  We address the problem of giving robust performance bounds based on the study
of the asymptotic behavior of the insensitive load balancing schemes when the
number of servers and the load scales jointly. These schemes have the desirable
property that the stationary distribution of the resulting stochastic network
depends on the distribution of job sizes only through its mean. It was shown
that they give good estimates of performance indicators for systems with finite
buffers, generalizing henceforth Erlang's formula whereas optimal policies are
already theoretically and computationally out of reach for networks of moderate
size. We study a single class of traffic acting on a symmetric set of processor
sharing queues with finite buffers and we consider the case where the load
scales with the number of servers. We characterize central limit theorems and
large deviations, the response of symmetric systems under those schemes at
different scales and show that three amplitudes of deviations can be
identified. A central limit scaling takes place for a sub-critical load; for
$\rho=1$, the number of free servers scales like $n^{ {\theta \over \theta+1}}$
($\theta$ being the buffer depth and $n$ being the number of servers) and is of
order 1 for super-critical loads. This further implies the existence of
different phases for the blocking probability, Before a (refined) critical load
$\rho_c(n)=1-a n^{- {\theta \over \theta+1}}$, the blocking is exponentially
small and becomes of order $ n^{- {\theta \over \theta+1}}$ at $\rho_c(n)$.
This generalizes the well-known Quality and Efficiency Driven (QED) regime or
Halfin-Whitt regime for a one-dimensional queue, and leads to a generalized
staffing rule for a given target blocking probability.
