  This paper studies the heavy-traffic joint distribution of queue lengths in
two stochastic processing networks (SPN), viz., an input-queued switch
operating under the MaxWeight scheduling policy and a two-server parallel
server system called the $\mathcal{N}$-system. These two systems serve as
representatives of SPNs that do not satisfy the so-called Complete Resource
Pooling (CRP) condition, and consequently exhibit a multidimensional State
Space Collapse (SSC). Except in special cases, only mean queue lengths of such
non-CRP systems is known in the literature. In this paper, we develop the
Transform method to study the joint distribution of queue lengths in non-CRP
systems. The key challenge is in solving an implicit functional equation
involving the Laplace transform of the heavy-traffic limiting distribution. For
the $\mathcal{N}$-system and a special case of an input-queued switch involving
only three queues, we obtain the exact limiting heavy-traffic joint
distribution in terms of a linear combination of two iid exponentials. For the
general $n\times n$ input-queued switch that has $n^2$ queues, under a
conjecture on uniqueness of the solution of the functional equation, we obtain
an exact joint distribution of the heavy-traffic limiting queue-lengths in
terms of a non-linear combination of $2n$ iid exponentials.
