Lindley's equation is well known within queueing theory and is as follows

$F(y) = - \int_0^\infty F(x)dH(y-x)$

However, many textbooks only consider the case where 0 $\le$ y $\le \infty$ (which results in a solution using Laplace transforms), and I am more interested in the situation where 0 $\le$ y $\le$ k.

This condition results in F(x) = 1 for x > k and H(-$\infty$)=0, making the previous equation

$ F(y) = - \int_0^k F(x)dH(y-x) - \int_k^\infty dH(y-x) $

$ = H(y-k) - \int_0^k F(x)dH(y-x) $

which I am having problems applying the Laplace transform to. Could anyone point me in the right direction for this?