A Batch Demands Queueing-inventory Model with Positive Replenishment Time

Abstract

This study examines an M/M/1 queueing model using a (s,Q) inventory system and batch demand environment. The customers may request one or more items with a maximum demand of B items. There is just one service provider and a limited number of places available for customers to wait. A customer who arrived in accordance with the Poisson process is forced to leave without service if the waiting space is full. The service time and the replenishment time are assumed to have independent exponential distributions. It is assumed that maximum batch demand B is smaller than or equal to reorder level s in order to avoid repeated replenishment in a service. With the help of an iterative process, we are able to derive the steady-state joint probability distribution of the number of customers in the system and the on-hand inventory level of the queueing-inventory model. The optimum values for waiting space (N), reorder level (s), and order quantity (Q) are found by establishing a number of stationary system performance measures and estimating the total expected cost function under an appropriate cost structure. Some numerical results for various model parameters are provided in order to explain the key performance measures of the system. We execute simulation results with ARENA software to validate our model. We also perform simulation results for the equivalent M/G/1 queueing-inventory model.