Cars, of random length $L$, arrive at a gate. The first car parks against the gate. The other arriving cars park behind at a distance uniformly distributed on $[0,1]$.
Let $N(t)$ be the number of cars parked at a distance $t$ from the gate. Find:
$$
\lim_{t\to \infty} E[N(t)]/t
$$
I have not been given a distribution for $L$.
If there was no space in between cars I could just use the continuous renewal equation. I'm not sure how to incorporate the space in between cars here.
I think the expected value of the spaces between cars is $1/2(N(t)-1)$. Assuming I have found this correctly, could I just add this expected value to the expected value of the general continuous renewal process?
Denote the distance between two cars as $U$. The first car takes $L + U$ of space. The second car takes $L + U$ of space, etc.
Using the notation of the Wikipedia page: the holding times $S_i$ are i.i.d. and distributed as $L + U$. The number of cars at distance $t$ is $N(t)$. $N(\cdot)$ is a renewal process. By the elementary renewal theorem
\begin{equation} \lim_{t \to \infty} \frac{\mathbb{E}[N(t)]}{t} = \frac{1}{\mathbb{E}[S_1]} = \frac{1}{\mathbb{E}[L] + \mathbb{E}[U]}. \end{equation}