product types, 1 and 2. Product type 1 is purchased from an external agency, whereas type 2 is manufactured in-house through a sequential batching process. The maximum stock position attainable by type 1 is S1 and that of type 2 is S2. Unit demands arise independently for the two products, where type 1 demand arrives following a Poisson process with rate λ1 and that for product B also follows a Poisson process with rate λ2. At the instance of the stock level of type 1 dropping to zero, it is replenished instantaneously to the maximum level S1, such that the stock level is never zero, and hence all demands for type 1 product are satisfied. The production machine attached to type 2 stops manufacturing immediately when its stock level reaches S2, and resumes immediately when the stock level drops to S2−1. In the event of the type 2 product not being available when demand arrives, it is substituted with the type 1 product with probability p. The production time for a single unit of type 2 is exponentially distributed with mean 1γ. We identify the underlying Markov process and analyse the performance of the interconnected inventory system.