An M/M/1 Queueing-Inventory System with Geometric Batch Demands and Lost Sales

This paper studies an M/M/1 queueing-inventory system with batch demands. Customers arrive in the system according to a compound Poisson process, where the size of the batch demands for each arrival is a random variable that follows a geometric distribution. The inventory is replenished according to the standard (s,S) policy. The replenishment time follows an exponential distribution. Two models are considered. In the first model, if the on-hand inventory is less than the size of the batch demands of an arrived customer, the customer takes away all the items in the inventory, and a part of the customer’s batch demands is lost. In the second model, if the on-hand inventory is less than the size of the batch demands of an arrived customer, the customer leaves without taking any item from the inventory, and all of the customer’s batch demands are lost. For these two models, the authors derive the stationary conditions of the system. Then, the authors derive the stationary distributions of the product-form of the joint queue length and the on-hand inventory process. Besides this, the authors obtain some important performance measures and the average cost functions by using these stationary distributions. The results are illustrated by numerical examples.