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A value-of-information trigger may look decisive while its estimate is still moving because both the possible future dataset and the queue outcomes inside that dataset are simulated. Nested Monte Carlo creates an outer layer for future evidence and an inner layer for operational consequences. The practical question is not whether the estimate has many decimal places. It is whether simulation error is small enough that another honest run would leave the collect, stop or recalibrate decision unchanged.
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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. Name the decision before measuring simulation error
Fix the operational decision first: collect another week, implement the current capacity, or reopen the ambiguity-radius review. Define the net-value trigger and every cost in common units before increasing simulation size.
A simulation standard error is useful only relative to a decision boundary. An error of 0.2 units may be harmless when conservative net value is 4.0, yet decisive when the trigger sits at zero.
CHAPTER 2 OF 12 · DEFINE THE DECISION
2. Separate outer, inner and model uncertainty
Outer draws represent possible future datasets. Inner draws estimate queue loss under each updated model and candidate capacity. Their sampling errors behave differently and should not be hidden inside one unexplained interval.
Model uncertainty is a third layer. More Monte Carlo effort can reduce numerical noise but cannot repair a wrong arrival model, an omitted abandonment mechanism or a cost that was never measured.
CHAPTER 3 OF 12 · WORK THE LEDGERS
3. Allocate a finite simulation budget
A contact centre has 120,000 queue replications available for a one-week value-of-information review. Pilot runs estimate which layer contributes most variance.
| Design | Outer datasets | Inner runs per action | Estimated trigger SE | Interpretation |
|---|---|---|---|---|
| A | 120 | 500 | 0.31 | Too few outer futures |
| B | 300 | 200 | 0.19 | Balanced pilot |
| C | 600 | 100 | 0.18 | Small gain |
| D | 1,200 | 50 | 0.27 | Inner comparisons too noisy |
The best allocation depends on variance, computation cost and the nonsmooth max or selection operation inside the value calculation. It should be estimated, not chosen from a universal ratio.
CHAPTER 4 OF 12 · WORK THE LEDGERS
4. Worked decision-stability calculation
Suppose the estimated conservative net value of another week is 0.46 cost units. Four independent simulation batches are used to check whether the trigger is genuinely positive.
| Batch | Net-value estimate | Monte Carlo SE | 95% simulation interval | Decision at zero |
|---|---|---|---|---|
| 1 | 0.42 | 0.24 | −0.05 to 0.89 | Unstable |
| 2 | 0.51 | 0.17 | 0.18 to 0.84 | Collect |
| 3 | 0.39 | 0.16 | 0.08 to 0.70 | Collect |
| 4 pooled | 0.46 | 0.11 | 0.24 to 0.68 | Collect |
This interval describes numerical uncertainty conditional on the simulation model. A separate sensitivity analysis is still required for model and cost assumptions.
CHAPTER 5 OF 12 · WORK THE LEDGERS
5. Use paired randomness for action differences
When comparing two capacities, simulate them under the same arrival and service streams where valid. Common random numbers can reduce the variance of their loss difference because both actions face the same difficult days.
Keep random-number streams auditable and preserve independence between genuinely independent batches. Pairing is a variance-reduction design, not permission to reuse outcomes selectively.
CHAPTER 6 OF 12 · TRANSFER THE METHOD
6. Require seed and allocation stability
Repeat the calculation with independent seeds and at least one credible outer-inner allocation. A stable trigger keeps the same operational recommendation and similar margin after these changes.
Do not demand identical estimates. The check asks whether reasonable numerical choices reverse collect versus stop, change the selected capacity or move the review date materially.
CHAPTER 7 OF 12 · TRANSFER THE METHOD
7. Changed-context transfer: emergency intake
Transfer the calculation to emergency intake, where outer draws represent future shift profiles and inner runs represent congestion, boarding and diversion under each staffing plan.
The transfer check raises the consequence of rare severe delays, retains patient-level or shift-level dependence and requires clinical gates before a positive average information value can justify waiting.
CHAPTER 8 OF 12 · TRANSFER THE METHOD
8. Connect triggers, action sets and recalibration
Start with the value-of-information trigger guide. Use the robust action-set guide to identify which capacity boundaries matter.
The radius-uncertainty guide handles calibration uncertainty. This article adds numerical error in the nested decision calculation rather than relabelling either problem.
CHAPTER 9 OF 12 · CHECK LIMITS AND EVIDENCE
9. Limits and common misconceptions
A tiny Monte Carlo standard error does not validate the queue model. It only says the chosen simulator has been averaged precisely under its own assumptions.
A single favourable seed is not evidence of stability. Nor should simulation continue until a preferred answer appears. State the precision target and stopping rule before the decisive run.
CHAPTER 10 OF 12 · CHECK LIMITS AND EVIDENCE
10. Practical evidence checklist
Archive the estimand, trigger, outer and inner sample sizes, seed schedule, variance-reduction method, batching rule, computation budget, standard-error estimator and every failed or rerun job.
Report model sensitivity beside simulation uncertainty. Record whether each credible rerun changes the collect, stop or recalibrate action and preserve the code version that produced the final ledger.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Frequently asked questions
Should the inner sample always be very large?
No. Too many inner runs can leave too few outer futures, while too few inner runs can bias or destabilise a nonlinear decision comparison. Use pilot variance and convergence checks.
Can bootstrap intervals replace independent simulation batches?
They can help quantify numerical error under a justified resampling design, but independent batches remain a useful diagnostic for seed, implementation and convergence problems.
When is the trigger stable enough?
When its simulation margin clears the operational boundary and credible changes in seed, allocation and numerical method leave the action unchanged.
CHAPTER 12 OF 12 · ASK AND CONTINUE
12. Sources and next reading
Koehler, Brown and Haneuse examine Monte Carlo error in simulation-based statistical analyses. Giles discusses multilevel Monte Carlo for nested expectations. Pasupathy and Schmeiser develop a sequential sampling procedure for stochastic programming. The queue ledgers here are original teaching examples.
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