  We consider a load balancing system comprised of a fixed number of single
server queues, operating under the well-known Join-the-Shortest Queue policy,
and where jobs/customers are impatient and abandon if they do not receive
service after some (random) amount of time. In this setting, we characterize
the centered and appropriately scaled steady state queue length distribution
(hereafter referred to as limiting distribution), in the limit as the
abandonment rate goes to zero at the same time as the load either converges to
one or is larger than one.
  Depending on the arrival, service, and abandonment rates, we observe three
different regimes of operation that yield three different limiting
distributions. The first regime is when the system is underloaded and its load
converges relatively slowly to one. In this case, abandonments do not affect
the limiting distribution, and we obtain the same exponential distribution as
in the system without abandonments. When the load converges to one faster, we
have the second regime, where abandonments become significant. Here, the system
undergoes a phase transition, and the limiting distribution is a truncated
Gaussian. Further, the third regime is when the system is heavily overloaded,
and so the queue lengths are very large. In this case, we show the limiting
distribution converges to a normal distribution.
  To establish our results, we first prove a weaker form of State Space
Collapse by providing a uniform bound on the second moment of the (unscaled)
perpendicular component of the queue lengths, which shows that the system
behaves like a single server queue. We then use exponential Lyapunov functions
to characterize the limiting distribution of the steady state queue length
vector.
