Let $ (N_t)_{t\in\mathbb{R}^+} $ be a homogeneous Poisson point process of intensity $\lambda>0$. I'm interested in determining the distribution of \begin{align*} S_t = \sum_{k=1}^{N_t} T_k, \end{align*} where $T_k = \inf \{t\ge 0, N_t\ge k\}$.
My attempt: Note that $S_t$ is not a conditioned Erlang distributed random variable, because $T_1,T_2,\dots$ are not independent. However, we can define $\Delta T_k = T_k-T_{k-1}$, with $T_0=0$, and rewrite $S_t$ as \begin{align*} S_t = \sum_{k=1}^{N_t} k \,\Delta T_{N_t+1-k}, \end{align*} where $\{\Delta T_k\}_k$ is a set of iid random variables that follow an exponential distribution of parameter $\lambda$.
Then, the characteristic function of $S_t \mid N_t = n$ is \begin{align*} \varphi_n(x) = \prod_{k=1}^{n} \frac{\lambda}{\lambda - ikx}, \end{align*} and the characteristic function of $S_t$ is given by $\varphi(x) = \mathbb{E}[\varphi_{N_t}(x)]$. I tried to perform the above calculations, but without success until now. Perhaps, there is a better approach to this problem. Thanks for all the help!