Two-Class M/M/1 Make-to-Stock Queueing Systems with Both Backlogs and Lost Sales

Abstract

We introduce a new simple allocation policy which is a very good approximation of the optimal allocation policy in an inventory system with a single product and two priority classes of customers. A production facility produces new items with exponentially distributed production times as long as the inventory level is below a base-stock level of inventory. We assume that customers arrive to the system according to a Poisson process. They may be satisfied, backlogged, or rejected depending on their priority, the inventory level upon their arrivals, availability of products in stock, and availability of a finite waiting area. We define a categorized cost function to investigate the efficiency of the new allocation policy and several known allocation policies in the literature. The system is modeled as a combination of one- and two-dimensional Birth-and-Death processes under four different allocation policies: Sharing with Minimum Allocation (SMA) policy, Complete Partitioning (CP) policy, Multilevel Rationing (MR) policy, and Lost Sales (LS) policy. By solving the model and deriving the relevant probabilities, we calculate the relative gap between each policy and the optimal policy. Based on the numerical results, we find that the SMA policy provides a very good approximation of the optimal policy, and is applicable in practical problems with high dimensions and static levels of inventory and waiting areas. We show that the MR and LS policies are special cases of the SMA policy. Therefore, their performances can be evaluated using the results obtained under the SMA allocation policy.