  We consider the FCFS $GI/GI/n$ queue in the Halfin-Whitt heavy traffic
regime, and prove bounds for the steady-state probability of delay (s.s.p.d.)
for generally distributed processing times. We prove that there exist
$\epsilon_1, \epsilon_2 > 0$, depending on the first three moments of the
inter-arrival and processing time distributions, such that the s.s.p.d. is
bounded from above by $\exp\big(-\epsilon_1 B^2\big)$ as the associated excess
parameter $B \rightarrow \infty$; and by $1 - \epsilon_2 B$ as $B \rightarrow
0$. We also prove that the tail of the steady-state number of idle servers has
a Gaussian decay. We provide explicit bounds in all cases, in terms of the
first three moments of the inter-arrival and service distributions, and use
known results to show that our bounds correctly capture various qualitative
scalings. \\\indent Our main proof technique is the derivation of new
stochastic comparison bounds for the FCFS $GI/GI/n$ queue, which are of a
structural nature, hold for all $n$ and times $t$, and significantly generalize
the work of \citet{GG.10c} (e.g. by providing bounds for the queue length to
exceed any given level, as opposed to any given level strictly greater than the
number of servers as acheived in \citet{GG.10c}). Our results do not follow
from simple comparison arguments to e.g. infinite-server systems or loss
models, which would in all cases provide bounds in the opposite direction.
