A novel approach for modeling a controllable queuing system using a Markovian environment is presented in the present article. The queueing nomenclatures, namely, working vacation, feedback policy, and customer impatience, are incorporated into the statistical modeling. The proposed congestion problem is examined from a production/service perspective with dynamic investigation, regardless of how it seems significant from an economic perspective. The suggested queueing-theoretic framework enables explicit modeling of control policies that can be implemented to regulate the rate of customer arrival and departure process. Using the Laplace transformation approach, the variation in the queue-size distribution over time is illustrated. To further confirm the significance of the proposed modeling, sensitivity analysis and several numerical simulations based on different system characteristics are also performed. Finally, tabular and graphical representations of the research findings obtained through Markovian modeling are exhibited.