  In this paper, we study a large system of $N$ servers each with capacity to
process at most $C$ simultaneous jobs and an incoming job is routed to a server
if it has the lowest occupancy amongst $d$ (out of N) randomly selected
servers. A job that is routed to a server with no vacancy is assumed to be
blocked and lost. Such randomized policies are referred to JSQ(d) (Join the
Shortest Queue out of $d$) policies. Under the assumption that jobs arrive
according to a Poisson process with rate $N\lambda^{(N)}$ where
$\lambda^{(N)}=\sigma-\frac{\beta}{\sqrt{N}}$, $\sigma\in\mb{R}_+$ and
$\beta\in\mb{R}$, we establish functional central limit theorems (FCLTs) for
the fluctuation process in both the transient and stationary regimes when
service time distributions are exponential. In particular, we show that the
limit is an Ornstein-Uhlenbeck process whose mean and variance depend on the
mean-field of the considered model. Using this, we obtain approximations to the
blocking probabilities for large $N$, where we can precisely estimate the
accuracy of first-order approximations.
