Reliability Functions and Failure Analysis — Risk Theory & Actuarial Math

Modern technical systems—airplanes, nuclear reactors, medical equipment—contain millions of components. Failure of any one can cost lives and billions of dollars. Reliability theory studies the probability of failure-free operation and offers quantitative methods for evaluating and designing systems with a specified level of safety. It is applied in aviation (DO-254), nuclear energy (IAEA SSG-3), automotive industry (ISO 26262), medicine (IEC 60601). Without it, civil aviation (catastrophe probability 10⁻⁹ per flight hour) and nuclear power would be impossible.

Basic Reliability Functions

For a component with time to failure $T$ (a random variable):

Reliability function: $R(t) = P(T > t)$ — the probability of failure-free operation up to moment $t$. Analogous to the survival function $S(x)$ in actuarial mathematics. $R(0) = 1$, $R(\infty) = 0$.

Failure distribution function: $F(t) = 1 − R(t) = P(T \leq t)$.

Failure density: $f(t) = F'(t) = −R'(t)$.

Hazard rate (failure rate): $h(t) = \dfrac{f(t)}{R(t)} = -\dfrac{d \ln R(t)}{dt}$.

Interpretation of $h(t)$: the instantaneous probability of failure at time $t$, given that the component has survived up to $t$. Analogous to $\mu(x)$ in survival theory. Inverse relationship: $R(t) = \exp\left(-\int_0^t h(s); ds\right)$.

Bathtub Curve

Real technical components often have $h(t)$ with three characteristic periods:

Infant mortality: $h(t)$ decreases. Hidden manufacturing defects manifest and are eliminated. Lasts days to months. Solution: "burn-in test"—the manufacturer tests components, rejecting defective ones.
Useful life: $h(t) \approx$ const $= \lambda$. Random failures (lightning, power surges, human factor). Lasts years.
Wear-out: $h(t)$ increases. Physical wear—metal fatigue, corrosion, electrolyte leakage. Solution: preventive replacement.

Parametric Models

1. Exponential: $R(t) = e^{−\lambda t}$, $h(t) = \lambda = \text{const}$.
Corresponds to "useful life." Memoryless: $P(T > s + t\mid T > s) = P(T > t)$.
MTBF (Mean Time Between Failures): $E[T] = 1/\lambda$.
Example: processor with $\lambda = 10^{-5}$/hour $\to$ MTBF $= 100,!000$ hours $\approx 11$ years.

2. Weibull: $R(t) = \exp\left(-\left(\dfrac{t}{\eta}\right)^\beta\right)$, $h(t) = \dfrac{\beta}{\eta}\left(\dfrac{t}{\eta}\right)^{\beta-1}$.

$\beta < 1$: $h$ decreases (infant mortality).
$\beta = 1$: exponential (useful life).
$\beta > 1$: $h$ increases (wear-out).
$\beta = 2$: Rayleigh distribution—linear growth of $h$.

$\eta$ — scale (characteristic time), $\beta$ — shape. Flexibility makes Weibull the reliability standard.

3. Gompertz: $R(t) = \exp\left(-\dfrac{B}{c}(c^t-1)\right)$. Analogous to actuarial model—for biological systems and humans.

4. Lognormal: $\ln T \sim N(\mu, \sigma^2)$. Used for failures due to metal fatigue (Bazovsky’s law).

Structural System Reliability

Complex systems consist of components. The structure of their connection determines system reliability $R_s$.

Series system. All components must work (failure of any $\to$ system failure):
$R_s = \prod_i R_i$.

Example: chain of 5 components with $R_i = 0.99$: $R_s = 0.99^5 = 0.951$. Each extra link reduces reliability.

Parallel system. Sufficient for at least one to work:
$R_s = 1 − \prod_i(1 − R_i)$.

Example: 3 parallel components with $R = 0.9$: $R_s = 1 − 0.1^3 = 0.999$. Redundancy is the main method to increase reliability.

k-out-of-n. System works if at least $k$ out of $n$ identical components work:
$R_s = \sum_{j=k}^n C(n, j) \cdot R^j \cdot (1 − R)^{n−j}$.

Example: 2-of-3 with $R = 0.95$: $R_s = 3 \cdot 0.95^2 \cdot 0.05 + 0.95^3 = 0.135 + 0.857 = 0.993$.

Fault Tree Analysis (FTA) and Reliability Block Diagrams (RBD)

FTA — fault tree analysis. Deductive method: top (undesirable) event $\to$ intermediate causes $\to$ basic events (component failures).

Logic gates:

AND: top event requires all inputs (product of probabilities).
OR: at least one input is sufficient.

Minimal cut sets. Minimal combinations of basic events leading to top event. If $n$ cut sets, each requires $k_j$ components: $P(\text{top}) \approx \sum_j \prod_{i \in S_j} q_i$ (for small $q$).

RBD — reliability block diagrams. Graphical representation of system structure in terms of series/parallel connected blocks. Equivalent to FTA, but more convenient for calculating $R_s$.

Numerical Example

System: A and B in series, C, D, E in parallel, then (AB) and (CDE) in parallel.
$R_A = 0.98$, $R_B = 0.95$, $R_C = R_D = R_E = 0.90$.

$R_{AB} = 0.98 \times 0.95 = 0.931$.
$R_{CDE} = 1 - (1 - 0.9)^3 = 1 - 0.001 = 0.999$.
$R_\text{system} = 1 - (1 - 0.931)\times (1 - 0.999) = 1 - 0.069 \times 0.001 = 0.99993$.

Importance analysis (Birnbaum importance): $I_B(i) = R_s | R_i = 1 - R_s | R_i = 0$.
For component A: $R_s | R_A = 1 = 1 - (1 - 0.95)\times 0.001 \approx 0.99995$;
$R_s | R_A = 0 = 1 - 1 \times 0.001 = 0.999$.
$I_B(A) \approx 0.001$—low importance (duplicated by parallel CDE branch).
For C: $I_B(C) \approx 0.069 \times 0.01 = 0.00069$.
The most important are A and B (series link determines operation of the AB branch, though the system still functions via CDE).

Real-World Applications

Aviation. Boeing 787, Airbus A350: each critical system (flight control, hydraulics, avionics) has triple redundancy. Calculated probability of FCS failure
lt;$ 10⁻⁹/hour.
Nuclear energy. WANO benchmark: probability of core meltdown
lt;$ $10^{-5}$/reactor-year. After Fukushima—revised standards, additional safety systems.
Automotive industry. ISO 26262 ASIL D (for airbag, ABS): requirement $R$(10 years)
gt; 1 - 10^{-9}$. Verified by FMEDA + FTA.
Medicine. Pacemakers (Medtronic, St. Jude): MTBF
gt;$ 8 years, calculated MTTF
gt;$ 12 years. Tested by accelerated life testing.
IT infrastructure. AWS, Azure SLA 99.99% ($\approx 53$ min downtime/year). Achieved by redundancy of data centers (multi-AZ deployment).

Assignment. System: 5 components in configuration from the problem: A-B in series, then in parallel with C-D-E (which are also parallel among themselves). $R_A = 0.98$, $R_B = 0.95$, $R_C = R_D = R_E = 0.90$. (a) Build the RBD. (b) Calculate $R_\text{system}$. (c) Find Birnbaum importance of each component. (d) Which component is most critical for improving $R_\text{system}$? (e) If the budget allows improving one component to $R = 0.99$, which would you choose to maximize $R_\text{system}$?