How is a simple birth process is time-homogeneous? Why is it that a simple birth process is time-homogeneous?
The incidence of a birth in a small time interval depends on the population size at the time of start of the interval. Doesn't this population size clearly depend on the time (since the population grows over time)?
Or is the time factor irrelevant, since another population (also undergoing simple birth process) at an earlier time might have a larger population? Even so, within the same population, population size still does depend on the time, so I'm still unsure how we can say that a simple birth process is time-homogeneous.

The notion of time-homogeneity of a stochastic process $(X_t)$ can refer to the invariance of its distribution, that is, to the fact that the distribution of $X_t$ does not depend on $t$ and more generally to the fact that, for every set $T$ of nonnegative time indices, the distribution of $X_{t+T}=(X_{t+s})_{s\in T}$ does not depend on $t$. 
Or it can refer to the invariance of its evolution, that is, to the fact that the conditional distribution of $X_{t+1}$ conditionally on $X_t$ does not depend on $t$ and more generally to the fact that, for every set $T$ of nonnegative time indices, the conditional distribution of $X_{t+T}$ conditionally on $X_t$ does not depend on $t$.
A simple birth process is time-homogeneous in the second sense (which is the most frequently used of the two).
