Bulk Queue Model with Multiphase Service and Repair with Balking and N Policy

This paper investigates a single-server queueing system in which customers arrive in batches at a constant rate. Each customer undergoes a mandatory first-stage service, followed by an optional second-stage service. The server is subject to breakdowns that can occur at any point during either service stage. A phased repair mechanism, consisting of both essential and optional repairs, is incorporated into the model. Upon completion of the essential repair, the server proceeds to the second repair phase with probability . Likewise, after the phase , it enters the phase with probability ​; otherwise, it leaves the repair system to resume service. In this manner, the server may undergo up to k repair phases, including the initial essential phase. Further, it is assumed that customers are impatient in nature may balk from the system if server is not available on their arrival. To analyze the system’s steady-state behavior, the study utilizes probabilistic reasoning alongside the supplementary variable technique. Key performance metrics are derived using the generating function method, and their validity is demonstrated through numerical examples.