Consider a queue $M/G/\infty$, starting with arrival time of calls as a PPP$(\Lambda)$ and lengths of calls as $iid$ random variables with common distribution $G$. The times when the calls terminate and free up lines also constitute a Poisson Process.
Find an explicit expression for the mean measure of this process. Simplify it when $G$ has a density $g$ and show thay in that case, it is an inhomogeneous Poisson process on $\mathbb{R}$. Give an expression for its intensity function.
Let $\overset{\overset{\sim}{\sim}}{\Lambda}$ be the mean measure of this process and $\overset{\sim}{\Lambda}$ be the mean measure of the marked process with calls and lengths. By marking theory, we know that $$\overset{\sim}{\Lambda}(A\times A')=\Lambda\otimes G(A\times A')=\Lambda(A)G(A')$$ and by transformation theory, we know that if $T$ is the transformation such that $T(x,y)=x+y$, $\overset{\overset{\sim}{\sim}}{\Lambda}=\overset{\sim}{\Lambda}(T^{-1}(A)).$ So, $$\overset{\overset{\sim}{\sim}}{\Lambda}((a,b])=\overset{\sim}{\Lambda}((x,y):x+y\in(a,b])=\Lambda\otimes G((x,y):x+y\in(a,b])=\int\int_{(x,y):x+y\in(a,b]}d(\Lambda(x))d(G(y))=\int_0^\infty d(G(x))\int_{a-x}^{b-x}d(\Lambda(y))=\int_0^\infty g(x)[\Lambda(b-x)-\Lambda(a-x)]dx$$
How do I simplify this?