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A useful rare-delay proposal is not automatically the proposal that deserves most of the simulation budget. Allocation must balance variance, cost, threshold coverage and the fact that changing mixture shares can change the estimator itself.
Open the chapter contents · Queueing Theory root · Gold Standard of Education · How X Works
Chapter contents
Choose the route closest to the question, or read from Chapter 1 for the complete argument. The chapters remain open on the page; these grouped routes are an index.
Route 1 · Frame the problem
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 · FRAME THE PROBLEM
1. Allocation is part of the estimator
Suppose a queue model has proposals for arrival bursts, slow service and failed recovery. A pilot run can reveal which proposal reaches a delay threshold efficiently, but the final sampling shares cannot be chosen by raw hit count alone. Importance weights, overlap and computing cost all matter.
In stratified estimators with fixed component contributions, a familiar cost-aware rule allocates in proportion to a component standard deviation divided by the square root of its per-draw cost. In multiple importance sampling, however, the mixture denominator also depends on the shares. Exact optimisation is therefore joint; the simple rule is a planning diagnostic, not a universal theorem.
CHAPTER 2 OF 12 · FRAME THE PROBLEM
2. Separate pilot, production and audit budgets
The pilot budget learns about proposal behaviour. The production budget estimates the target under a declared allocation. The audit budget repeats, perturbs or independently checks the result. Spending every run on the apparent pilot winner leaves no protection against pilot noise or an unseen tail mechanism.
Freeze the production rule before inspecting the final decision. If later adaptation is allowed, record the batches and use an estimator whose validity covers that adaptation. Calling an allocation “adaptive” does not by itself correct reuse of the same outcomes for selection and inference.
CHAPTER 3 OF 12 · WORK THE LEDGERS
3. A three-proposal pilot ledger
Consider 600 pilot paths from each proposal. The diagnostic standard deviations refer to weighted contributions for a 60-minute delay event, while cost is measured in baseline simulation-time units.
| Proposal | Pilot paths | Weighted SD | Cost per path |
|---|---|---|---|
| Arrival burst | 600 | 0.012 | 1.0 |
| Service slowdown | 600 | 0.028 | 1.4 |
| Recovery failure | 600 | 0.045 | 2.2 |
| Total | 1,800 | — | — |
Recovery failure is expensive, yet its weighted contribution varies most. Giving it no additional paths would leave a noisy component; giving it everything would abandon coverage of other mechanisms.
CHAPTER 4 OF 12 · WORK THE LEDGERS
4. A worked cost-aware planning share
As a transparent first pass, compute score σ/√c. The scores are 0.012, about 0.0237 and about 0.0303. Normalising them gives planning shares near 18%, 36% and 46% for the remaining path budget.
| Proposal | σ/√cost | Planning share | Paths from 7,200 |
|---|---|---|---|
| Arrival burst | 0.0120 | 18% | 1,296 |
| Service slowdown | 0.0237 | 36% | 2,592 |
| Recovery failure | 0.0303 | 46% | 3,312 |
| Total | 0.0660 | 100% | 7,200 |
The calculation is useful because every input is visible. It is not a licence to report false precision. Round the counts, retain minimum coverage and evaluate the predicted variance using the estimator that will actually be run.
CHAPTER 5 OF 12 · WORK THE LEDGERS
5. Coverage floors protect against pilot blindness
A short pilot can miss a rare route entirely. Set a minimum allocation for every proposal needed to cover baseline support or a decision threshold. Defensive baseline sampling may also receive a fixed share even when its pilot variance appears low.
Coverage floors should follow mechanism and support arguments, not an arbitrary equal-split tradition. After applying the floors, allocate only the remaining budget. If the floor consumes most of the budget, the proposal set may be too large for the available evidence.
CHAPTER 6 OF 12 · TRANSFER THE METHOD
6. Optimise across a threshold ladder
A share vector that is excellent for a 60-minute delay can be poor for 20 or 120 minutes. Define a loss across the thresholds that matter: maximum relative variance, a weighted sum of variances, or probability of making the wrong capacity decision.
Do not let the largest estimated probability dominate merely because it is easiest to estimate. Put decision weights and tolerances on the scale of the operational question, then show each threshold’s effective evidence separately.
CHAPTER 7 OF 12 · TRANSFER THE METHOD
7. Changed-context transfer: emergency call routing
Transfer the allocation logic to an emergency call-routing simulation with proposals for a demand surge, long handling times and dispatch feedback. A pilot can guide production effort, while coverage floors retain each plausible overload route.
The transfer fails if altered mechanisms change dependent parts of the path without a valid joint likelihood. Safety-critical staffing also needs domain constraints, service targets and consequence analysis beyond Monte Carlo precision.
CHAPTER 8 OF 12 · TRANSFER THE METHOD
8. Connect pilot allocation to the MIS branch
The multiple-proposal MIS guide defines valid combination weights after several proposals are sampled. The mixture-proposal guide explains why different mechanisms need overlapping support.
Use Pareto-tail diagnostics and smoothed weights to audit whether the chosen budget produced reliable evidence rather than a small number of dominant paths.
CHAPTER 9 OF 12 · CHECK LIMITS AND EVIDENCE
9. Limits and misconceptions
Pilot-estimated variances are noisy, especially for rare events. The apparent optimum can overfit one seed. Mixture-share optimisation may be nontrivial because changing a share changes both the samples and the denominator used in balance weighting.
More paths do not repair missing support, a wrong baseline queue model or a reversed likelihood ratio. An allocation with the smallest estimated Monte Carlo error can still answer the wrong operational question.
CHAPTER 10 OF 12 · CHECK LIMITS AND EVIDENCE
10. Practical evidence checklist
Archive pilot seeds, proposal definitions, log densities, per-path costs, threshold contributions and the allocation calculation. State every floor, cap, rounding rule and stopping rule before production.
After production, compare predicted and observed component variance, effective counts, largest weights, replicate estimates and conclusions under nearby allocations. Keep the audit budget independent enough to reveal pilot overfitting.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Frequently asked questions
Should the cheapest proposal receive most paths?
Only if its information per unit cost supports that choice. A cheap proposal that misses the important tail contributes little to the decision.
Can raw tail hits determine the shares?
No. Hits occur under altered proposal laws. Valid weighted variance and overlap matter more than raw counts.
Must the allocation stay fixed forever?
No, but each adaptive batch must follow a declared method whose estimator and uncertainty calculation match the adaptation.
CHAPTER 12 OF 12 · ASK AND CONTINUE
12. Sources and next reading
He and Owen study optimal mixture weights and sequential estimation of mixture shares. Veach and Guibas introduce multiple importance sampling and balance weighting. Owen and Zhou develop safe and effective importance sampling. The queue ledgers above are original teaching examples.
Continue through the Queueing Theory root, the multiple-proposal MIS guide and likelihood-ratio diagnostics. The Gold Standard of Education and How X Works provide the wider route.
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