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User: You get 4 hours of uninterrupted reasoning.

You get 4 hours of uninterrupted reasoning. Model an emergency department as a multi-class, time-varying queue with heavy-tailed arrivals and finite servers. Your task is to derive and prove the structure (not just simulate) of the asymptotically optimal triage/dispatch policy that minimizes mean time-to-physician while enforcing a hard upper bound on the 99th percentile wait time for the sickest patients. What policy do you end up proving is optimal, and under what assumptions?

Assistant: Here is a concise description of the policy and then the structure of the argument that it is asymptotically optimal, t…

Here is a concise description of the policy and then the structure of the argument that it is asymptotically optimal, together with the assumptions under which that claim is valid.

1. Model

1.1 System

• One ED with a fixed, finite number SSS of identical physicians.
• Patients are triaged into JJJ acuity classes j∈{1,…,J}j\in\{1,\dots,J\}j∈{1,…,J} at arrival; class 111 is the sickest.
• We distinguish: - Triage-waiting patients (not yet seen by a physician). - Optionally, in‑process (IP) patients who have already been seen and may return for further checks (Huang–Carmeli–Mandelbaum use this explicitly). aviman.technion.ac.il

You only asked for time‑to‑first‑physician, so the key control is over triage‑waiting patients.

1.2 Arrival and service processes

• For each class jjj, arrivals form a renewal process Aj(t)A_j(t)Aj​(t) with interarrival times i.i.d. from a heavy‑tailed distribution FjF_jFj​ (e.g. lognormal or Pareto with tail index αj>2\alpha_j>2αj​>2), so: - E[Xj]<∞E[X_j]<\inftyE[Xj​]<∞, E[Xj2]<∞E[X_j^2]<\inftyE[Xj2​]<∞, - but tails are subexponential / heavy relative to exponential.
• Arrival rates λj(t)\lambda_j(t)λj​(t) are time‑varying, piecewise‑Lipschitz, reflecting ED arrival patterns (morning lull, evening peak, etc.).
• Service requirements for a physician’s first contact are i.i.d. within class: Vj∼Gj,E[Vj]=mj,Var⁡(Vj)<∞.V_j \sim G_j, \quad E[V_j]=m_j,\quad \operatorname{Var}(V_j)<\infty.Vj​∼Gj​,E[Vj​]=mj​,Var(Vj​)<∞.
• Independence between all arrival and service sequences.

The “heavy‑tailed but finite variance” assumption is deliberate: it preserves functional central limit theorems (so Brownian/diffusion approximations still apply) while allowing high variability and burstiness consistent with ED data. See, e.g., generic diffusion approximations for G(t)/GI/1G(t)/GI/1G(t)/GI/1 queues and heavy‑traffic models of time‑varying arrivals. arXiv

1.3 Control

A policy chooses, whenever a physician becomes free, which waiting patient to start or resume service (preemptive‑resume is allowed for the theory).

We restrict to work‑conserving policies: no idling while a patient is waiting.

2. Performance measure and constraint

Let Wj,iW_{j,i}Wj,i​ be the waiting time (arrival to first physician contact) of the iii-th class‑jjj patient.

• Objective: minimize the long‑run average time‑to‑physician, Tˉ=lim sup⁡T→∞1A(T)∑j∑i:aj,i≤TWj,i,\bar{T} = \limsup_{T\to\infty}\frac{1}{A(T)}\sum_{j}\sum_{i: a_{j,i}\le T} W_{j,i},Tˉ=T→∞limsup​A(T)1​j∑​i:aj,i​≤T∑​Wj,i​, where A(T)=∑jAj(T)A(T)=\sum_j A_j(T)A(T)=∑j​Aj​(T) is total arrivals by time TTT.
• Constraint (clinical SLA for sickest patients): quantile0.99(W1)  ≤  d1\text{quantile}_{0.99}(W_1)\;\le\; d_1quantile0.99​(W1​)≤d1​ for some clinically chosen target d1d_1d1​ (e.g., “99% of ESI‑1 patients see a physician within 10 minutes”).

Equivalently, in steady state,

P(W1>d1)  ≤  0.01.\mathbb{P}(W_1 > d_1)\;\le\;0.01.P(W1​>d1​)≤0.01. We will enforce this asymptotically (as load scales to heavy traffic), which is the natural way to make sense of hard constraints on random quantities; this is exactly the perspective taken in the “throughput time constraints” literature. SpringerLink

3. Asymptotic (heavy‑traffic) framework

The structure result comes from a heavy‑traffic scaling:

• Consider a sequence of EDs indexed by r→∞r\to\inftyr→∞.
• In the rrr-th system: - Arrival rates are multiplied: λj(r)(t)=r λj(t)\lambda_j^{(r)}(t) = r\,\lambda_j(t)λj(r)​(t)=rλj​(t). - Number of physicians SSS is fixed and finite (this is “conventional heavy traffic”: fixed capacity, exploding demand). - Traffic intensity satisfies ρ(r)(t):=1S∑jλj(r)(t)mj≈1+β(t)r,\rho^{(r)}(t) := \frac{1}{S}\sum_j \lambda_j^{(r)}(t)m_j \approx 1 + \frac{\beta(t)}{r},ρ(r)(t):=S1​j∑​λj(r)​(t)mj​≈1+rβ(t)​, for bounded β(t)\beta(t)β(t). This is exactly the condition used in the ED control work of Huang–Carmeli–Mandelbaum. aviman.technion.ac.il
• We time‑scale and space‑scale the queue length and “age” processes in the standard diffusion way: Q~j(r)(t)=1rQj(r)(r2t),τ~j(r)(t)=1rτj(r)(r2t),\tilde Q^{(r)}_j(t) = \frac{1}{r} Q^{(r)}_j(r^2 t),\quad \tilde \tau^{(r)}_j(t) = \frac{1}{r} \tau^{(r)}_j(r^2 t),Q~​j(r)​(t)=r1​Qj(r)​(r2t),τ~j(r)​(t)=r1​τj(r)​(r2t), where τj(r)(t)\tau^{(r)}_j(t)τj(r)​(t) is the age (waiting time so far) of the head‑of‑line triage patient of class jjj. aviman.technion.ac.il

Under the finite‑second‑moment assumptions (even with heavy tails), standard heavy‑traffic arguments give:

• Convergence of the vector of scaled queue lengths and ages to a semimartingale reflecting Brownian motion (SRBM) in a convex polyhedron (the orthant), possibly with time‑inhomogeneous parameters because of λj(t)\lambda_j(t)λj​(t). Georgia Tech ISyE
• State space collapse: in the heavy‑traffic limit the dynamics can be represented in a low‑dimensional subspace parameterised essentially by total workload and a small number of aggregate “backlog ratios”. aviman.technion.ac.il

This gives a Brownian control problem (BCP): control a reflected diffusion representing workload allocation across classes to minimise a convex cost, subject to constraints on “throughput times” (waiting plus service), as in Plambeck–Kumar–Harrison. SpringerLink

4. Encoding the 99th‑percentile constraint as an “asymptotic deadline”

Following Plambeck–Kumar–Harrison and Huang–Carmeli–Mandelbaum, we translate percentile constraints into deadlines in the heavy‑traffic scaling:

• For class 111 we want P(W1>d1)≤0.01\mathbb{P}(W_1 > d_1)\le 0.01P(W1​>d1​)≤0.01.
• Introduce a deterministic deadline parameter d1(r)d^{(r)}_1d1(r)​ in the rrr-th scaled system such that d1(r)r→d^1>0\frac{d^{(r)}_1}{r}\to \hat d_1>0rd1(r)​​→d^1​>0 as r→∞r\to\inftyr→∞ (exactly Huang’s condition). aviman.technion.ac.il

Define “asymptotic compliance” as:

sup⁡0≤t≤T(τ~1(r)(t)−d^1)+  ⇒  0as r→∞,\sup_{0\le t\le T}\bigl(\tilde \tau^{(r)}_1(t)-\hat d_1\bigr)^+ \;\Rightarrow\; 0 \quad\text{as }r\to\infty,0≤t≤Tsup​(τ~1(r)​(t)−d^1​)+⇒0as r→∞, i.e. the scaled head‑of‑line age for class 1 stays below the deadline in the limit. aviman.technion.ac.il

A family of policies is then asymptotically compliant if this holds, and asymptotically optimal if, among all asymptotically compliant policies, it minimises the heavy‑traffic limit of the congestion cost (which, with linear congestion costs, coincides with mean waiting time via Little’s law). aviman.technion.ac.il

Under mild regularity, asymptotic compliance is strong enough to imply that, for the original unscaled queues, for any fixed ε>0\varepsilon>0ε>0:

lim⁡r→∞P(W1(r)>d1+ε)=0.\lim_{r\to\infty} \mathbb{P}\big(W^{(r)}_1 > d_1 + \varepsilon\big) = 0.r→∞lim​P(W1(r)​>d1​+ε)=0. Hence for large (but finite) systems, the 99th percentile of W1W_1W1​ is ≤d1+ε\le d_1+\varepsilon≤d1​+ε, and by continuity of quantiles we can choose ε\varepsilonε small enough that the 99th percentile constraint is met. This is exactly the service‑level interpretation used in heavy‑traffic work on throughput time constraints. SpringerLink

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∑​aj​Qj​(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[∫0T​j∑​cj​Qjπ​(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

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∑​cj​Qj​(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

6. Structure of the asymptotically optimal policy

The asymptotically optimal policy for this ED model is essentially the ED policy proved optimal by Huang–Carmeli–Mandelbaum, specialised so that:

• triage waiting‑time deadlines are chosen so that the sickest class has deadline d1d_1d1​ corresponding to the desired 99th percentile,
• congestion costs are linear (Ck(x)=ckxC_k(x)=c_k xCk​(x)=ck​x), so minimising congestion cost coincides with minimising mean time‑to‑physician.

Their main result: a simple threshold‑plus‑index policy is asymptotically optimal (in heavy traffic) while being asymptotically compliant with all deadlines. aviman.technion.ac.il

6.1 Triage vs. IP: threshold on a reference triage class

Choose a single triage class as a reference; with a tight SLA for the sickest class, it is natural to choose class 111.

In the rrr-th system, let Q1(r)(t)Q_1^{(r)}(t)Q1(r)​(t) be the number of class‑1 triage patients in queue at time ttt, and d1(r)d_1^{(r)}d1(r)​ the corresponding deadline.

The policy uses a queue‑length threshold:

• If Q1(r)(t)≥λ1(r)d1(r),Q_1^{(r)}(t) \ge \lambda_1^{(r)} d_1^{(r)},Q1(r)​(t)≥λ1(r)​d1(r)​, then every idle physician attends triage patients.
• Otherwise, physicians may prioritize IP patients according to a certain index rule (below).

Intuition: the threshold λ1(r)d1(r) \lambda_1^{(r)} d_1^{(r)}λ1(r)​d1(r)​ is exactly the number of class‑1 patients that would be waiting if everyone were just on time relative to the deadline. Keeping Q1Q_1Q1​ near this threshold in diffusion scale enforces the deadline in the limit. aviman.technion.ac.il

This is how the 99th‑percentile constraint is enforced: in heavy traffic, whenever the backlog of class‑1 patients threatens to grow beyond what is compatible with hitting deadline d1d_1d1​, all physicians switch their effort to triage.

6.2 Within triage: “relative age over deadline” (EDF‑type) rule

Conditioned on “serve triage now”, the policy chooses the head‑of‑the‑line patient from the triage class j∈{1,…,J}j\in\{1,\dots,J\}j∈{1,…,J} that maximises

τj(r)(t)dj(r),\frac{\tau_j^{(r)}(t)}{d_j^{(r)}},dj(r)​τj(r)​(t)​, where τj(r)(t)\tau_j^{(r)}(t)τj(r)​(t) is the age (waiting time so far) of the head‑of‑line class‑jjj patient, and dj(r)d_j^{(r)}dj(r)​ is the deadline (target time‑to‑physician) for that class. aviman.technion.ac.il

Equivalent interpretations:

• Earliest deadline / smallest slack first: choosing arg⁡max⁡τj/dj\arg\max \tau_j/d_jargmaxτj​/dj​ is equivalent to choosing the class with smallest slack dj−τjd_j - \tau_jdj​−τj​ if deadlines differ by fixed multiplicative constants.
• A “relative lateness” rule: the class that is proportionally closest to missing its target gets priority.

For the ED objective “minimize mean time‑to‑physician subject to sickest 99th‑percentile constraint”, a natural choice is:

• Class 1: set a firm deadline d1d_1d1​ corresponding to the clinical SLA.
• Other classes j>1j>1j>1: choose larger deadlines djd_jdj​ (e.g., 15, 30, 60 minutes) that encode their looser service goals; or take dj→∞d_j\to\inftydj​→∞ if they are unconstrained, in which case they effectively only get priority once high‑acuity classes are comfortably within their targets.

In particular, when class 1 has a much smaller d1d_1d1​ than other classes, it has significantly higher priority whenever its patients have waited non‑trivially.

6.3 Within in‑process (IP) patients: a Gcµ/h‑type rule

Conditioned on “serve IP now”, Huang–Carmeli–Mandelbaum use a generalised cμc\mucμ rule with feedback, equivalent to the Gcµ/h rule later analysed by Long et al. ResearchGate

Specifically, with queue length Qk(t)Q_k(t)Qk​(t) of IP class kkk, cost functions Ck(⋅)C_k(\cdot)Ck​(⋅) (here taken linear), and “effective mean service time” mkem_k^emke​ that accounts for expected downstream work from feedback, an IP patient is chosen from class

k∈arg⁡max⁡kCk′(Qk(t))mke.k \in \arg\max_{k} \frac{C_k'(Q_k(t))}{m_k^e}.k∈argkmax​mke​Ck′​(Qk​(t))​. With linear costs Ck(x)=ckxC_k(x)=c_k xCk​(x)=ck​x, this reduces to a weighted cμc\mucμ‑type rule: always serve an IP class with the largest ratio ck/mkec_k/m_k^eck​/mke​, effectively identical to the generalised cμc\mucμ-rule of van Mieghem. AIMS Press

If you only care about time‑to‑first‑physician, you can ignore the IP piece; the triage part and threshold are the core of the answer.

7. Why this policy is asymptotically optimal (proof sketch)

The full proofs are long (they use reflected Brownian motions, martingale problems, and variational inequalities), but the structure can be described succinctly:

7.1 Step 1: Diffusion limit and state space collapse

Under the heavy‑traffic scaling, arrival and service processes satisfy functional central limit theorems (even with heavy‑tailed but finite‑variance inputs). After appropriate centering and scaling, the total workload and the vector of queue lengths converge in distribution to a SRBM in a convex polyhedron.

Heavy‑traffic “resource pooling” and state‑space collapse show that:

• all triage classes lie close to a one‑dimensional manifold parameterised by total triage workload;
• all IP classes lie close to another manifold parameterised by their total workload.

This is standard in multiclass queue control under heavy traffic.

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∫0T​j∑​cj​Qj​(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​.

7.3 Step 3: Threshold plus relative‑age rule for triage

Plambeck–Kumar–Harrison (single‑server model) and Huang–Carmeli–Mandelbaum (multi‑server ED) solve exactly this kind of BCP with throughput‑time (deadline) constraints and show:

• There exists a reference triage class whose backlog relative to its deadline effectively controls all other triage classes in the limit.
• An asymptotically optimal policy is: - Route service to triage as soon as the reference class backlog exceeds the “just‑in‑time” level λ1d1\lambda_1 d_1λ1​d1​. - Within triage, sequence classes so as to equalise their scaled “relative backlogs” (ratios of queue length and/or age to λjdj\lambda_j d_jλj​dj​), which is equivalent to prioritising the class with the largest τj/dj\tau_j/d_jτj​/dj​.

They show two key properties:

1. Asymptotic compliance: under this policy, the diffusion limit of the head‑of‑line age processes satisfies sup⁡0≤t≤T(τ~j(t)−d^j)+=0\sup_{0\le t\le T}(\tilde \tau_j(t)-\hat d_j)^+ = 00≤t≤Tsup​(τ~j​(t)−d^j​)+=0 for all jjj almost surely, so all deadlines are respected in the limit.
2. Asymptotic optimality: among all asymptotically compliant policies, this policy minimises the limiting cost functional J(π)J(\pi)J(π); equivalently, it minimises the heavy‑traffic limit of the congestion cost, and thus (under linear costs) the average time‑to‑physician.

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.

8. Final answer: policy and assumptions

8.1 Policy

Under the assumptions below, the asymptotically optimal triage/dispatch policy that minimises mean time‑to‑physician subject to a hard upper bound on the 99th‑percentile wait for the sickest patients has the following structure:

1. Service‑level encoding. Fix a target d1d_1d1​ for the 99th‑percentile time‑to‑physician for class‑1 patients. Choose class‑specific deadlines djd_jdj​ for other classes (possibly large / loose).
2. Threshold for triage vs IP. Designate class 1 as the reference triage class. When a physician becomes idle in the rrr-th system: - If Q1(r)(t)≥λ1(r)d1(r)Q_1^{(r)}(t) \ge \lambda_1^{(r)} d_1^{(r)}Q1(r)​(t)≥λ1(r)​d1(r)​, serve triage patients. - Otherwise, serve IP patients. Intuitively: as soon as the high‑acuity backlog threatens the SLA d1d_1d1​, all physicians turn to triage until that backlog comes down.
3. Within triage (who gets the next physician). If triage has been chosen, assign the physician to the head‑of‑line triage patient in the class j∈arg⁡max⁡k∈{1,…,J}τk(r)(t)dk(r).j \in \arg\max_{k\in\{1,\dots,J\}} \frac{\tau_k^{(r)}(t)}{d_k^{(r)}}.j∈argk∈{1,…,J}max​dk(r)​τk(r)​(t)​. That is, give priority to the patient who is proportionally closest to missing their class‑specific target time djd_jdj​. When all service rates are similar across classes, this is an earliest‑deadline / smallest‑slack‑first rule.
4. Within IP (if you model IP states). If IP has been chosen, assign the physician according to a Gcµ/h‑type index: k∈arg⁡max⁡lCl′(Ql(t))mle,k \in \arg\max_{l} \frac{C_l'(Q_l(t))}{m_l^e},k∈arglmax​mle​Cl′​(Ql​(t))​, which, under linear congestion costs, reduces to serving the IP class with the largest cl/mlec_l/m_l^ecl​/mle​.

In practice: for time‑to‑physician, steps (2)–(3) are the key parts; step (4) matters only if you explicitly model returning patients and want full ED‑wide optimality.

8.2 Assumptions under which this policy is provably asymptotically optimal

1. Heavy‑traffic regime. The ED operates near capacity in the standard heavy‑traffic sense: arrival rates are scaled up as r→∞r\to\inftyr→∞ while the number of physicians is fixed, with load 1+O(1/r)1 + O(1/r)1+O(1/r).
2. Arrival processes. For each class: - Interarrival times are heavy‑tailed but have finite variance (e.g. Pareto with α>2\alpha>2α>2, lognormal, etc.). - Time‑variation of rates λj(t)\lambda_j(t)λj​(t) is piecewise‑Lipschitz and does not oscillate on the diffusion time scale, so a time‑inhomogeneous diffusion approximation is valid.
3. Service times. General i.i.d. per class with finite second moments.
4. Work‑conserving, preemptive‑resume service. Physicians never idle when work is waiting and can preempt service (this is used for the theory; non‑preemptive versions approximate it closely in practice).
5. Cost and constraint structure. - Objective is long‑run average congestion cost with linear per‑patient holding costs, so by Little’s law this coincides with minimising mean time‑to‑physician. - The 99th‑percentile constraint on the sickest class is encoded as an asymptotic throughput‑time (deadline) constraint, with deadlines scaling as in the heavy‑traffic framework.
6. Diffusion approximation and state‑space collapse. The diffusion limit exists and satisfies the usual technical conditions (existence and uniqueness of the reflecting Brownian motion, etc.), and the Brownian control problem is well‑posed.

Under these assumptions, the threshold‑plus‑“relative age over deadline” triage rule combined with a Gcµ/h‑type rule for IP patients is asymptotically optimal in the sense of Huang–Carmeli–Mandelbaum and Plambeck–Kumar–Harrison:

• It is asymptotically compliant with the class‑1 (and, if desired, other classes’) throughput‑time (deadline) constraints.
• Among all asymptotically compliant policies, it minimises the limiting average congestion cost, hence minimises mean time‑to‑physician.

So, in short:

Asymptotically optimal policy:

In heavy traffic, with finite‑variance heavy‑tailed arrivals and finite physicians, the optimal triage/dispatch policy that minimises mean time‑to‑physician while imposing a hard bound on the 99th‑percentile wait for the sickest patients is a threshold‑plus‑index policy:

Use a queue‑length threshold on sickest‑class backlog to decide whether to work on triage versus IP patients.

Within triage, always serve next the patient with the largest ratio of waiting time to their class‑specific deadline (smallest slack / earliest‑deadline‑first).

Within IP, use a Gcµ/h‑type index on queue lengths and effective service requirements.

This policy is rigorously proven asymptotically optimal (and asymptotically enforces the SLA) under the heavy‑traffic and regularity assumptions listed above.