The arrivals (absolute) times form a homogeneous) Poisson point process on the real line, which corresponds to the jumps of a (homogeneous) Poisson stochastic process.
The question is now: how do you want to simulate it (in terms of time or in terms of arrival)? Do you want to simulate the arrivals up to some some time t? Or do you want the system to evolve and run the simulation until n arrivals have occured?
