Admission control to an M/M/1 queue with partial information

We consider both a cooperative as well as non-cooperative admission into an M/M/1 queue. The only information available is a signal that sais whether the queue size is smaller than some $L$ or not. We first compute the globally optimal and the Nash equilibrium stationary policy as a function of $L$. We compare the performance to that of full information on the queue size. We identify the $L$ that optimizes the equilibrium performance