Reliability and performance modeling of an M[X]/G(a,b)/1 retrial queue with re-service and Balking behavior under working vacation policy

This study investigates a bulk arrival retrial queueing model characterized by two-phase service and a working vacation policy. Customers arrive in batches according to a Poisson process. If the server is unavailable, some customers may balk, while the remaining customers join the orbit and retry for service after a random period of time. The server provides a required first service phase followed by a second service phase; during this regular service period, customers who are unsatisfied or have specific needs may return for re-service. When the orbit is empty, the active server takes a working vacation, providing service to customers at a slower rate. This mechanism improves performance by reducing waiting times during low-traffic periods. The model employs supplementary variable techniques to determine system attributes, including probability generating functions for the mean queue length and mean orbit size. Numerical examples illustrate the impact of balking, vacation, and phase-type service on significant metrics.