We study single and multistage inventory systems with stochastic lead times. We study a class of stochastic lead time processes, which we refer to as exogenous lead times. This class of lead time processes includes as special cases all lead time models from existing literature (such as Kaplan's lead times with no order crossing or independent and identically distributed lead times with order crossing, among others) but is a substantially broader class. For a system with an exogenous lead time process, we provide a method to determine base-stock levels and to compute the cost of a given base-stock policy. The method relies on relating the cost of a base-stock policy to the cost of a threshold policy in a related single-unit, single-customer problem. This single-unit method is exact for single-stage systems and for multistage systems under certain conditions. If the conditions are not satisfied, the method obtains near-optimal base-stock levels and accurate approximations of cost for multistage systems.
