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A queue rarely has only one service threshold. Once analysts monitor 30-, 60- and 90-minute delays, a separate interval at each threshold and each look creates many chances to overstate certainty. The useful object is a band whose guarantee matches both dimensions of the decision.
Open the chapter contents · Queueing Theory root · Gold Standard of Education · How X Works
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 evidence
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 EVIDENCE
1. One curve creates two kinds of multiplicity
A delay curve records the probability that waiting time exceeds each threshold. Looking across many thresholds creates value multiplicity; looking after many simulation batches creates time multiplicity. A method that protects only one dimension does not automatically protect the other.
The decision claim must say whether coverage is pointwise, simultaneous across a declared threshold grid, uniform over all thresholds, time-uniform, or uniform over both time and threshold. Those are different guarantees, not interchangeable labels.
CHAPTER 2 OF 12 · DEFINE THE EVIDENCE
2. Define the family before examining it
Choose the queue policy, delay variable, threshold set, monitoring times and decision rules before viewing the estimates. The family might be three operational thresholds or the complete cumulative distribution function. A post hoc threshold selected because it looks worst belongs to the protected family too.
If importance sampling is used, the band must cover the weighted distribution estimator produced by the actual adaptive proposal. A Bernoulli formula for one unweighted threshold does not become a curve band merely by repetition.
CHAPTER 3 OF 12 · WORK THE LEDGERS
3. A simultaneous delay-curve ledger
Consider a final monitored batch with three operational delay thresholds. The values below are an original teaching example from a jointly protected procedure.
| Delay threshold | Estimated exceedance | Simultaneous band | Operational cap |
|---|---|---|---|
| 30 minutes | 8.0% | 6.8%–9.4% | 10.0% |
| 60 minutes | 3.0% | 2.3%–3.9% | 4.0% |
| 90 minutes | 1.2% | 0.8%–1.8% | 1.5% |
The first two upper endpoints are below their caps. The 90-minute endpoint is not, so an all-threshold approval rule must continue or return an inconclusive result.
CHAPTER 4 OF 12 · WORK THE LEDGERS
4. Worked familywise decision
Suppose the overall error budget is 5% and three separate anytime-valid sequences are used. A simple Bonferroni design assigns at most 5% ÷ 3 = 1.67% error to each sequence. The union bound then limits the probability that any sequence ever misses its target to at most 5%, without requiring independence.
| Component | Allocated error | Upper band vs cap | Component result |
|---|---|---|---|
| 30-minute curve point | 1.67% | 9.4% < 10.0% | Pass |
| 60-minute curve point | 1.67% | 3.9% < 4.0% | Pass |
| 90-minute curve point | 1.67% | 1.8% > 1.5% | Not yet |
| Family decision | At most 5% total | All must pass | Continue |
Bonferroni is transparent but can be conservative. A jointly constructed time-and-value-uniform band can use dependence across thresholds more efficiently when its assumptions match the data.
CHAPTER 5 OF 12 · WORK THE LEDGERS
5. Read the shape, not only the crossings
Threshold estimates are nested: the probability of waiting more than 90 minutes cannot exceed the probability of waiting more than 60 minutes. A coherent band should respect or be projected onto this monotone structure without invalidating coverage.
Watch where the band is widest. Sparse tail evidence, unstable likelihood ratios or few effective paths often make the far tail decisive. A smooth-looking curve does not repair a support gap in the simulator.
CHAPTER 6 OF 12 · TRANSFER THE METHOD
6. Anytime-valid curves and stopping
A time-uniform, value-uniform band supports flexible looks and threshold inspection within its stated domain. Stop only under a predeclared rule such as every upper boundary falling below its cap, or a resource ceiling being reached.
A guarantee can remain valid when the study stops adaptively, yet the operational choice can still be poor if caps were chosen after the data or if the queue policy changed. Separate statistical optional stopping from policy revision.
CHAPTER 7 OF 12 · TRANSFER THE METHOD
7. Changed-context transfer: emergency-department waits
Transfer the design to emergency-department waiting-time simulations with thresholds tied to triage categories. Use a protected family for the declared categories and a separate governance process for selecting the service caps.
The transfer check asks whether arrivals, priority rules and service-time distributions are stable enough for the estimator, whether category thresholds were fixed before monitoring and whether one patient episode contributes to several dependent outcomes.
CHAPTER 8 OF 12 · TRANSFER THE METHOD
8. Connect the rare-delay branch
The time-uniform confidence-sequence guide develops the single-target stopping logic. The sequential budget guide separates allocation changes from evidence stopping.
Use defensive mixture proposals when one proposal must cover several delay thresholds without abandoning common support.
CHAPTER 9 OF 12 · CHECK LIMITS AND EVIDENCE
9. Limits and common misconceptions
Familywise control does not guarantee that every component interval is equally narrow, and it does not identify a scientifically meaningful threshold. False-discovery control answers a different question from simultaneous confidence coverage and may not suit an all-threshold safety rule.
A band derived for independent paths may fail under common random numbers, recycled trajectories or adaptive resampling. A value-uniform band can be trivial in difficult nonstationary settings; always-valid does not mean always-informative.
CHAPTER 10 OF 12 · CHECK LIMITS AND EVIDENCE
10. Practical evidence checklist
Archive the threshold family, monitoring schedule, global error budget, component allocation or joint-band theorem, queue policy, estimator, weights, path dependence and stop rule. Save every look and every threshold considered, not only the final graph.
Check nesting, tail effective sample size, largest weights, replicate calibration and the result under a wider threshold grid. State whether the band protects a fixed grid, all real thresholds, all times, or a combination.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Frequently asked questions
Can I add a new threshold during monitoring?
Only if the procedure was designed to cover it, such as a value-uniform band. Otherwise the error budget and decision rule need a fresh analysis.
Is Bonferroni always required?
No. It is one transparent construction. A theorem that directly gives joint time-and-value coverage may be more efficient.
Does the lowest point estimate determine approval?
No. The decision uses the relevant simultaneous boundary and the declared all-threshold or componentwise rule.
CHAPTER 12 OF 12 · ASK AND CONTINUE
12. Sources and next reading
Mineiro and Howard develop time-uniform confidence bands for cumulative distribution functions, including importance-weighted extensions. Howard, Ramdas, McAuliffe and Sekhon develop time-uniform nonparametric confidence sequences. Howard and Ramdas study sequential quantile estimation. The queue ledgers above are original teaching examples.
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