A Supplementary Variable Approach for a Non-Markovian Bulk Queue with Rework and Differentiated Server Breakdowns

This paper presents a comprehensive study of a non-Markovian bulk service queueing model incorporating inspection, rework, differentiated server breakdowns, a standby server mechanism, and multiple vacation policies. The proposed model is developed to analyze complex real-world production and manufacturing systems where items arrive in batches and are processed in groups under uncertain operating conditions. In many industrial environments, defective or spoiled items may occur during service, leading to additional operational costs and delays. To address this issue, the model integrates inspection and rework processes, which help identify defective items and restore them to acceptable quality standards, thereby reducing wastage and improving overall system efficiency. The study further considers two types of server breakdowns, namely soft and hard failures, which represent different levels of service interruption. To ensure continuity of service during such breakdown situations, a standby server is incorporated into the system. This standby support significantly reduces interruption time and enhances the reliability and stability of the service mechanism. In addition, the inclusion of multiple vacation policies allows the server to take controlled idle periods whenever the system becomes empty, reflecting realistic operational practices in manufacturing and communication systems. Using the supplementary variable technique, the steady-state behavior of the proposed queueing system is analyzed in detail. Numerical illustrations are also provided to demonstrate the practical applicability and effectiveness of the proposed model.