Analytical and simulation studies of GI/M/1 queue integrated with an (s,Q) inventory policy

This study investigates a single server GI/M/1 queueing system attached with an (s,Q) inventory policy. The system has finite number of waiting rooms for the arrival of demands. Demands arrive in the manner of arbitrary distribution and they are served according to the exponential distribution. Every demand needs a single item from the stock. An order for items is placed when the stock level falls to a predetermined threshold s. An outside source replenishes the stock of the items according to an exponential distribution. The arrival demand waits in the queue for replenishment when stock level is zero. The paper derives the joint probability distribution of the number of demands and the stock level at the pre-arrival epoch. To determine the joint probability distribution at an arbitrary epoch, we obtain a relation between the system length and stock level distribution at the pre-arrival epoch, as well as the residual time of arrival demand as a supplementary variable at the arbitrary epoch. We then determine the probability density function of the waiting time distribution for a demand who joined the queue. We construct the total cost function with the help of performance measures. We numerically determine the optimal values for waiting space, reordering threshold, and order quantity. Numerical results support our analytical findings. The validation of our model is confirmed through simulation experiments. Additionally, a simulation study is conducted for the GI/G/1 queueing-inventory model to review the cost analysis for various parameters.