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

===== 7.2 Step 2: Brownian control problem =====

The limiting control problem becomes:
* Control the allocation of instantaneous service rate among triage classes jjj and IP classes kkk as a function of the current state (Qj,τj,Qk)(Q_j,\tau_j,Q_k)(Qj,τj,Qk).
* Objective: minimise E∫0T∑jcjQj(t)+∑kCk(Qk(t)) dt\mathbb{E}\int_0^T \sum_j c_j Q_j(t) + \sum_k C_k(Q_k(t)) \, dtE∫0Tj∑cjQj(t)+k∑Ck(Qk(t))dt subject to the reflected Brownian dynamics and the hard constraint that the class‑1 age process τ1(t)\tau_1(t)τ1(t) stays below d^1\hat d_1d^1 (asymptotically).

Using Lagrange multipliers for the constraint and standard arguments for convex stochastic control, the problem decomposes into:
* A workload allocation problem across triage vs IP.
* A triage sequencing problem across classes jjj.
* An IP sequencing problem across IP classes kkk.

The KKT conditions of the associated static optimisation (the “inner loop” of the diffusion control) yield index structures: at any state, the marginal value of putting one more unit of service capacity into class jjj is a monotone function of τj/dj\tau_j/d_jτj/dj; similarly, for IP classes, the marginal value is proportional to Ck′(Qk)/mkeC_k'(Q_k)/m_k^eCk′(Qk)/mke.