This paper proposes a new discrete-time Geo/G/1 queueing model under the control of bi-level randomized (p, N₁, N₂)-policy. That is, the server is closed down immediately when the system is empty. If N₁ (≥1) customers are accumulated in the queue, the server is activated for service with probability p (0≤ p≤1) or still left off with probability (1-p). When the number of customers in the system becomes N₂ (≥ N₁), the server begins serving the waiting customers until the system becomes empty again. For the model, firstly, we obtain the transient solution of the queue size distribution and the explicit recursive formulas of the stationary queue length distribution by employing the total probability decomposition technique. Then, the expressions of its probability generating function of the steady-state queue size and the expected steady-state queue size are presented. Additionally, numerical examples are conducted to discuss the effect of the system parameters on some performance indices. Furthermore, the steady-state distribution of queue length at epochs n^-, n and outside observer's observation epoch are explored, respectively. Finally, we establish a cost function to investigate the cost optimization problem under the constraint of the average waiting time. And the presented model provides a less expected cost as compared to the traditional N-policy.