  We study the convergence of the $M/G/1$ processor-sharing, queue length
process in the heavy traffic regime, in the finite variance case. To do so, we
combine results pertaining to L\'{e}vy processes, branching processes and
queuing theory. These results yield the convergence of long excursions of the
queue length processes, toward excursions obtained from those of some reflected
Brownian motion with drift, after taking the image of their local time process
by the Lamperti transformation. We also show, via excursion theoretic
arguments, that this entails the convergence of the entire processes to some
(other) reflected Brownian motion with drift. Along the way, we prove various
invariance principles for homogeneous, binary Crump-Mode-Jagers processes. In
the last section we discuss potential implications of the state space collapse
property, well known in the queuing literature, to branching processes.
