Let $(\alpha_i)_{i\in\Bbb N}$ be a collection of i.i.d. random variables on $[0,\infty)$ with density to lebesgue measure. Then, for $t>0, $ let $N(t) := \max\{ k \geq 1 : \sum_{i=1}^k \alpha_i \leq t\}$.
I am interested if one can say something about $\Bbb E[N(t)]$. I think, heuristically, because $$\Bbb E \left[ \sum_{i=1}^k \alpha_i\right] \leq t \Leftrightarrow k \leq \frac t {\Bbb E[ \alpha_1]}$$ we could expect that \begin{align}\tag{1}\label{this} \Bbb E[N(t)] \sim \frac t {\Bbb E[ \alpha_1]}\end{align}
Lately I observed that for $\alpha_1 \sim \text{Exp}(\lambda)$ this holds with equality, more precisely $$\Bbb E[N(t)] = \lambda t$$ while $N(t)\sim$ Poi$(\lambda t)$ is a Poisson-Process. Further this holds for $\alpha_1 \sim$ $\chi^2(2)$ with $$\Bbb E[N(t)] = \frac t 2$$
Is there any approach to the behaviour of (\ref{this}) already? I could not really figure it out, and especially in the calculations for the equations above I used the explicit properties of the distributions of $\alpha_1$. And my first attempt looked like this: \begin{align} \Bbb E[N(t)] = \sum_{k=1}^\infty \Bbb P (N(t) \geq k) = \sum_{k=1}^\infty \Bbb P \left(\sum_{j=1}^k \alpha_j \leq t \right) = \;\;? \end{align} but didn't go further.