Diffusion Approximation for an Input-queued Switch Operating under a Maximum Weight Matching Policy

For N ≥ 2, we consider an N × N input-queued switch operating under a maximum weight matching policy. We establish a diffusion approximation for a (2N − 1)-dimensional workload process associated with this switch when all input ports and output ports are heavily loaded. The diffusion process is a semimartingale reflecting Brownian motion living in a polyhedral cone with N² boundary faces, each of which has an associated constant direction of reflection. Our proof builds on our own prior work [13] on an invariance principle for semimartingale reflecting Brownian motions in piecewise smooth domains and on a multiplicative state space collapse result for switched networks established by Shah and Wischik in [19].