We consider a storage process $X(t)$ having a compound Poisson process as input and general release rules, and a nonnegative additive functional $Z(t) = \int^t_0 f(X(s)) ds$. Under the situation that the input rate is equal to the maximal output rate, it is shown for a suitable class of functions of $f$ that an appropriate normalization of the process $Z(t)$ converges weakly to a process which is represented as a constant times the local time of a Bessel process at zero.
Keigo Yamada. "A Limit Theorem for Nonnegative Additive Functionals of Storage Processes." Ann. Probab. 13 (2) 397 - 413, May, 1985. https://doi.org/10.1214/aop/1176992999