  We consider a queueing system with $n$ parallel queues operating according to
the so-called "supermarket model" in which arriving customers join the shortest
of $d$ randomly selected queues. Assuming rate $n\lambda_{n}$ Poisson arrivals
and rate $1$ exponentially distributed service times, we consider this model in
the heavy traffic regime, described by $\lambda_{n}\uparrow 1$ as $n\to\infty$.
We give a simple expectation argument establishing that majority of queues have
steady state length at least $\log_d(1-\lambda_{n})^{-1} - O(1)$ with
probability approaching one as $n\rightarrow\infty$, implying the same for the
steady state delay of a typical customer.
  Our main result concerns the detailed behavior of queues with length smaller
than $\log_d(1-\lambda_{n})^{-1}-O(1)$. Assuming $\lambda_{n}$ converges to $1$
at rate at most $\sqrt{n}$, we show that the dynamics of such queues does not
follow a diffusion process, as is typical for queueing systems in heavy
traffic, but is described instead by a deterministic infinite system of linear
differential equations, after an appropriate rescaling. The unique fixed point
solution of this system is shown explicitly to be of the form
$\pi_{1}(d^{i}-1)/(d-1), i\ge 1$, which we conjecture describes the steady
state behavior of the queue lengths after the same rescaling. Our result is
obtained by combination of several technical ideas including establishing the
existence and uniqueness of an associated infinite dimensional system of
non-linear integral equations and adopting an appropriate stopped process as an
intermediate step.
