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Not every delay threshold carries the same consequence. A missed 30-minute promise may trigger staffing action while a 47-minute display point is descriptive. Unequal error budgets can reflect that difference, but only when the weights are declared before the evidence and the joint guarantee remains explicit.
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 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. A weight is a protection choice
For thresholds t1 through tm, a weighted Bonferroni design can assign local error levels alpha_j=alpha w_j, where each w_j is non-negative and the weights sum to one. The familywise error remains at most alpha by the union bound.
Larger w_j gives threshold j a less stringent local confidence level and usually a narrower interval. That is not free precision: it spends more of the shared error budget at that threshold and leaves less elsewhere.
CHAPTER 2 OF 12 · DEFINE THE DECISION
2. Importance must be fixed before outcomes
Weights may come from service contracts, safety tiers, customer impact or a predeclared loss function. They must not be increased because a preliminary curve looks close to a cap unless the adaptive rule itself has a valid guarantee.
Separate decision importance from anticipated statistical power. A high-cost threshold can deserve protection even if the event is rare; a common event does not automatically deserve the largest weight.
CHAPTER 3 OF 12 · WORK THE LEDGERS
3. A four-threshold allocation ledger
A queue team protects four delay thresholds with total alpha=0.05. The 30- and 60-minute promises govern staffing and escalation, so they receive most of the budget.
| Threshold | Operational role | Weight w | Local alpha |
|---|---|---|---|
| 15 min | Dashboard | 0.10 | 0.005 |
| 30 min | Service promise | 0.35 | 0.0175 |
| 60 min | Escalation trigger | 0.40 | 0.0200 |
| 120 min | Rare-delay audit | 0.15 | 0.0075 |
The design documents why 60 minutes receives 0.40. Without that rationale, weights can become a disguised way to favour the desired conclusion.
CHAPTER 4 OF 12 · WORK THE LEDGERS
4. Worked decision-focused interval comparison
Suppose a simple normal approximation gives estimate plus or minus z times standard error. Use illustrative critical values 2.81, 2.38, 2.33 and 2.67 for the four two-sided local levels.
| Threshold | Estimate | SE | Illustrative half-width | Operational cap |
|---|---|---|---|---|
| 15 min | 18.0% | 1.2% | 3.37% | 25% |
| 30 min | 10.4% | 1.0% | 2.38% | 12% |
| 60 min | 5.9% | 0.8% | 1.86% | 6% |
| 120 min | 1.8% | 0.5% | 1.34% | 2% |
The next move is not to reassign weight after seeing the ambiguity. It is to gather targeted evidence, improve the estimator or use a predeclared sequential design.
CHAPTER 5 OF 12 · WORK THE LEDGERS
5. Combine weighting with monotone structure carefully
Queue exceedance probabilities are non-increasing with threshold. A coverage-preserving order closure can intersect the weighted simultaneous set with that monotone constraint.
Do not independently smooth the centres and then attach the old intervals. The threshold-grid and monotone-band guide separates logical closure from arbitrary smoothing.
CHAPTER 6 OF 12 · TRANSFER THE METHOD
6. Reserve budget for unplanned inspection
A practical design can reserve a small exploration weight for secondary thresholds or use a continuum-wide band whose guarantee already covers threshold inspection. The reservation must be part of the original family.
If every new threshold receives weight without reducing another, the advertised total error budget is no longer the same. Version the threshold family and restart the guarantee when the operational contract changes.
CHAPTER 7 OF 12 · TRANSFER THE METHOD
7. Changed-context transfer: emergency arrivals
Transfer the method to emergency-department arrival-to-triage times. Safety-critical thresholds may receive more protection than dashboard thresholds, but weights should be set with clinical governance rather than chosen by analysts alone.
The transfer check handles time-of-day clusters, repeated visits, policy changes and any adaptive stopping. A weighted band cannot correct biased timestamps or a changed definition of arrival.
CHAPTER 8 OF 12 · TRANSFER THE METHOD
8. Connect the confidence branch
The simultaneous-threshold guide defines the protected family. The time-uniform guide covers repeated looks and optional stopping.
Use the importance-weight diagnostics guide when simulation path weights, rather than error-budget weights, are unstable. The two kinds of weight solve different problems.
CHAPTER 9 OF 12 · CHECK LIMITS AND EVIDENCE
9. Limits and common misconceptions
Unequal alpha allocation does not make the estimator more representative, remove dependence or validate a queue model. It changes the distribution of inferential protection across a declared family.
A high decision weight is not a posterior probability that the threshold matters. It is a design choice whose scientific and operational rationale should be visible.
CHAPTER 10 OF 12 · CHECK LIMITS AND EVIDENCE
10. Practical evidence checklist
Archive the threshold family, total alpha, weights, weight rationale, local procedures, dependence assumptions, any correlation adjustment, curve estimator, caps, monitoring schedule and revision policy.
Show simultaneous coverage type, interval width at every threshold, decisions under equal and unequal weights, and sensitivity to plausible weight sets. Preserve low-priority results instead of hiding wide intervals.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Frequently asked questions
Can the weights be learned from data?
Only with a procedure whose independence, sample-splitting or cross-weighting conditions justify that learning. Outcome-driven changes without protection invalidate the original claim.
Must the weights sum to one?
In the basic weighted Bonferroni construction, yes. Other closed or graphical procedures have their own bookkeeping rules.
Why not spend everything on one threshold?
That can be coherent for one primary decision, but it leaves no joint protection for the others and should be described as a single-threshold analysis.
CHAPTER 12 OF 12 · ASK AND CONTINUE
12. Sources and next reading
Bretz and colleagues develop graphical multiple-comparison procedures that separate weighting strategy from the test procedure. Ignatiadis and Huber present cross-weighted multiple testing with finite-sample control under stated independence conditions. Mineiro and Howard develop time-uniform confidence bands for cumulative distribution functions. The queue ledgers above are original teaching examples.
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