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We consider the problem of admission control to a multiserver finite buffer queue under partial information. The controller cannot see the queue but is informed immediately if an admitted customer is lost due to buffer overflow. Turning away (i.e., blocking) customers is costly and so is losing an admitted customer. The latter cost is greater than that of blocking. The controller's objective is to minimize the average cost of blocking and rejection per incoming customer. Lin and Ross  studied this problem for multiserver loss systems. We extend their work by allowing a finite buffer and the arrival process to be of the renewal type. We propose a control policy based on a novel state aggregation approach that exploits the regenerative structure of the system, performs well, and gives a lower bound on the optimal cost. The control policy is inspired by a simulation technique that reduces the variance of the estimators by not simulating the customer service process. Numerical experiments show that our bound varies with the load offered to the system and is typically within 1% and 10% of the optimal cost. Also, our bound is tight in the important case when the cost of blocking is low compared to the cost of rejection and the load offered to the system is high. The quality of the bound degrades with the degree of state aggregation, but the computational effort is comparatively small. Moreover, the control policies that we obtain perform better compared to a heuristic suggested by Lin and Ross. The state aggregation technique developed in this article can be used more generally to solve problems in which the objective is to control the time to the end of a cycle and the quality of the information available to the controller degrades with the length of the cycle.