Tail inflation
A rare common-cause branch can dominate the probability of many simultaneous failures.
Reliability
Component Failure Simulation Calculator
Run a seeded component-failure Monte Carlo model with an explicit common-cause branch.
MONTE CARLO SCENARIO
Expose the tail created by a shared failure mechanism
For reliability teams testing redundancy thresholds when independent component risk is not the whole story.
Threshold probability-
Mean failures-
Any failure-
P95 failure count-
Analytic mixture mean-
CURRENT DECISION RECORD
Seeded failure-count histogram
Every row is regenerated from the active inputs and carried into Copy, TXT, and the page-specific PDF payload.
| Failed components | Simulated trials | Trial share (%) |
|---|
CURRENT CALCULATION PROCESS
Formula, substitution, intermediate values, and reconciliation
F = N with probability q; otherwise F ~ Binomial(N,p)
Waiting for valid inputs.
HOW TO USE
Run a reproducible scenario
- Define the number of simultaneously exposed components.
- Enter independent per-component failure probability for one mission.
- Enter a separate common-cause probability for an all-component shock.
- Choose a decision threshold, iteration count, and recorded seed.
- Rerun with alternate seeds and compare simulation error with the analytic mean.
FOUNDATIONS
Simulation essentials
- Trial
- One complete component-group mission.
- Bernoulli event
- Each independent component either fails or survives.
- Common cause
- A shared event that fails the complete entered group in this simplified mixture.
- Seed
- A reproducibility key for the pseudo-random sequence.
- Monte Carlo error
- Finite-run variation that shrinks as iterations increase.
DEEP ANALYSIS
Why tails, checks, and thresholds matter
Mean check
The mixture mean N[q+(1-q)p] provides an independent deterministic check.
Threshold choice
Set it from the consequence or redundancy rule before looking at simulated results.
CASES
Independent and shared-shock behavior
Independent electronics batch
Low p and q=0 produces a familiar binomial-like histogram concentrated near zero.
Common-cause boundary
Even a small q places mass at all components failed and can sharply raise the threshold probability.
TERMS
Monte Carlo vocabulary
- Iteration
- One simulated mission.
- Histogram
- Counts of trials by failed-component total.
- P95
- The 95th percentile failure count.
- Mixture model
- A model combining common-cause and independent branches.
- Threshold exceedance
- Failure count at or above the entered limit.
- Pseudo-random
- Deterministic numbers that emulate random draws from a seed.
EVIDENCE RECORD
Inputs and lineage to preserve
Archive component count, independent probability, common-cause probability, threshold, iterations, seed, model revision, and alternate-seed results. Preserve the engineering basis for p and q separately so reviewers can reproduce the mixture and challenge the shared-shock assumption.
MODEL LIMITS
Where the seeded mixture stops
- Common-cause branch fails every entered component.
- Independent branch uses identical constant probabilities.
- No repair, time ordering, load sharing, or cascading failure.
- Simulation output is seed- and iteration-dependent.
SOURCES
Probability and reliability references
FAQ
Simulation-specific questions
Is the seed random?
It initializes a deterministic xorshift32 stream so the same inputs reproduce the same table.
Why not use only the analytic binomial?
The common-cause branch changes the tail and simulation makes that mixed outcome tangible.
Does q overlap p?
The model applies q first; independent failures run only when the common-cause event does not occur.
How many iterations are enough?
Enough depends on the tail probability and precision needed; compare seeds and report Monte Carlo uncertainty.
Can partial common-cause failures be modeled?
Not here; this page uses an explicit all-component shock branch.
Is P95 a worst case?
No. Five percent of simulated trials can be at or above it, subject to ties.
IMPORTANT SIMULATION NOTE
A common-cause branch must come from engineering evidence
The seeded result illustrates the entered mixture. It does not establish the true shared-shock probability, independence of the remaining component failures, or safety of the threshold.