Stochastic analysis of a system of two interconnected inventories

Abstract

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.