Many periodic review inventory studies make the simplifying assumption that orders do not crossover, that is, that they are delivered in the same sequence as they were issued. In many real-life situations, however, long international shipping routes result with frequent crossovers. Accordingly, we investigate a periodic review inventory system with backlogged compound Poisson demand in which order may crossover. We develop an exact formula for the window fill rate, i.e. the probability for customers to receive service within their tolerable wait. Evaluating the window fill rate using the exact formula is very time consuming and therefore an efficient approximation formula that assumes that orders do not crossover is considered. Lead times of actual global supply lines demonstrate that this approximation results in considerable overstocking when customers' tolerable wait is high, or understocking when the tolerable wait is low. Replacing the actual lead times distribution with the effective lead times distribution improves the approximation formula's accuracy considerably. This result allows us to show how the window fill rate depends on the model parameters and make practical observations about the trade-off between the tolerable wait and stock levels needed to maintain a required level of window fill rate.

许多定期审查库存研究做出了简单的假设，即订单不会交叉，也就是说，它们的交付顺序与发出的顺序相同。然而，在许多现实生活中，漫长的国际航线会导致频繁的交叉。因此，我们调查了一个具有积压的复合泊松需求的定期审查库存系统，其中订单可能会交叉。我们为窗口填充率制定了一个精确的公式，即客户在可容忍的等待内接受服务的概率。使用精确公式评估窗口填充率非常耗时，因此需要考虑假设订单不交叉的有效近似公式。实际全球供应线的交货时间表明，当客户的可容忍等待时间较长时，这种近似会导致相当多的库存积压，或者当可容忍等待时间较短时导致库存不足。用有效提前期分布替换实际提前期分布可显着提高近似公式的准确性。该结果使我们能够展示窗口填充率如何取决于模型参数，并对维持所需窗口填充率水平所需的可容忍等待和库存水平之间的权衡进行实际观察。