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Lindley's integral equation is as follows
$$W(y)=\int_{u=-\infty}^{y}W(y-u)dC(u),$$for $y\ge0$;
$$W^{-}(y)=\int_{u=-\infty}^{y}W(y-u)dC(u)$$,for $y<0$.
So we have
$$W(y)+W^{-}(y)=\int_{u=-\infty}^{y}W(y-u)dC(u)$$,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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