In Assemble-To-Order (ATO) systems, situations may arise in which customer demand must be backlogged due to a shortage of some components, leaving available stock of other components unused. Such unused component stock is called remnant stock. Remnant stock is a consequence of both component ordering decisions and decisions regarding allocation of components to end-product demand. In this article, we examine periodic-review ATO systems under linear holding and backlogging costs with a component installation stock policy and a First-Come-First-Served (FCFS) allocation policy. We show that the FCFS allocation policy decouples the problem of optimal component allocation over time into deterministic period-by-period optimal component allocation problems. We denote the optimal allocation of components to end-product demand as multimatching. We solve the multi-matching problem by an iterative algorithm. In addition, an approximation scheme for the joint replenishment and allocation optimization problem with both upper and lower bounds is proposed. Numerical experiments for base-stock component replenishment policies show that under optimal base-stock policies and optimal allocation, remnant stock holding costs must be taken into account. Finally, joint optimization incorporating optimal FCFS component allocation is valuable because it provides a benchmark against which heuristic methods can be compared.