  This paper introduces and analyzes the notion of throughput suboptimality for
many-server queueing systems in heavy traffic. The queueing model under
consideration has multiple customer classes, indexed by a finite set
$\mathcal{I}$, and heterogenous, exponential servers. Servers are dynamically
chosen to serve customers, and buffers are available for customers waiting to
be served. The arrival rates and the number of servers are scaled up in such a
way that the processes representing the number of class-$i$ customers in the
system, $i\in\mathcal{I}$, fluctuate about a static fluid model, that is
assumed to be critically loaded in a standard sense. At the same time, the
fluid model is assumed to be throughput suboptimal. Roughly, this means that
the servers can be allocated so as to achieve a total processing rate that is
greater than the total arrival rate. We show that there exists a dynamic
control policy for the queueing model that is efficient in the following strong
sense: Under this policy, for every finite $T$, the measure of the set of times
prior to $T$, at which at least one customer is in the buffer, converges to
zero in probability as the arrival rates and number of servers go to infinity.
On the way to prove our main result, we provide a characterization of
throughput suboptimality in terms of properties of the buffer-station graph.
