This study focuses on optimizing the efficiency of single-lane roundabouts for connected automated vehicles (CAVs). While roundabouts are vital for traffic management, their performance declines significantly under high traffic volumes. By leveraging vehicle-to-vehicle communication and the cooperative decision-making capabilities of CAVs, this research develops a microscopic control model that not only minimizes sojourn times (maximizes efficiency) but also provides precise trajectory control for each individual vehicle. Besides, a theoretical M/G/1/k queueing model is employed to calculate the expected sojourn time, serving as a baseline to compare CAV-controlled scenarios with human-driven conditions. The results demonstrate that the proposed microscopic control model significantly reduces the expected sojourn time for vehicles entering the roundabout. For a single-lane roundabout with a capacity of 17 vehicles per leg, compared to uncontrolled scenarios simulated in VISSIM, the control model achieves empirical efficiency improvements of 26%, 80%, 91%, and 90% under arrival rates of 200, 400, 600, and 800pcu/h per leg, respectively. These empirical findings align closely with theoretical predictions from the M/G/1/k queueing model, which estimate efficiency gains of 18%, 78%, 89%, and 90%. Moreover, the control model demonstrates robustness under extreme traffic conditions, maintaining high efficiency and passenger comfort. This research provides valuable insights into the integration of CAVs into traffic systems and contributes both practically and theoretically to the modernization of roundabout traffic management.