Editing Openai/6931fd85-fcec-8011-8057-6c5f7152feee (section)

===== 7.4 Step 4: Lifting back to the original queue =====

Finally, standard interchange‑of‑limits and stability arguments (as in Harrison’s and Atar–Mandelbaum–Reiman’s work on multiclass queues) show that if a policy is asymptotically optimal for the diffusion model and the diffusion model is a valid heavy‑traffic approximation, then the corresponding sequence of prelimit policies is asymptotically optimal for the original queues.

Under the finite‑variance heavy‑tailed assumption and time‑varying but “slow” arrival rates (piecewise‑Lipschitz), the diffusion approximation is valid on any fixed time horizon, and the asymptotic compliance w.r.t. deadlines translates into achieving arbitrarily tight service‑level targets for percentiles such as the 99th.