On a single item fill rate for a finite horizon

We investigate expressions for expected item fill rate in a periodic inventory system. The typical treatment of fill rate found in many operations management texts assumes infinite horizon, independent and stationary demand. For the case when the horizon is finite, we show that the expected value of the actual fill rate is greater than the value given by the infinite horizon expression. The implication of our results is that an inventory manager in a finite horizon situation who uses the infinite horizon expression to set stocking levels will achieve a higher than desired expected fill rate at greater than necessary inventory expense.