Loss rate for a general L\'evy process with downward periodic barrier

In this paper we consider a general L\'{e}vy process $X$ reflected at downward periodic barrier $A_t$ and constant upper barrier $K$ giving a process $V^K_t=X_t+L^A_t-L^K_t$. We find the expression for a loss rate defined by $l^K=\mathbb{E} L^K_1$ and identify its asymptotics as $K\to\infty$ when $X$ has light-tailed jumps and $\mathbb{E}X_1<0$.