Simulate simple non-homogeneous Poisson proces

Question

Consider a Poisson process whose conditional intensity is

$$\lambda(t) = \alpha e^{-t}$$

starting at time $t=0$ for some parameter $\alpha>0$.

I would like to simulate arrival/event/failure times as efficiently as possible. (I am not sure what the standard term is for the times you get from a Poisson process.) Is there a fast way to do this?

Answer 1

"Amusing" downvote to an otherwise perfectly valid answer.

If one knows how to simulate a homogenous Poisson process $(N(t))$ with intensity $\alpha$ on the time interval $[0,1]$, one can then consider the process $(X(t))$ defined, for every $t\geqslant0$, by $$X(t)=N(1-\mathrm e^{-t}).$$