# Wiener-Hopf Integral/Lindley's Equation

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?

-
what function is H? – Pietro Majer Nov 12 '12 at 12:45
H is the c.d.f. of the difference of the interarrival and service times – P. Browning Nov 12 '12 at 17:26
@ P. Browning : Hi what is $F$, and how would you apply Laplace's Tranform in the unbounded case successfully ? Best regards. – The Bridge Nov 13 '12 at 17:18
@ The Bridge : $F$ is meant to be the waiting time distribution in the case of queueing theory. The solution can be found in Gross and Harris's Fundamentals of Queueing theory. It involves defining a new function, $Z_t^-$ to account for $y \le 0$. It amounts to $Z_t^-(s)+Z_t(s)=Z_t(s)A^*(-s)B^*(s)$ where $H^*(s)=A^*(-s)B^*(s)$ – P. Browning Nov 13 '12 at 19:25