In this paper we consider an Assemble-to-Order system with multiple end-products. Demands for an end-product follow a Poisson process and each end-product requires a fixed set of components. We are interested in the order fill rates, i.e., the percentage of demands for which all requested components are available from stock. Requested components that are not in stock are supplied via an emergency shipment and the demand for these components is lost for the stockpoint under consideration. The component lead times are deterministic and may differ per component. The inventory of each component is controlled via a base stock policy. We show that the system decomposes into subsystems which can be analyzed independently. Each subsystem can be approximated by a subsystem with exponentially distributed lead times, for which an exact evaluation exists. For big subsystems, however, this method requires considerable computational effort. Therefore, we formulate a simple and accurate approximation for the order fill rates. Our approximation uses two estimates of which one generally gives an underestimation of the order fill rate and the other one an overestimation. A weighing factor is used to combine these two estimates into an approximate value. The approximation is shown to be accurate and requires little computational effort.