Lindley's integral equation is as follows
$$W(y)=\int_{u=\infty}^{y}W(yu)dC(u),$$for $y\ge0$;
and
$$W^{}(y)=\int_{u=\infty}^{y}W(yu)dC(u)$$,for $y<0$.
So we have
$$W(y)+W^{}(y)=\int_{u=\infty}^{y}W(yu)dC(u)$$,for all $y\in \Re.$
Take laplace transform on both sides, we have
$\mathcal{L}[W(y)]+\mathcal{L}[W^{}(y)]=\mathcal{L}[W(y)]\mathcal{L}[a(y)]\mathcal{L}[b(y)].$
In which $a(y)$ and $b(y)$ are the PDF of interarrival time and service time, and $W(y)$ is the PDF of the waiting time in queuing.
However, in my case, both $a(y)$ and $b(y)$ aer something like
\begin{equation}
a(y)=
\begin{cases}
0.15 &\mbox{,if y}\in[0,1)\\
0.25 &\mbox{,if y}\in[1,2)\\
0.40 &\mbox{,if y}\in[2,3]\\
0 &\mbox{,if y}\notin[0,3]\\
\end{cases}
\end{equation}.
I've read through many textbooks, and I can't find a way to solve the $W(y)$ out.Can someone help me?
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