Uniqueness of a Constrained Variational Problem and Large Deviations of Buffer Size

We show global uniqueness of the solution to a class of constrained variational problems, using scaling properties. This is used to establish the essential uniqueness of solutions of a large deviations problem in multiple dimensions. The result is motivated by models of buffers, and in particular the probability of, and typical path to overflow in the limit of small buffers, which we analyze.