Diffusion approximation for an input-queued switch operating under a maximum weight matching policy

Abstract

For $N\geq 2$, we consider an $N\times N$ 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 $N^{2}$ 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].