Inventory Models with MAP Demands and Random Replenishment

Combining the study of queuing with inventory is extremely common and such systems are mentioned as queuing-inventory systems within the literature. These systems occur naturally in practice and are studied extensively within the literature. The inventory systems considered within the literature generally include (s,S)-type. However, during this paper we glance at opportunistic-type inventory replenishment during which there's an independent point process that's wont to model events that are called opportunistic for replenishing inventory. When a chance (to replenish) occurs, a probabilistic rule that depends on the inventory level is employed to work out whether to avail it or not. Assuming that the purchasers arrive consistent with a Markovian arrival process, the stress for inventory occur in batches of varying size, the stress require random service times that are modeled employing a continuous-time phase-type distribution, and therefore the point process for the opportunistic replenishment may be a Poisson process, we apply matrix-analytic methods to review two of such models. In one among the models, the purchasers are lost when at arrivals there's no inventory and within the other model, the purchasers can enter into the system albeit the inventory is zero but the server has got to be busy at that moment. However, the purchasers 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 a few possible future work is highlighted.