  The present paper establishes an explicit multi-dimensional state space
collapse (SSC) for parallel-processing systems with arbitrary compatibility
constraints between servers and job types. This breaks major new ground beyond
the SSC results and queue length asymptotics in the literature which are
largely restricted to complete resource pooling (CRP) scenarios where the
steady-state queue length vector concentrates around a line in heavy traffic.
The multi-dimensional SSC that we establish reveals heavy-traffic behavior
which is also far more tractable than the pre-limit queue length distribution,
yet exhibits a fundamentally more intricate structure than in the
one-dimensional case, providing useful insight into the system dynamics. In
particular, we prove that the limiting queue length vector lives in a
$K$-dimensional cone of which the set of spanning vectors is random in general,
capturing the delicate interplay between the various job types and servers. For
a broad class of systems we provide a further simplification which shows that
the collection of random cones constitutes a fixed $K$-dimensional cone,
resulting in a $K$-dimensional SSC. The dimension $K$ represents the number of
critically loaded subsystems, or equivalently, capacity bottlenecks in
heavy-traffic, with $K=1$ corresponding to conventional CRP scenarios. Our
approach leverages probability generating function (PGF) expressions for
Markovian systems operating under redundancy policies.
