Irreducibility of joint inventory positions in an assemble-to-order system under (r, nQ) policies

In a typical assemble-to-order system, a customer order may request multiple items, and the order may not be filled if any of the requested items are out of stock. A key customer service measure when unfilled orders are backordered is the order-based backorder level. To evaluate this crucial performance measure, a fundamental question is whether the stationary joint inventory positions follow an independent and uniform distribution. In this context, this is equivalent to the irreducibility of the Markov chain formed by the joint inventory positions. This article presents a necessary and sufficient condition for the irreducibility of such a Markov chain through a set of simultaneous Diophantine equations. This result also leads to sufficient conditions that are more general than those in the published reports.