This paper proposes a randomized policy for the control of arrivals in a finite-buffer GI/M/1 system with a single server. When the capacity of system is full, no new arrivals are permitted to enter the system. When the number of customers in the system decreases to threshold F, a new arriving customer is allowed to join the system with probability p. The system requires an exponential startup time before allowing customers to enter the system. The startup process may not be successful and is then restarted once again. Using the supplementary variable technique in a recursive process, we obtain the stationary distribution of the system size. Various performance measures of the system are developed. We also create a cost model based on the system performance measures and cost elements. The optimal threshold, optimal capacity and optimal startup rate of the system are determined to minimize the expected cost per unit time. Finally, we provide numerical examples to conduct a sensitivity analysis.