Since Beta is for fixed blocks and Poisson is for fixed time, I would guess we have to make the (N-1)/N correction like this to your Poisson equations to convert it to fixed blocks:
E[bw] = n \frac{b-1}{b}
Var[bw] = \frac{n^2}{b} \frac{b}{b-1}
In other words, if Beta is used (fixed-blocks), then make the above correction to Poisson which should be the same as using Erlang (or Gamma). If Poisson is used (fixed time interval), use Bionomial in place of Beta.