Bounds on performance measures for the (r, q) lost sales system with Poisson demand and constant lead time

We demonstrate that the fraction of sales lost for the (r, q) system under consideration can be conveniently bounded in a manner suitable for quick, back-of-the-envelope estimates. We assume that customer demand arises from a Poisson process with one unit demanded at a time, that all demand occurring during a stockout is lost, and that lead time is constant. In addition, we allow the situation where multiple replenishment orders may be simultaneously outstanding. We show that the difference between the upper and lower bounds on the the fraction of sales lost appears likely to be no more than 6:5% (0:065), and is typically significantly better than that. In particular, the bounds appear to be very useful when the reorder point is relatively high or when the mean lead time demand is significantly different from the reorder point. In addition, we relate the common performance measures for the system, and so enable bounding the total operating costs.