 Solution

A rate-two Poisson process starts at $N_0=0$, has independent increments and stationary increments, and has $N_{t+h}-N_t\sim \mathrm{Poisson}(2h)$ for $h\ge 0$ when time is measured in minutes. Equivalently it is a counting process with independent stationary increments satisfying, uniformly in the start time,as $h\downarrow 0$. Thus an average rate alone is insufficient; the independence and increment assumptions are part of the model.