How do I generate a Poisson Process?

Question

Original Question:

I want to generate a Poisson process. If the number of arrivals by time t is N(t) and I have a Poisson distribution with parameter λ how do I generate N(t)? How would I do this in C++?

Clarification:

I originally wanted to generate the process using a Poisson distribution. But, I was confused about what parameter from the process I needed; I thought I could use N(t) but that tells me how many arrivals have occurred on the interval (0,t] which wasn't what I wanted. So, then I thought I could use N(t2)-N(t1) to get the number of arrivals on the interval [t1,t2]. Since N(t)~Poisson(t x λ) I could use Poisson(t2 x λ)-Poisson(t1 x λ) but I don't want the number of arrivals in an interval.

Rather, I want to generate the explicit times that arrivals occur at.

I could do this by making the interval [t2,t1] sufficiently small so that each interval has only one arrival (which occurs as |t2-t1| -> 0).

Answer 1

Here's sample code for generating Poisson samples using C++ TR1.

If you want a Poisson process, times between arrivals are exponentially distributed, and exponential values can be generated trivially with the inverse CDF method: -k*log(u) where u is a uniform random variable and k is the mean of the exponential.