Performance evaluation of a two-echelon inventory system with network lost sales

Inventory systems are largely analyzed in the literature under the common assumption of backorders due to the complexity of lost sales. In this paper, we consider a two-echelon inventory system composed of a central warehouse and multiple local warehouses subject to lost sales. The demand faced by each local warehouse is a Poisson process and the stock in the warehouses is controlled according to a continuous review base-stock policy. This system has been analyzed in the literature under deterministic or exponential lead-times at the central warehouse, deterministic lead times at the local warehouses and approximate performance evaluations have been proposed for two cases: (1) the demand is lost if no items are available in the local warehouse, the central warehouse, or in the pipeline in between (i.e., a waiting time threshold for incoming demand equal to the local warehouse lead time), and (2) when there is a waiting time threshold less than the local warehouse lead time. Based on a queuing network representation of the system, we extend the performance analysis of the system in the first case by considering generally distributed lead times both at the central and local warehouses and by providing the exact closed-form expressions for the inventory performance measures. In the second case, we provide new approximate solutions under generally distributed lead times at the central warehouse. We numerically show that our exact and approximate solutions perform equally or better than those presented in the literature under deterministic lead times.