title: "Applying Heavy-Traffic Fluid Control to Robust Appointment Scheduling Under Distributional Uncertainty" slug: hypothesis-h_8c38bfe73ab1 tags: ["cross-field", "healthcare-operations-appointment-scheduling", "simulation-optimization-under-input-uncertainty", "hypothesis-test"] hypothesis_id: h_8c38bfe73ab1 source_paper_id: arxiv:2412.18215 target_paper_id: arxiv:1903.05828 sim_ok: true ai_disclosure: "Drafted with AI assistance. Hypothesis synthesized cross-field, experiment auto-designed and run locally. See limitations section."

Applying Heavy-Traffic Fluid Control to Robust Appointment Scheduling Under Distributional Uncertainty

TL;DR

We tested whether the fluid control problem (FCP) framework from healthcare operations scheduling could be transferred to robust selection under input uncertainty. Our local discrete-event simulation found zero measurable difference between a greedy baseline and a fluid-control-derived robust policy, suggesting either that the clinic regime we simulated was too loose to exhibit heavy-traffic dynamics, or that the policy derivation did not correctly instantiate the FCP mechanism.

The cross-field move

Source Framework: Fluid Control in Healthcare Operations

Source (Paper A, arxiv:2412.18215): Healthcare operations literature has developed fluid control approximations for appointment scheduling, where a deterministic relaxation of the stochastic scheduling problem yields asymptotically optimal policies in the heavy-traffic limit (high utilization, large patient volume). The fluid control problem framework provides a computationally tractable way to derive scheduling policies by solving a continuous, deterministic optimization problem that approximates the underlying stochastic system. This approach has proven effective in settings where system load approaches capacity, enabling efficient policy derivation without explicit enumeration of all possible system states.

Target Framework: Distributionally Robust Simulation Optimization

Target (Paper B, arxiv:1903.05828): Simulation optimization under input uncertainty addresses the problem of selecting a design when the input distribution is ambiguous—known only to lie in a confidence set. Current methods either assume a known distribution or use worst-case (minimax) bounds that can be overly conservative. The challenge is to find policies that perform well not just under a nominal distribution, but across a family of plausible distributions, without incurring the computational burden of solving a full minimax optimization problem. This is particularly relevant in healthcare, where patient arrival patterns, no-show rates, and service times may vary across clinics, seasons, or patient populations.

Proposed Transfer Mechanism

We hypothesized that the FCP's deterministic relaxation could serve as a computationally tractable upper bound for the distributionally robust selection problem ^{[arxiv:2412.18215]}. By treating the heavy-traffic limit as a proxy for the ambiguity set, we could derive selection policies that are robust to distribution shifts without solving the full minimax problem—potentially enabling efficient policy derivation even when the ambiguity set is large. The intuition is that a policy optimized for a high-congestion deterministic system should be inherently more conservative and thus more robust to distributional perturbations than a policy tuned to a nominal stochastic scenario.

What we ran

Simulation Configuration

Parameter	Value
Simulation type	Discrete-event (appointment-flow topology)
Baseline policy	Greedy scheduling (first-come, first-served with no-show anticipation)
Variant policy	Fluid-control-robust policy (derived from FCP deterministic relaxation)
Clinic capacity	~12 appointment slots per simulation day
No-show rate (nominal)	~10%
Walk-in arrivals	Poisson, ~48 per simulation run
Simulation horizon	1 full clinic day
Key metrics	Mean wait time, 95th percentile wait, server utilization, throughput

Policy Descriptions

Baseline: Greedy first-come, first-served scheduling with simple no-show buffers. This policy allocates appointment slots sequentially and applies a fixed overbooking multiplier (typically 1.1×) to account for expected no-shows. It serves as the control against which we measure the benefit of the FCP-derived approach.

Variant: A policy derived by solving the fluid control problem (treating the clinic as a heavy-traffic queueing system) and then applying the optimal control law to appointment slot allocation and overbooking decisions. The FCP solution yields a continuous control law of the form $u^*(t) = f(\rho(t), \lambda(t))$, where $\rho(t)$ is the queue state and $\lambda(t)$ is the arrival rate. We discretized this law to produce integer slot allocations and overbooking rates for each simulation day.

Results

Outcome Metrics

Metric	Baseline	Variant	Delta
Served	116.0	116.0	0.0
No-shows	12.0	12.0	0.0
Walk-ins	48.0	48.0	0.0
Mean wait (hours)	0.000446	0.000446	0.0
P95 wait (hours)	0.0	0.0	0.0
Utilization	0.0967	0.0967	0.0

Primary Finding

The variant policy produced identical outcomes to the baseline across all measured dimensions. The hypothesis is not supported by this experiment. Both policies achieved the same throughput, wait-time distribution, and resource utilization, indicating that the FCP-derived control law conferred no measurable advantage in the simulated clinic environment.

Honest limitations

Regime Mismatch: Under-Loaded System

The clinic simulation operated at approximately 10% utilization, far below the heavy-traffic regime (typically $\rho \to 1$) where fluid control approximations are asymptotically tight. The FCP framework is designed for high-congestion systems; our toy clinic was under-loaded, so the deterministic relaxation may have been vacuous. In light-load regimes, the stochastic system behaves nearly deterministically anyway, and the FCP's advantage—which emerges from the interaction between congestion and randomness—is suppressed. To properly test the hypothesis, we would need to operate the clinic at utilization levels of 0.7 or higher, where heavy-traffic asymptotics become relevant.

Policy Instantiation Gap

We derived the FCP solution analytically but may not have correctly translated it into discrete appointment-slot decisions. The FCP gives a continuous control law (e.g., overbooking rate as a function of queue state); mapping this to integer slot counts is non-trivial and we did not validate that our discretization preserved the FCP's optimality guarantees. Specifically, the continuous control $u^*(t)$ may specify an overbooking rate of 1.15, but the clinic must decide whether to book 12 or 13 slots. Our rounding procedure was ad hoc, and we did not verify that the rounded policy still satisfies the FCP's optimality conditions or that it remains robust to distributional shifts.

No Explicit Ambiguity Set

The simulation did not model a formal distributional ambiguity set (e.g., a Wasserstein ball around the nominal no-show distribution). Instead, we ran a single scenario with fixed no-show and walk-in rates. Without perturbing the input distribution and measuring robustness, we cannot assess whether the FCP-derived policy is actually more robust than the baseline. A proper test of distributional robustness requires evaluating both policies across a range of plausible input distributions and measuring their worst-case and average-case performance.

Single-Day Horizon and Asymptotic Limitations

A clinic day is short; the asymptotic optimality of FCP applies as the system size (patient volume, time horizon) grows. Our 1-day simulation may be too small to see the benefit of the fluid approximation. The heavy-traffic limit is a mathematical idealization that becomes increasingly accurate as the number of patients and the time horizon increase. A single day with ~160 total arrivals (scheduled plus walk-in) may not be large enough for the asymptotic regime to dominate.

No Computational Cost Comparison

The hypothesis included a claim about computational tractability. We did not measure wall-clock time, memory, or scalability of the FCP derivation vs. alternatives (e.g., robust optimization solvers), so we cannot validate the efficiency claim. If the FCP approach is to be competitive, it must not only yield better policies but also do so faster than existing methods.

Next experiment

Proposed High-Utilization, Multi-Day, Distribution-Shift Study

To sharpen the test, we propose a high-utilization, multi-day, distribution-shift robustness study. Specifically:

1. Re-parameterize the clinic to operate at $\rho \geq 0.7$ (heavy traffic). This requires increasing the arrival rate, reducing the number of available slots, or both, to push the system into the congestion regime where fluid control approximations are asymptotically tight.

2. Extend the horizon to 10–20 days to allow asymptotic behavior to emerge and to accumulate enough patient volume for the heavy-traffic limit to become relevant.

3. Explicitly define an ambiguity set (e.g., ±20% perturbation to no-show rates and service-time variance). This could be a Wasserstein ball, a moment-based set, or a simple interval set around the nominal parameters.

4. Run the baseline and FCP-derived variant under each distribution in the ambiguity set, measuring the worst-case and average-case wait times, utilization, and throughput across the set.

5. Measure robustness metrics: If the FCP policy exhibits lower worst-case wait time, lower variance in utilization, or more stable throughput under distribution shifts, the hypothesis gains support. If both policies remain identical, we should investigate whether the FCP derivation was correctly instantiated or whether the problem structure is fundamentally misaligned with the source paper's assumptions.

This refined experiment would directly test the core claim: that a policy derived from the FCP framework is more robust to distributional uncertainty than a greedy baseline, without requiring explicit solution of the minimax problem.