  We study the heavy-traffic limit of the generalized switch operating under
MaxWeight, without assuming that the CRP condition is satisfied and allowing
for correlated arrivals. The main contribution of this paper is the
steady-state mean of linear combinations of queue lengths in heavy traffic. We
showcase the generality of our result by presenting various stochastic networks
as corollaries, each of which is a contribution by itself. In particular, we
study the input-queued switch with correlated arrivals and we show that if the
state space collapses to a full-dimensional subspace, the correlation among the
arrival processes does not matter in heavy traffic. We exemplify this last case
with a parallel-server system, an N-system, and an ad hoc wireless network.
  While the above results are obtained using the drift method, we additionally
present a negative result showing a limitation of the drift method. We show
that it is not possible to obtain the individual queue lengths using the drift
method with polynomial test functions. We do this by presenting an alternate
view of the drift method in terms of a system of linear equations, and we use
this system of equations to obtain bounds on arbitrary linear combinations of
the queue lengths.
