The distribution of the content of finite dams

We shall consider the following model of finite dams: In the time interval (0, ∞) water is flowing into a dam (reservoir) in accordance with a random process. Denote by χ ( u ) the total quantity of water flowing into the dam in the time interval (0, u ). The capacity of the dam is a finite positive number m. If the dam becomes full, the excess water overflows. If the dam is not empty, there is a continuous release at a constant unit rate. Denote by η ( t ) the content of the dam at time t.