  This paper studies a class of load balancing algorithms for many-server ($N$
servers) systems assuming finite buffer with size $b-1$ (i.e. a server can have
at most one job in service and $b-1$ jobs in queue). We focus on steady-state
performance of load balancing algorithms in the heavy traffic regime such that
the load of system is $\lambda = 1 - N^{-\alpha}$ for $0<\alpha<0.5,$ which we
call sub-Halfin-Whitt regime ($\alpha=0.5$ is the so-called the Halfin-Whitt
regime). We establish a sufficient condition under which the probability that
an incoming job is routed to an idle server is one asymptotically. The class of
load balancing algorithms that satisfy the condition includes
join-the-shortest-queue (JSQ), idle-one-first (I1F), join-the-idle-queue (JIQ),
and power-of-$d$-choices (Po$d$) with $d=N^\alpha\log N$. The proof of the main
result is based on the framework of Stein's method. A key contribution is to
use a simple generator approximation based on state space collapse.
