An optimal mechanism charging for priority in a queue

We derive a revenue-maximizing scheme that charges customers who are homogeneous with respect to their waiting cost parameter, for a random fee in order to become premium customers. This scheme incentivizes all customers to purchase priority, each at his/her drawn price. This is repeated for the case where customers are heterogeneous. The mechanisms are based on the fact that once some customers get priority, its value of the other gets even higher.