Say I have two event streams, modeled as Poisson processes. These streams omit events at rates $\lambda_1$ and $\lambda_2$ respectively, so the count of observed events in $t$ time is $\lambda_1t$ and $\lambda_2t$. For some desired confidence, how long do I need to observe in order to be confident that either $\lambda_1 > \lambda_2$ or vice versa?
Intuition suggests that the answer will be some distribution that's a function of $\lambda_1$, $\lambda_2$, $t$, and my desired confidence. I did some searching and just found tests for inequality of $\lambda_1$ and $\lambda_2$, not a specific parameterized distribution.