Failure Rate Estimation

Failure-Rate Estimation (systems-engineering-safety/arp4761a/failure-rate-estimation)

Use when the task is estimating or demonstrating a system failure rate from test or service data: the point estimate, the exact confidence upper-bound, the zero-failure demonstration rule, the test-hours needed to demonstrate a target rate, and the MTBF bounds. This leaf produces the rate numbers that feed the FTA, Markov, and PSSA/SSA quantitative models; it does not run those models.

Domain quick reference

• ARP4761A quantitative analyses (FTA, Markov, PSSA/SSA) need a failure rate per basic event. Rates come from service data, bench tests, or demonstration campaigns, modeled as a Poisson process with constant rate lambda per flight hour.
• Point estimate: lambda_hat = n / T with n failures in T test-hours. Worked: 5 failures in 1,000,000 h gives 5e-6 per hour and an MTBF point value of 200,000 h.
• Exact Poisson upper confidence bound: lambda_upper = chi2(confidence, 2n + 2) / (2 T). Worked: n = 0, T = 1,000,000 h, 80 percent confidence gives chi2(0.80, 2) = 3.21888 over 2e6 = 1.60944e-6 per hour.
• Zero-failure rule: a test with zero failures demonstrates -ln(1 - confidence) / T. Worked: 1,000,000 h demonstrates 1.60944e-6 per hour at 80 percent, 2.30259e-6 at 90 percent, 0.91629e-6 at 60 percent confidence.
• Test-hours planning: T = chi2(confidence, 2n + 2) / (2 * target_rate) with n the allowed failures. Worked: a 1e-6 per hour rate at 80 percent confidence with zero allowed failures needs 1.60944e6 test-hours; at 95 percent with two allowed failures it needs 6.29579e6 test-hours.
• MTBF lower confidence bound: 2 T / chi2(confidence, 2n + 2). Worked: n = 1, T = 1,000,000 h, 95 percent gives 2e6 / 9.48773 = 210,797 h.
• Acceptance probability of a plan: P(X <= k) = Q(k + 1, lambda * T) with X Poisson. Worked: lambda * T = 1.0 gives P(0 failures) = 0.3679 and P(<= 1 failure) = 0.7358.
• Maximum confidence demonstrated by a zero-failure test of T hours against a target rate: 1 - exp(-target_rate * T). Worked: 1.60944e6 h against 1e-6 per hour demonstrates 80.0 percent confidence.
• Chi-square quantiles used above (standard tables): chi2(0.80, 2) = 3.21888, chi2(0.95, 2) = 5.99146, chi2(0.95, 4) = 9.48773, chi2(0.95, 6) = 12.59159.

Workflow

1. Collect the failure count n and the exposure test-hours T from test or service data; confirm the constant-rate (Poisson) assumption holds for the exposure period.
2. Compute the point estimates with point_estimate_failure_rate(n, T) and mtbf_estimate(n, T) (the latter requires n >= 1).
3. Derive the exact upper-bound with poisson_rate_upper_bound(n, T, confidence); for n = 0 cross-check against zero_failure_demonstrated_rate(T, confidence).
4. Plan the demonstration campaign with test_time_to_demonstrate( target_rate, confidence, allowed_failures).
5. Bound the MTBF with mtbf_lower_bound(n, T, confidence).
6. Check the acceptance probability of the chosen plan with poisson_cdf(rate, T, k) and state the demonstrated confidence with confidence_from_zero_failure_test(T, target_rate).
7. Feed the rate (prefer the conservative upper-bound) into the FTA/Markov models and compare against the PSSA/SSA probability requirement for the failure condition.

Pitfalls

• Routing here, not to markov-analysis: Markov analysis models a system with given transition rates, availability, and MTTF; this leaf produces or demonstrates the rates from data before any model runs.
• Routing here, not to fta-fmea: fault tree cut-set computation consumes rates as inputs; the statistical estimate of those rates is this leaf's job.
• Routing here, not to verification-planning: assigning the verification method (test, analysis, demonstration, inspection) to each requirement belongs to the ARP4754A verification-planning leaf; the confidence math behind a demonstrated rate claim is this leaf.
• Quoting the point estimate as the claimed rate: with n = 0 the point estimate is zero, which is never the demonstrated rate; use the zero-failure upper-bound instead.
• Quoting 1 / lambda_hat as a demonstrated MTBF without the chi-square lower-bound, which understates the uncertainty for small n.
• Swapping confidence levels: chi2(0.95, 2) = 5.99146 is 86 percent larger than chi2(0.80, 2) = 3.21888, so the required test-hours scale by the same factor.
• Confusing statistical confidence with reliability: "80 percent confidence" does not mean "80 percent probability of no failure".
• Applying the zero-failure rule to a repairable system that actually failed and was repaired during the exposure; that is the two-state repair model in markov-analysis.
• Planning with allowed_failures = 0 but accepting failures during the run: re-plan with the real allowed count, or the demonstration claim is void.
• Forgetting the constant-rate assumption: infant mortality and wear-out data violate the Poisson model and the chi-square bound is not valid.
• Double-counting evidence: the same test-hours cannot both size the estimate and serve as independent verification of the requirement.

Behavior contract (gate 3)

The estimation and demonstration logic is exercised by the gate 3 contract test: scripts/test_failure_rate_estimation.py against scripts/failure_rate_estimation_logic.py (stdlib unittest, offline). Run: python3 scripts/test_failure_rate_estimation.py

Compliance

• Standards referenced, not reproduced: ARP4761A quantitative analysis methods and ARP4754A verification context are proprietary (SAE); summary-only per standards-map.yaml and brief 06.
• The chi-square and Poisson bounds are common reliability statistics (Abramowitz and Stegun table 26.7 class methods), paraphrased; no verbatim standard text.
• compliance: STANDARDS-REF, gated: false.