Multi-dimensional state space collapse in non-complete resource pooling scenarios

The present paper establishes an explicit multi-dimensional state space collapse for parallel-processing systems with arbitrary compatibility constraints between servers and job types. This breaks major new ground beyond the state space collapse 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 state space collapse 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. Specifically, we prove that the limiting queue length vector lives in a K-dimensional cone which explicitly captures the delicate interplay between the various job types and servers. 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.