We know that when the arriving rate of a Poisson process $X(t)$ becomes constant, then the process becomes a homogeneous Poisson process.
I have trouble understanding what "a constant arriving rate" means. My confusion largely arises from the fact that any ensemble $X(t_0)$ ($t_0$ is a constant) is in fact a random variable, implying the number of points arriving between $[0, t_0]$ is random.
Given that I can't even say for sure how many points arrive between $[0, t]$, how can I say I have "a constant arriving rate"?
In real life, what is an example case where we have a constant arriving rate, but we can't say for sure how many points will arrive for a known period of time? It just sounds contradictory to me.