Order-fulfillment performance analysis of an assemble-to-order system with unreliable machines

We study a multi-item, multiple classes of demand, assemble-to-order system. The inventory of each item is kept at the item level and controlled by the base-stock policy with a finite capacity. Each item is replenished by an independent unreliable machine. Each type of demand arrives according to a Poisson process with an individual rate and requires a subset of the items. When the item requirements of an arriving demand cannot be satisfied entirely, two kinds of stockout may occur, namely total-order-service and partial-order-service. We formulate the system as a queuing network and deduce that it is a quasi-birth-death process. Applying the matrix-geometric solution approach, we derive the exact joint steady-state distribution of on-order inventories, based on which we compute the order-based and item-based the fill rate within a time window and the service level. We present numerical examples to show how system performance varies with changes in system parameters to show the importance of taking machine failures into consideration.