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

==== 5. Brownian control problem and Lagrangian structure ====

Let W(t)W(t)W(t) denote the total workload process in the diffusion limit, and Qj(t)Q_j(t)Qj(t) the (limit) queue length of class jjj. Under state‑space collapse, there are constants aj>0a_j>0aj>0 such that

W(t)≈∑jajQj(t).W(t) \approx \sum_j a_j Q_j(t).W(t)≈j∑ajQj(t).
We write a cost functional (for the diffusion limit) as

J(π)=E[∫0T∑jcjQjπ(t) dt],J(\pi) = \mathbb{E}\Bigg[ \int_0^T \sum_{j} c_j Q_j^\pi(t)\,dt \Bigg],J(π)=E[∫0Tj∑cjQjπ(t)dt],
where cjc_jcj are per‑class holding‑cost weights. With all cj=1c_j=1cj=1, this is proportional to the sum of mean waiting times by Little’s law; more generally, cjc_jcj can weight acuity classes differently. AIMS Press<ref>{{cite web|title=AIMS Press|url=https://www.aimspress.com/article/doi/10.3934/math.2025196?viewType=HTML|publisher=AIMS Press|access-date=2025-12-04}}</ref>

We want to minimise J(π)J(\pi)J(π) subject to the asymptotic class‑1 deadline constraint. A standard way to handle this is a Lagrangian relaxation: introduce a large penalty κ\kappaκ for violating the class‑1 deadline, leading to an unconstrained problem with modified instantaneous cost:

∑jcjQj(t)+κ⋅(τ~1(t)−d^1)+.\sum_j c_j Q_j(t) + \kappa \cdot \Bigl(\tilde \tau_1(t) - \hat d_1\Bigr)^+.j∑cjQj(t)+κ⋅(τ~1(t)−d^1)+.
In the limit κ→∞\kappa\to\inftyκ→∞, any optimal solution of this penalised problem must be asymptotically compliant with the deadline constraint. Plambeck–Kumar–Harrison show that the resulting BCP yields an index‑type rule; Huang–Carmeli–Mandelbaum adapt this for EDs with deadlines and IP feedback and prove asymptotic optimality of a concrete policy family. aviman.technion.ac.il<ref>{{cite web|title=aviman.technion.ac.il|url=https://link.springer.com/article/10.1023/A%3A1017983532376|publisher=link.springer.com|access-date=2025-12-04}}</ref>