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An ambiguity-set radius decides how far the robust queue model may move from its empirical distribution. Too small, and a plausible delay pattern sits outside the set. Too large, and the decision may buy costly capacity against scenarios the evidence does not support. Calibration makes that trade visible on held-out queue episodes.
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Chapter contents
Choose the route closest to the decision, or read from Chapter 1. The chapters remain open; these grouped routes are a navigable index.
Route 1 · Define the decision
Route 2 · Work the ledgers
Route 3 · Transfer the method
Route 4 · Check limits and evidence
Route 5 · Ask and continue
CHAPTER 1 OF 12 · DEFINE THE DECISION
1. The radius is a design choice with consequences
A Wasserstein or related ambiguity set contains distributions within a declared distance of a nominal empirical law. The radius controls the set’s reach; it is not a universal constant and it should not be selected because a round number looks cautious.
In a queueing decision, a wider set can raise worst-case delay or staffing cost. That protection is useful only when it improves performance under credible shifts rather than merely making every recommendation more expensive.
CHAPTER 2 OF 12 · DEFINE THE DECISION
2. Separate coverage from operational regret
Coverage asks whether held-out empirical distributions, summary features or target risks are compatible with the set. Operational regret asks how much worse the chosen staffing or capacity action performs than the best action that hindsight would have selected for that holdout episode.
A radius can achieve generous coverage while producing poor decisions. Record both quantities, because statistical inclusion and decision quality answer different questions.
CHAPTER 3 OF 12 · WORK THE LEDGERS
3. A radius-calibration ledger
Suppose a call queue is calibrated on historical days and tested on twelve held-out weeks. The loss includes staffing cost and a penalty for the 60-minute delay threshold.
| Radius | Holdout coverage | Mean holdout regret | Worst-week regret |
|---|---|---|---|
| 0.02 | 58% | 8.7 | 21.4 |
| 0.05 | 83% | 5.1 | 13.2 |
| 0.09 | 92% | 5.6 | 11.1 |
| 0.16 | 100% | 9.8 | 10.7 |
The table does not dictate one answer. If extreme-week protection is critical, 0.09 may be preferred to 0.05; if routine staffing cost dominates, the smaller choice may be defensible.
CHAPTER 4 OF 12 · WORK THE LEDGERS
4. Worked coverage–conservatism choice
A governance rule requires at least 90% holdout coverage, mean regret below 6.0 and worst-week regret below 12.0. Only radius 0.09 passes all three gates.
| Gate | Required | Radius 0.09 | Result |
|---|---|---|---|
| Coverage | At least 90% | 92% | Pass |
| Mean regret | Below 6.0 | 5.6 | Pass |
| Worst-week regret | Below 12.0 | 11.1 | Pass |
Predeclare the gates before viewing the final test period. Repeatedly tuning against the same holdout set turns it into training data and makes the reported performance optimistic.
CHAPTER 5 OF 12 · WORK THE LEDGERS
5. Use nested calibration when tuning many choices
Split data into model fitting, radius calibration and final evaluation. When data are scarce, use nested resampling that preserves season, campaign and incident structure rather than shuffling individual arrivals.
The nominal distribution, ground metric, feature scaling and radius interact. Changing the metric after seeing failures means the radius must be recalibrated; the old value has lost its interpretation.
CHAPTER 6 OF 12 · TRANSFER THE METHOD
6. Stress the shifts that matter to queues
Holdouts should include peak intensity, over-dispersion, service-time tails, abandonment, retries and dependence across time. A set calibrated only on quiet weeks can give impressive average coverage and still miss the operational branch that matters.
Report performance by shift type. One aggregate coverage number can hide systematic failure on holidays, outages or low-staffing periods.
CHAPTER 7 OF 12 · TRANSFER THE METHOD
7. Changed-context transfer: emergency intake capacity
Transfer the workflow to emergency intake, where the loss places more weight on upper-tail wait and diversion. Preserve whole days or shifts as holdout units so serial dependence and staffing regimes remain intact.
The transfer check revises the cost scale, protected thresholds and acceptable regret. It does not carry over a radius numerically from the call queue, even if both use Wasserstein distance.
CHAPTER 8 OF 12 · TRANSFER THE METHOD
8. Connect the ambiguity and priority branches
The Wasserstein ambiguity guide introduces radius-controlled robust capacity. The elicitation guide separates uncertain priorities from statistical uncertainty.
Use the regret-robust weighting guide when threshold importance is itself disputed.
CHAPTER 9 OF 12 · CHECK LIMITS AND EVIDENCE
9. Limits and common misconceptions
Empirical coverage of a small holdout sample is noisy. Twelve weeks cannot establish 92% population coverage with precision, so include uncertainty intervals and repeat the check when new regimes arrive.
A larger radius is not automatically safer. It may protect implausible distributions, mask a poor ground metric and impose avoidable cost.
CHAPTER 10 OF 12 · CHECK LIMITS AND EVIDENCE
10. Practical evidence checklist
Archive the nominal sample, distance metric, scaling, candidate radii, split logic, coverage definition, regret function, capacity solutions and results by shift type.
Keep the final evaluation untouched until selection is complete. Record which failure would trigger recalibration and whether the response should change the radius, the metric or the underlying queue model.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Frequently asked questions
Should the radius be chosen only for coverage?
No. Coverage is one constraint; regret, cost and worst-case decision stability show whether that coverage is operationally useful.
Can cross-validation select the radius?
Yes, if folds preserve the dependence structure and the final assessment uses data not consumed by selection.
Does one radius fit every season?
Not necessarily. Regime-specific or contextual radii can be justified, but each added flexibility increases calibration and validation demands.
CHAPTER 12 OF 12 · ASK AND CONTINUE
12. Sources and next reading
Esfahani and Kuhn develop data-driven distributionally robust optimisation with Wasserstein distance. Schuurmans and Patrinos propose cost-aware ambiguity sets designed to reduce conservatism. Zhou and colleagues study held-out diagnostic transport for ambiguity-set calibration. The queue calibration ledgers above are original teaching examples.
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