  Throughput sub-optimality (TSO), introduced in Atar and Shaikhet [Ann. Appl.
Probab. 19 (2009) 521-555] for static fluid models of parallel queueing
networks, corresponds to the existence of a resource allocation, under which
the total service rate becomes greater than the total arrival rate. As shown in
Atar, Mandelbaum and Shaikhet [Ann. Appl. Probab. 16 (2006) 1764-1804] and Atar
and Shaikhet (2009), in the many server Halfin-Whitt regime, TSO implies null
controllability (NC), the existence of a routing policy under which, 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 at the scaling
limit. The present paper investigates the question whether the converse
relation is also true and TSO is both sufficient and necessary for the NC
behavior. In what follows we do get the affirmation for systems with either two
customer classes (and possibly more service pools) or vice-versa and state a
condition on the underlying static fluid model that allows the extension of the
result to general structures.
