A two-stage bulk-service queueing model with rework, closedown and multiple vacations

In the manufacturing industry, the quality of products is of great significance in the preservation of the brand image. Quality inspection is defined as a procedure for ensuring the compliance, reliability, uniformity, and standard of a specific product. Through effective inspection, manufacturers are able to detect problems that require correction and prevent errors to ensure quality products for customers. To study such scenarios in queueing systems, we examine a two-stage bulk service single server with batch arrivals, incorporating rework, system closedown, and multiple vacation periods. Once a batch of fabric completes the first stage of regular service, the server proceeds to the second inspection stage for the same batch. If the inspection stage identifies defects, the server returns the batch for rework in the regular service stage with a specific probability. The server initiates regular service for the next batch with a specified probability when inspection yields no defects, and the queue length exceeds $``a"$. Conversely, if the queue length is below $``a"$ and no defects are present, the server may enter a closedown state with a given probability. After closedown, the server can take multiple vacations of random duration. Using the supplementary variable technique, we develop a model to analyze the probability-generating function of the queue size at any given moment. Additionally, we explore performance metrics through numerical examples, provide a cost analysis, and present the findings with graphical representations.