Distribution of the first hitting time for a Poisson point process

Question

Consider a $\mathbb{R}_+^2$-valued Poisson point process $(e_s,\, s \geq 0)$ with intensity measure $\mu(\mathrm{d}x)\mathrm{d}y$ where $\mu$ is a finite measure on $\mathbb{R}_+$. Let $f \colon \mathbb{R}_+ \to \mathbb{R}_+$. Writing $e_s = (x_s,y_s)$, define $T = \inf\{s\geq 0\colon\, y_s\leq f(s)\}$. It is not hard to see that $$\mathbb{P}(T= \infty) = \mathbb{P}\left(\mathrm{Card}\{s\colon \, y_s \leq f(s)\}=0\right)=\exp\left(-\mu(\mathbb{R}_+)f(s)\right).$$ Now I would like to determine the joint distribution of $(T,x_T)$ on the event $\{T<\infty\}$. I know that if $f\equiv c$ is constant then a classical result on Poisson point processes gives that $T$ is exponentially distributed, $x_T$ has distribution $\mu(\mathrm{d}x)/\mu(\mathbb{R}_+)$ and they are independent. What can be said about the general case when $f$ is not constant? Does a result of this type exist in the literature?