We consider the notoriously difficult discrete-time inventory model with stochastic demands, a constant lead time, and lost sales. We show that the effective state space is a relatively manageable compact set. Then, we test various plausible heuristics. We find that several perform reasonably well, although none is perfect. However, the standard base-stock policy (a direct analogue of the optimal policy for a backlog system.) performs badly. We also show that the optimal cost is increasing in the lead time. © 2008 INFORMS.