A reflected fBm limit for fluid models with ON/OFF sources under heavy traffic

We consider a family of non-deterministic fluid models that can be approximated under heavy traffic conditions by a multidimensional reflected fractional Brownian motion (rfBm). Specifically, we prove a heavy traffic limit theorem for multi-station fluid models with feedback and non-deterministic arrival process generated by a large enough number of heavy tailed ON/OFF sources, say NN. Scaling in time by a factor rr and in state space conveniently, and letting NN and rr approach infinity (in this order) we prove that the scaled immediate workload process converges in some sense to a rfBm.