  We consider a system of $N$ parallel single-server queues with unit
exponential service rates and a single dispatcher where tasks arrive as a
Poisson process of rate $\lambda(N)$. When a task arrives, the dispatcher
assigns it to a server with the shortest queue among $d(N)$ randomly selected
servers ($1 \leq d(N) \leq N$). This load balancing strategy is referred to as
a JSQ($d(N)$) scheme, marking that it subsumes the celebrated Join-the-Shortest
Queue (JSQ) policy as a crucial special case for $d(N) = N$.
  We construct a stochastic coupling to bound the difference in the queue
length processes between the JSQ policy and a scheme with an arbitrary value of
$d(N)$. We use the coupling to derive the fluid limit in the regime where
$\lambda(N) / N \to \lambda < 1$ as $N \to \infty$ with $d(N) \to\infty$, along
with the associated fixed point. The fluid limit turns out not to depend on the
exact growth rate of $d(N)$, and in particular coincides with that for the JSQ
policy. We further leverage the coupling to establish that the diffusion limit
in the critical regime where $(N - \lambda(N)) / \sqrt{N} \to \beta > 0$ as $N
\to \infty$ with $d(N)/(\sqrt{N} \log (N))\to\infty$ corresponds to that for
the JSQ policy. These results indicate that the optimality of the JSQ policy
can be preserved at the fluid-level and diffusion-level while reducing the
overhead by nearly a factor O($N$) and O($\sqrt{N}/\log(N)$), respectively.
