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 $V_t^K=X_t+L_t^A-L_t^K$. We find the expression for a loss rate defined by $l^K=\mathbb{E}{L}_1^K$ and identify its asymptotics as $K\to \mathrm{\infty }$ when $X$ has light-tailed jumps and $\mathbb{E}{X}_1<0$.