in the fluid (or continuum) limit, you should allow $k$ to vary continuously and consider instead of $\pi_k$ (the probability of a queue of exactly length $k$) the probability $\pi(k)dk$ that a queue has length between $k$ and $k+dk$; the difference between A and B is then whether $\pi(k)$ varies continuously or as a step function, and in the large-$N$ limit these two descriptions become equivalent.