Validity of heavy-traffic steady-state approximations in multiclass queueing networks: The case of queue-ratio disciplines

A class of stochastic processes known as semi-martingale reflecting Brownian motions (SRBMs) is often used to approximate the dynamics of heavily loaded queueing networks. In two influential papers, Bramson (1998) and Williams (1998) laid out a general and structured approach for proving the validity of such heavy-traffic approximations, in which an SRBM is obtained as a diffusion limit from a sequence of suitably normalized workload processes. However, for multiclass networks it is still not known in general whether the steady-state distribution of the SRBM provides a valid approximation for the steady-state distribution of the original network. In this paper we study the case of queue-ratio disciplines and provide a set of sufficient conditions under which the above question can be answered in the affirmative. In addition to standard assumptions made in the literature towards the stability of the pre- and post-limit processes and the existence of diffusion limits, we add a requirement that solutions to the fluid model are attracted to the invariant manifold at linear rate. For the special case of static-priority networks such linear attraction is known to hold under certain conditions on the network primitives. The analysis elucidates some interesting connections between stability of the pre- and post-limit processes, their respective fluid models and state-space collapse, and identifies the respective roles played by all of the above in establishing validity of heavy-traffic steady-state approximations.