exponential, memoryless interarrival times

Instead of counting events, watch the clock between them. The time you wait from one arrival to the next is an interarrival time. For a Poisson process these gaps are the heart of the matter: each one is a fresh, independent draw from the same exponential distribution, and that single fact generates the whole process. If you can simulate exponential waits, you can simulate the entire Poisson process just by adding them up.

Concretely, in a Poisson process of rate lambda the interarrival times T1, T2, T3, ... are independent and identically distributed, each exponential with rate lambda, so each has mean 1/lambda and the probability of waiting longer than t is P(T > t) = e^(-lambda*t). The defining magic is memorylessness: P(T > s + t given T > s) = P(T > t). In words, if you have already waited s minutes with nothing arriving, the remaining wait has the SAME distribution as a fresh wait — the process does not 'remember' that you have been waiting, and it is never 'due'. The exponential is the only continuous distribution with this property, which is exactly why it is forced on us by the Poisson assumptions.

This is liberating and counterintuitive at once. It means a Poisson process restarts from scratch at any instant you care to look, which makes calculations clean. But it clashes with intuition: a bus you have waited ages for is not more likely to come soon. And it warns you when the model is wrong: if waiting longer really does make the next event more (or less) likely — machines that wear out, customers who give up — the gaps are not exponential and you need a renewal process with some other interarrival distribution instead.