The heavy-traffic diffusion limit of symmetric queues

Consider an M/G/1 queue operating under a symmetric service discipline in the sense of Kelly: at queue length $n$ the job in position $i$ receives the fraction $\gamma(n,i)$ of the server, and an arriving job joins position $i$ with probability $\gamma(n+1,i)$. Zwart conjectured that, for every such discipline and every service law with finite second moment, the queue length started empty and scaled diffusively in heavy traffic converges to the same reflected Brownian motion, the workload limit divided by $h=\mathbb E B^2/(2\mathbb E B)$. This was known for processor sharing and for last-come-first-served. We prove the conjecture for every symmetric discipline and every service law with $\mathbb E B^2<\infty$. The proof is a state-space collapse $\sup_{s\le r^2T}|W_r(s)-hQ_r(s)|=o_{\mathbb P}(r)$. It uses only the product-form stationary law of the queue with the original service requirements kept as marks, the regenerative structure at empty times, and a Bernstein bound for jobs grouped into dyadic classes of original requirement, which can be summed over the classes because the number of busy cycles up to time $r^2T$ is of order $r$.