Let $$Y(t) = \sum_{n=1}^{N(t)} X_n e^{-\alpha(t-T_n)}$$
be a renewal-reward process such that $X_n\sim \Gamma(2,1)$ and the arrival times $T_n$ correspond to a Poisson process of rate $1$ whose occurrences are counted in $N(t)$.
I understand this is a standard renewal-reward process as defined in Wikipedia, where the rewards depend on the arrival times.
I am interested in calculating the expected first passage time (i.e. the first instant in which $Y(t)$ exceeds some constant $K$). Namely, the expected value of $$T_K = \inf\{t\,:\, Y(t) >K \}$$
Is there a closed form for this expression, given the complexity of the process? I know that $\mathbb{P}(T_K\geq t)=\mathbb{P}(Y(t)<K)$, but the density of $Y(t)$ seems to be something quite difficult to integrate in order to find a closed form for $\mathbb{E}[T_K]$.
Any ideas or any sources I could use to know how to proceed?