  We study a multi-server model with $n$ flexible servers and $n$ queues,
connected through a bipartite graph, where the level of flexibility is captured
by the graph's average degree, $d_n$. Applications in content replication in
data centers, skill-based routing in call centers, and flexible supply chains
are among our main motivations.
  We focus on the scaling regime where the system size $n$ tends to infinity,
while the overall traffic intensity stays fixed. We show that a large capacity
region and an asymptotically vanishing queueing delay are simultaneously
achievable even under limited flexibility ($d_n \ll n$). Our main results
demonstrate that, when $d_n\gg \ln n$, a family of expander-graph-based
flexibility architectures has a capacity region that is within a constant
factor of the maximum possible, while simultaneously ensuring a diminishing
queueing delay for all arrival rate vectors in the capacity region. Our
analysis is centered around a new class of virtual-queue-based scheduling
policies that rely on dynamically constructed job-to-server assignments on the
connectivity graph. For comparison, we also analyze a natural family of modular
architectures, which is simpler but has provably weaker performance.
