system-reliability

name	system-reliability
description	System reliability engineering — fault tree analysis (AND/OR gates, minimal cut sets, Boolean reduction), event tree analysis, FMEA/FMECA (severity/occurrence/detection RPN), reliability block diagrams (series/parallel/k-of-n), Weibull system reliability, common cause failures (β-factor), MTTF/MTBF/availability, spare parts optimization, MIL-HDBK-217, IEC 60300, and RAMS analysis.
metadata	{"priority":7,"promptSignals":{"phrases":["system reliability","fault tree analysis","FMEA","reliability block diagram","MTTF MTBF","common cause failure"],"minScore":3}}

System Reliability Engineering — Complete Skill

Reliability Fundamentals

Reliability Function

Reliability R(t): probability that system performs intended function for duration t under stated conditions R(t) = P(T > t) = 1 − F(t) [F(t) = CDF; T = time to failure RV]

Failure rate λ(t): λ(t) = f(t) / R(t) [f(t) = PDF of failure; conditional failure rate at time t given survival to t] λ(t) constant → exponential distribution (memoryless; electronics baseline) λ(t) increasing → wear-out phase (mechanical fatigue, corrosion) λ(t) decreasing → infant mortality (manufacturing defects, burn-in)

Bathtub curve zones: Zone I (λ decreasing): t = 0 to t_1; infant mortality; screening/burn-in Zone II (λ constant): t_1 to t_2; useful life; exponential model valid Zone III (λ increasing): t > t_2; wear-out; Weibull β > 1

Mean Time Metrics

MTTF (Mean Time to Failure) — non-repairable: MTTF = ∫₀^∞ R(t) dt = E[T] [exact for any distribution] Exponential: MTTF = 1/λ [constant failure rate] Weibull: MTTF = (1/λ) × Γ(1 + 1/β) [Γ = gamma function]

MTBF (Mean Time Between Failures) — repairable: MTBF = MTTF + MTTR [MTTR = mean time to repair] For λ constant (MTTR << MTTF): MTBF ≈ MTTF = 1/λ

Availability (steady-state): A = MTBF / (MTBF + MTTR) = MTTF / (MTTF + MTTR) [fraction of time system is operational] Example: MTBF = 1,000 h; MTTR = 8 h → A = 1,000/(1,008) = 0.9921 (99.21%)

Weibull Distribution

Two-parameter Weibull: F(t) = 1 − exp(−(t/η)^β) [η = characteristic life (scale); β = shape parameter] R(t) = exp(−(t/η)^β) λ(t) = (β/η) × (t/η)^(β−1) [increasing if β > 1; constant if β = 1; decreasing if β < 1] MTTF = η × Γ(1 + 1/β) [Γ(1.5) = √π/2 ≈ 0.886]

Typical β values: β < 1: infant mortality (burn-in defects; electronic connections) β = 1: random failures (exponential; independent of age; electronics useful life) 1 < β < 2: early wear-out (rolling bearings, seals) β ≈ 3.5: fatigue (β = 3.5 mimics normal distribution) β > 4: extreme wear-out (abrasion, corrosion, creep)

Weibull probability plot: Transform: ln(ln(1/(1-F))) vs. ln(t) → straight line Slope = β (shape); intercept = −β×ln(η) (scale) Median rank: F(tᵢ) ≈ (i − 0.3)/(N + 0.4) [Benard's approximation; i = rank order]

Example: 10 components tested; failures at t = 50, 80, 120, 180, 240, 310, 400 hours (7 failures; 3 censored) Plot → slope β = 1.8; η = 280 h R(t = 100) = exp(−(100/280)^1.8) = exp(−0.137) = 0.872 (87.2%) MTTF = 280 × Γ(1.556) = 280 × 0.889 = 249 h

Reliability Block Diagram (RBD)

Series Configuration

All components must function (weakest-link): R_s(t) = Π_i R_i(t) [product of all component reliabilities]

Exponential components: R_s = exp(−Σᵢ λᵢ × t) = exp(−λ_s × t) [λ_s = Σλᵢ = sum of failure rates] MTTF_s = 1/λ_s = 1/(Σλᵢ)

Example: 3-component series: λ₁ = 100 FIT, λ₂ = 200 FIT, λ₃ = 50 FIT [1 FIT = 10⁻⁹ failures/hour] λ_s = 350 FIT; MTTF = 1/(350×10⁻⁹) = 2.857×10⁶ h = 326 years R_s(10 years = 87,600 h) = exp(−350e-9 × 87,600) = exp(−0.0307) = 0.9697 (96.97%)

Parallel (Redundant) Configuration

System fails only if ALL components fail: F_s(t) = Π_i F_i(t) [product of all component unreliabilities] R_s(t) = 1 − Π_i (1 − R_i(t))

Two identical parallel: R_s = 2R − R² = 1 − (1−R)² MTTF_s = MTTF × (1 + 1/2) = 1.5/λ [for 2 identical exponential components]

n identical parallel: MTTF_s = Σᵢ₌₁ⁿ (1/(i×λ)) [harmonic series; 3 identical → MTTF = 1/λ × (1 + 1/2 + 1/3) = 1.833/λ]

k-of-n (Voting) System

System works if at least k of n components work: R_s = Σⱼ₌ₖⁿ C(n,j) × R^j × (1−R)^(n−j) [binomial; identical components]

2-of-3 (majority vote; triplex): R_s = 3R² − 2R³ [fails if 2 or 3 fail] At R = 0.9: R_s = 3(0.81) − 2(0.729) = 2.430 − 1.458 = 0.972 (better than single 0.9; worse than 2-parallel 0.99)

MTTF_2oo3 = 5/(6λ) for identical exponential components

Complex Systems

Cut set method: Minimal cut set (MCS): minimum set of component failures causing system failure R_s = 1 − P(union of all MCS events) Upper bound (inclusion-exclusion first order): R_s ≈ 1 − Σᵢ P(MCSᵢ) [conservative; exact only if MCS independent]

Fault Tree Analysis (FTA)

Construction

Top event: undesired system event (e.g., "Pump fails to deliver fluid") Gates: AND (all inputs must occur); OR (any input suffices); INHIBIT; PRIORITY AND Basic events: individual component failures (assign failure probability)

AND gate: P(output) = Π_i P(inputᵢ) [intersection] OR gate: P(output) = 1 − Π_i (1 − P(inputᵢ)) ≈ Σ P(inputᵢ) for small probabilities

Minimal cut sets: Algorithm (Boolean reduction): convert FTA to cut sets → minimize by Boolean absorption Example: Top = A × (B + C) = AB + AC → MCS₁ = {A,B}; MCS₂ = {A,C}

Quantification: Each basic event assigned: q_i = 1 − exp(−λᵢ × t) or q_i = λᵢ × τ (constant demand failure) P(Top) = 1 − Π(1 − P(MCSᵢ)) or exact via binary decision diagram (BDD)

Importance measures: Birnbaum: I_B(i) = ∂P_s/∂q_i [marginal reliability importance; how much system reliability changes with component i reliability] Fussell-Vesely: I_FV(i) = P(any MCS containing i fails) / P(Top) [fraction of system failure probability involving component i] Critical Importance Factor (CIF): I_CIF = I_FV × q_i/(1−R_s)

FMEA and FMECA

FMEA Process (IEC 60812)

Worksheet columns:

1. Component/function
2. Failure mode (how it can fail: open circuit, short, corrosion, fracture, leakage...)
3. Failure cause (why: fatigue, overvoltage, wrong assembly, contamination...)
4. Failure effect (local → next higher level → system level)
5. Detection method (how failure is detected before/during operation)
6. Current controls (design features preventing/detecting failure)
7. Risk assessment: Severity (S), Occurrence (O), Detection (D)

Risk Priority Number (RPN): RPN = S × O × D [S: 1–10; O: 1–10; D: 1–10; RPN range 1–1,000]

Rating	Severity	Occurrence	Detection
1–2	No/minor effect	< 1 per 10⁶	Almost certain
3–4	Minor/low	1 per 100,000	High probability
5–6	Moderate	1 per 10,000	Moderate
7–8	High	1 per 1,000	Low
9–10	Catastrophic/safety	> 1 per 100	None

Action threshold: RPN > 100 or S ≥ 8 (any occurrence) → corrective action required AIAG standard: prioritize by S first, then O, then D; RPN alone insufficient

FMECA: adds Criticality Analysis Criticality number: C_i = β × α × λₚ × t [β = conditional probability of failure mode; α = failure mode ratio; λₚ = part failure rate; t = mission time]

Common Cause Failures (CCF)

Definition: failure of multiple redundant components due to single shared cause Examples: common environment (humidity, vibration), common manufacturing defect, common maintenance error, shared software bug

β-Factor Model

Two identical parallel components: q₁ = q₂ = q (independent failure probability) q_CCF = β × q [β = fraction of failures that are CCF; typical β = 0.05–0.20]

System unreliability with CCF: q_s = q² (independent contribution) + β × q (CCF contribution) For small q: q_s ≈ β × q [CCF dominates! redundancy provides little benefit if β > 0]

Typical β values: Safety instrumented systems: β = 0.02–0.10 (IEC 61511) Mechanical valves: β = 0.05–0.15 Electronic relays: β = 0.01–0.05

Example: Single component: q = 0.01; R_single = 0.99 Two parallel (independent): q_s = q² = 0.0001 → R = 0.9999 Two parallel with β = 0.05: q_s = 0.01² + 0.05×0.01 = 0.0001 + 0.0005 = 0.0006 → R = 0.9994 CCF reduces 100× improvement to 17× improvement

MGL (Multiple Greek Letters) model: extension for 3+ redundancy; β, γ, δ parameters

Event Tree Analysis (ETA)

Event tree: forward induction from initiating event through system responses Initiating event → safety system 1 response (success/fail) → safety system 2 → outcome

Quantification: Branch probability: conditional on all previous events P(outcome k) = P(IE) × Π_i P(branchᵢ) Expected frequency = Σ P(IE) × P(outcome) × consequence

Linking FTA and ETA: FTA supplies branch failure probabilities for each safety system in event tree Risk = Σ (initiating event frequency × conditional event tree probability × consequence severity)

Spare Parts and Maintenance Optimization

Poisson spare parts model: If failures arrive at rate λ (constant) over period T: N_failures ~ Poisson(λT) P(N_failures ≤ S) = Σₙ₌₀^S (λT)ⁿ × exp(−λT) / n! [S = stock level] Required stock S for target fill rate (service level) ≥ 0.95: find S from Poisson CDF

Optimal replacement interval (age replacement): Minimize total cost rate: C(tp) = [C_PM + C_CM × R(tp) / R(tp)] over tp For Weibull (β > 1): optimum tp* exists where marginal cost rate = average cost rate

CBM (Condition-Based Maintenance): monitor degradation indicator; replace when threshold exceeded Reduces unnecessary PM; extends life; requires sensor and prognostics

Standards and References

Standard	Scope
IEC 60300-3-1	Dependability management: reliability analysis techniques
IEC 60812	FMEA procedure
IEC 61025	FTA standard
MIL-HDBK-217F	Electronic component failure rates (FIT values)
MIL-STD-1629A	FMECA for military systems
IEC 61511	Functional safety — SIS (CCF β-factor requirements)
NUREG-0492	NRC fault tree handbook
ReliaSoft Weibull++	Software for Weibull analysis and RBD

Output

Provide: system description (function; components; success criteria; mission time t [h]; operating environment), reliability model type (RBD series/parallel/k-of-n; FTA; or both — state basis for choice), component failure rates (λᵢ [FIT or /h] from MIL-HDBK-217/field data; Weibull β and η if wear-out; data source: handbook/field/test), system reliability calculation (formula + step-by-step result; R_s at mission t = [value]; R_s with redundancy = [value]), MTTF/MTBF (MTTF = 1/λ_s [h]; MTBF with repair = [h]; availability A = MTTF/(MTTF+MTTR) = [%]), fault tree minimal cut sets (list top 5 MCS; Birnbaum importance I_B for each; dominant contributors), FMECA (top 5 highest-criticality failure modes; RPN; recommended action; revised RPN after action), CCF (β factor applied; q_s with and without CCF; residual risk after redundancy [%]; CCF mitigation: diversity/separation/procedure), maintenance strategy (optimal replacement interval tp [h]; spare parts stock S for 95% service level over [T] hours; PM vs. CM cost ratio), and applicable standard (IEC 60300/60812/61025; MIL-HDBK-217 for failure rates; IEC 61511 for CCF in safety systems).