  We re-visit the global - relative to control policies - stability of
multiclass queueing networks. In these, as is known, it is generally
insufficient that the nominal utilization at each server is below 100%. Certain
policies, although work conserving, may destabilize a network that satisfies
the nominal-load conditions; additional conditions on the primitives are needed
for global stability (stability under any work-conserving policy). The
global-stability region was fully characterized for two-station networks in
[13], but a general framework for networks with more than two stations remains
elusive. In this paper, we offer progress on this front by considering a subset
of non-idling control policies, namely queue-ratio (QR) policies. These include
as special cases all static-priority policies. With this restriction, we are
able to introduce a complete framework that applies to networks of any size.
Our framework breaks the analysis of robust QR stability (stability under any
QR policy) into (i) robust state-space collapse and (ii) robust stability of
the Skorohod problem (SP) representing the fluid workload. Sufficient
conditions for both are specified in terms of simple optimization problems. We
use these optimization problems to prove that the family of QR policies
satisfies a weak form of convexity relative to policies. A direct implication
of this convexity is that: if the SP is stable for all static-priority policies
(the "extreme" QR policies), then it is also stable under any QR policy. While
robust QR stability is weaker than global stability, our framework recovers
necessary and sufficient conditions for global stability in specific networks.
