My problem is this:
I have $1000$ machines which turn on after a delay upon receiving the 'on' signal. The delay is set to be a random number of minutes between $0$ and $15$. So for any single machine, upon receiving the 'on' signal, the delay is any number of minutes drawn at random between $0$ and $15$.
I have to find out how many machines could be 'on' at any given time after they all receive the on signal at once.
For example, I model a simulation of these machines from time $t=0$ to $t=10$. At $t=1$ the 'on' signal is sent to every machine. I have to approximately model the number of machines on at each time step from $t=1$ to $t=10$.
My thinking was that since the delay has a uniform distribution from $0-15$, I would just model the number of machines on directly proportional to the cumulative distribution function of $p=15$. Which should just be a linear relationship? (aka a straight, diagonal line with minutes $0-15$ on the $x$ axis and $\#$ of machines on in the $y$ axis).
Thank you for your time!