Supplementary variable technique approach on retrial queue with feedback and backup server during Bernoulli's lengthy vacation and random repairs

This paper explores a steady-state retrial queueing system with a single server, coupled with a backup server. Client arrivals occur in batches following a compound Poisson process. If the server is idle, one of the clients from the batch receives service immediately, and the remaining clients enter the orbit. Otherwise, the entire incoming batch joins the orbit. Unsatisfied clients after receiving service may become feedback clients. After the completion of each service, the server may enter a vacation period of random length. The server has the option to extend its vacation with certain probability. Breakdowns of the main server can occur at any point during the busy state. Repairs of the server commence without delay. During vacation and repair time of the main server, clients receive service from a backup server. Performance metrics, such as the mean of the system size, mean of the orbit size, server availability and server failure frequency are computed. Cost analysis has been carried out with the cost parameters. The stochastic decomposition law is verified for the proposed model. Numerical results are also presented to validate the impact of parameters on the system.