Transient scrutiny of $M^X/G(a,b)/1$ queueing system with feedback, balking and two phase of service subject to server failure under Bernoulli vacation

The transient scrutiny of a batch arrival feedback queueing system with balking and two stages of varying service with contrasting levels of service subjected to Bernoulli vacation has been examined in this study. Customers also have the option to decline services and leave the service area if the server is unable to fulfill their request when they arrive. The server may continue to serve the customers, if any, after each service with probability $\omega$, or it may undergo a vacation with probability $(1-\omega )$. The service channel may fail temporarily when the server is operating in any phase of service, which is then directed straight to the repair process. The model's steady state results and time-dependent probability generating functions in terms of their Laplace transforms have been derived. The mean queue length and the average time spent in the queue are explicitly determined as performance indicators in the various system states. A few unique cases and specific circumstances have also been presented. Finally, the effect of different parameters on the system's efficiency is then numerically analyzed.