This paper presents a comprehensive analysis of four single-server queueing inventory models (QI models ) in which customers can demand single or bulk units of a commodity. Inventory is managed using the (s, S) replenishment policy with exponentially distributed lead time, while customer arrivals follow Poisson process and service times are exponentially distributed. Each customer may demand up to a units, where \(a < s\) . Model 1 deals with a QI system with multiple queues: queues are formed based on the demand size of the customers, so that the \( i^{th}\) queue contains customers who demand \( i\) number of units ( \(i = 1,2,\dots ,a.\) ). Priority is given to customers with smaller demands. Model 2 is a restriction of Model 1, in which all customers are blocked during stockout. Model 3 considers a single-queue single-server queueing inventory system in which demand quantities are revealed only when customers are taken for service; all categories of customers are to be in the same queue. Model 4 is a modified version of Model 3 - blocking all customers when inventory level drops to zero. By formulating these models as quasi-birth-and-death (QBD) processes and applying the matrix-geometric solution method, we derive stability conditions, stationary distribution of the process and various system performance measures. In addition, we investigate server idle times and the durations of inventory cycles. Through several numerical experiments, we illustrate how variations in the system parameters affect the performance characteristics, providing insights into the operational dynamics of each model.
