Combining the study of queuing with inventory is very common and such systems are referred to as queuing-inventory systems in the literature. These systems occur naturally in practice and have been studied extensively in the literature. The inventory systems considered in the literature generally include $(s,S)$-type. However, in this paper we look at opportunistic-type inventory replenishment in which there is an independent point process that is used to model events that are called opportunistic for replenishing inventory. When an opportunity (to replenish) occurs, a probabilistic rule that depends on the inventory level is used to determine whether to avail it or not. Assuming that the customers arrive according to a Markovian arrival process, the demands for inventory occur in batches of varying size, the demands require random service times that are modeled using a continuous-time phase-type distribution, and the point process for the opportunistic replenishment is a Poisson process, we apply matrix-analytic methods to study two of such models. In one of the models, the customers are lost when at arrivals there is no inventory and in the other model, the customers can enter into the system even if the inventory is zero but the server has to be busy at that moment. However, the customers are lost at arrivals when the server is idle with zero inventory or at service completion epochs that leave the inventory to be zero. Illustrative numerical examples are presented, and some possible future work is highlighted.

中文翻译：

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具有MAP需求和随机补货机会的排队库存模型

将排队与库存的研究相结合是非常普遍的，这种系统在文献中被称为排队库存系统。这些系统在实践中自然存在，并且在文献中已经进行了广泛的研究。文献中考虑的清单系统通常包括$（s，\mathrm{小号}）$-类型。但是，在本文中，我们着眼于机会型库存补充，其中存在一个独立的点过程，该过程用于对事件进行建模，这些事件称为机会补充库存。当出现机会（补货）时，将使用取决于库存水平的概率规则来确定是否要利用它。假设客户按照马尔可夫的到达过程到达，则对库存的需求发生在不同大小的批次中，需求需要使用连续时间阶段类型分布建模的随机服务时间，以及机会性补货的积分过程是一个泊松过程，我们应用矩阵分析方法来研究其中的两个模型。在其中一个模型中，当到达时没有库存时，客户会流失；而在另一个模型中，即使库存为零，客户也可以进入系统，但此时服务器必须很忙。但是，当服务器闲置零库存时或在使库存为零的服务完成时期，客户会在到达时迷路。给出了说明性的数值示例，并突出了一些可能的未来工作。