We consider a controlled M/M/1 queueing system where customers may be subject to two potential rejections. The first occurs upon arrival and is dependent on the number of customers in the queue and the service rate of the customer currently in service. The second, which may or may not occur, occurs immediately prior to the customer receiving service. That is, after each service completion the customer in the front of the queue is assessed and the service rate of that customer is revealed. If the second decision-maker recommends rejection, the customer is denied service with a fixed probability. We show the existence of long-run average optimal monotone switching-curve policies. Further, we show that the average reward is increasing in the probability that the second decision-maker's recommendation of rejection is honored. Applications include call centers with delayed classifications and manufacturing systems when the server is responsible for multiple tasks.
