We introduce a single location periodic review inventory control problem with lost sales and fractional lead times. We model the optimal inventory control problem as a stochastic dynamic program and analyze properties of the objective function as well as the optimal policy. We present upper and lower bounds on the optimal policy. These bounds can readily be used in easily computable heuristics. In addition to these heuristics, we also analyze properties of the system when order-up-to S policies are used. We prove the convexity of the cost function with respect to the order-up-to parameter S. We use this convexity property to determine the best order-up-to levels for this system using bisection search. Our computational investigation reveals that the deviation from the optimal cost is less than 1.5% on an average for both the “upper-bound heuristic” and the best order-up-to S policy.