In this article, we provide a comprehensive analyses of two continuous review lost sales inventory system based on different replenishment policies, namely $(s,S)$ and $(s,Q)$. We assume that the arrival times of demands form a Poisson process and that the demand sizes have i.i.d. exponential distribution. We assume that the items in stock may obsolete after an exponential time. The lead time for replenishment is exponential. We also assume that the excess demands and the demands that occurred during stock out periods are lost. Using the system point method of level crossing and integral equation method, we derive the steady-state probability distribution of inventory level explicitly. After deriving some system performance measures, we computed the total expected cost rate. We also provide numerical examples of sensitivity analyses involving different parameters and costs. As a result of our numerical analysis, we provide several insights on the optimal $(s,S)$ and $(s,Q)$ policies for inventory systems of obsolescence items with positive lead times. The better policy for maintaining inventory can be quantified numerically.

在本文中，我们对两种基于不同补货政策的连续审查损失销售库存系统进行了全面分析，即$(s,S)$和$(s,Q)$。我们假设需求的到达时间形成泊松过程，并且需求大小具有独立同分布指数分布。我们假设库存商品可能会在指数时间后过时。补货的提前期是指数级的。我们还假设超额需求和缺货期间发生的需求都会丢失。利用水平交叉系统点法和积分方程法，明确推导了库存水平的稳态概率分布。在得出一些系统性能指标后，我们计算了总预期成本率。我们还提供涉及不同参数和成本的敏感性分析的数值示例。根据我们的数值分析，我们提供了关于最优$(s,S)$的一些见解，以及$(s,Q)$具有正交货时间的过时物品库存系统政策。更好的维持库存的政策可以用数字来量化。