  Randomized load-balancing algorithms play an important role in improving
performance in large-scale networks at relatively low computational cost. A
common model of such a system is a network of $N$ parallel queues in which
incoming jobs with independent and identically distributed service times are
routed on arrival using the join-the-shortest-of-$d$-queues routing algorithm.
Under fairly general conditions, it was shown by Aghajani and Ramanan that as
$N\rightarrow\infty$, the state dynamics converges to the unique solution of a
countable system of coupled deterministic measure-valued equations called the
hydrodynamic equations. In this article, a characterization of invariant states
of these hydrodynamic equations is obtained and, when $d=2$, used to construct
a numerical algorithm to compute the queue length distribution and mean virtual
waiting time in the invariant state. Additionally, it is also shown that under
a suitable tail condition on the service distribution, the queue length
distribution of the invariant state exhibits a doubly exponential tail decay,
thus demonstrating a vast improvement in performance over the case $d=1$, which
corresponds to random routing, when the tail decay could even be polynomial.
Furthermore, numerical evidence is provided to support the conjecture that the
invariant state is the limit of the steady-state distributions of the
$N$-server models. The proof methodology, which entails analysis of a coupled
system of measure-valued equations, can potentially be applied to other
many-server systems with general service distributions, where measure-valued
representations are useful.
