Glossary and Formula Reference

A quick-reference guide to the key symbols and formulas used throughout this book. Each entry links back to the chapter where it is first introduced. Symbols are listed in order of first introduction.

Key Symbols

Symbol	Name	Definition	First used
$\beta$	Shape parameter	Controls whether the failure rate increases, decreases, or stays constant over time	1 Reliability, Availability, and Maintainability (RAM)
$\eta$	Scale parameter (characteristic life)	Time at which $F(t) = 63.2\%$ for any $\beta$	3 Life Data Analysis
$\lambda$	Failure rate	Failures per unit time (constant rate — exponential model)	1 Reliability, Availability, and Maintainability (RAM)
$\rho(t)$	Intensity / recurrence rate	Time-varying failure rate for repairable systems	6 Repairable Systems Analysis
$R(t)$	Reliability function	Probability of survival to time $t$	1 Reliability, Availability, and Maintainability (RAM)
$F(t)$	Cumulative distribution function	Probability of failure by time $t$; $F(t) = 1 - R(t)$	3 Life Data Analysis
MTTF	Mean Time To Failure	Expected life for non-repairable items	1 Reliability, Availability, and Maintainability (RAM)
MTBF	Mean Time Between Failures	Expected time between failures for repairable systems	1 Reliability, Availability, and Maintainability (RAM)
MTTR	Mean Time To Repair	Expected time to restore a failed system	1 Reliability, Availability, and Maintainability (RAM)
$AF$	Acceleration Factor	Ratio of life at use conditions to life at elevated stress	5 Accelerated Life Testing
$E_a$	Activation energy	Energy barrier for thermally activated failure (eV)	5 Accelerated Life Testing
$k$	Boltzmann constant	$8.617 \times 10^{-5}$ eV/K	5 Accelerated Life Testing
$n$	Stress exponent	Sensitivity of life to non-thermal stress (Power Law)	5 Accelerated Life Testing
$t_0$	Failure-free period	Minimum time before any failure can occur (3-parameter Weibull)	3 Life Data Analysis
AD	Anderson-Darling statistic	Goodness-of-fit measure; lower values indicate a better fit	3 Life Data Analysis
MCF	Mean Cumulative Function	Expected cumulative failures per system over time (non-parametric)	6 Repairable Systems Analysis
$\beta_{\text{CCF}}$	CCF beta-factor	Fraction of total failure rate attributable to common cause failures	2 Reliability Block Diagrams

Key Formulas

RAM — 1 Reliability, Availability, and Maintainability (RAM)

\[R = 1 - \frac{\text{failed time}}{\text{total time}}\]

\[A = 1 - \frac{\text{unavailable time}}{\text{total time}} = \frac{\text{MTTF}}{\text{MTTF} + \text{MTTR}}\]

\[\lambda = \frac{\text{failures}}{\text{total time}}, \qquad \text{MTTF} = \frac{1}{\lambda}, \qquad \text{MTBF} = \frac{1}{\lambda}\]

\[R(t) = e^{-\lambda t} \qquad \text{(exponential model)}\]

\[B_n \text{ life}: \text{time } t \text{ such that } F(t) = n/100\]

Weibull Distribution — 3 Life Data Analysis

\[R(t) = \exp\!\left[-\left(\frac{t}{\eta}\right)^{\!\beta}\right], \qquad F(t) = 1 - R(t)\]

\[F(\eta) = 1 - e^{-1} \approx 63.2\% \quad \text{for any } \beta\]

3-parameter Weibull with failure-free period $t_0$:

\[R(t) = \exp\!\left[-\left(\frac{t - t_0}{\eta}\right)^{\!\beta}\right]\]

Reliability Block Diagrams — 2 Reliability Block Diagrams

\[R_{\text{series}} = \prod_{i=1}^{n} R_i\]

\[R_{\text{parallel}} = 1 - \prod_{i=1}^{n}(1 - R_i)\]

\[R_{\text{k-out-of-n}} = \sum_{i=k}^{n} \binom{n}{i} R^i (1-R)^{n-i}\]

Beta-factor CCF model:

\[\lambda_{\text{CCF}} = \beta_{\text{CCF}} \cdot \lambda_{\text{total}}, \qquad \lambda_{\text{ind}} = (1 - \beta_{\text{CCF}}) \cdot \lambda_{\text{total}}\]

Accelerated Life Testing — 5 Accelerated Life Testing

Arrhenius (temperature):

\[AF = \exp\!\left[\frac{E_a}{k}\left(\frac{1}{T_{\text{use}}} - \frac{1}{T_{\text{stress}}}\right)\right]\]

Power Law (non-thermal stress):

\[AF = \left(\frac{S_{\text{stress}}}{S_{\text{use}}}\right)^{n}\]

\[\text{Life}_{\text{use}} = AF \times \text{Life}_{\text{stress}}\]

Reliability Growth Analysis — 4 Reliability Growth Analysis

Duane cumulative MTBF ($K$ = scale parameter estimated from data):

\[\text{CMTBF}(t) = K \cdot t^{\beta - 1}\]

Crow-AMSAA cumulative failures ($\lambda_0$ = scale parameter; $\beta$ = shape parameter):

\[N(t) = \lambda_0 \cdot t^{\beta}\]

$\beta$ (Crow-AMSAA)	Meaning
$< 1$	Failures decreasing — reliability improving
$= 1$	Constant rate — stable
$> 1$	Failures increasing — reliability worsening

Warning

Note: Crow-AMSAA $\beta$ and Duane $\beta$ have opposite interpretations — Duane $\beta > 1$ means improving, Crow-AMSAA $\beta > 1$ means worsening. See 4 Reliability Growth Analysis for details.

Repairable Systems — 6 Repairable Systems Analysis

Power Law intensity (recurrence rate):

\[\rho(t) = \lambda \beta t^{\beta - 1}\]

Instantaneous MTBF at time $t$:

\[\text{MTBF}(t) = \frac{1}{\rho(t)}\]

# Glossary and Formula Reference {#sec-glossary .unnumbered} A quick-reference guide to the key symbols and formulas used throughout this book. Each entry links back to the chapter where it is first introduced. Symbols are listed in order of first introduction. ## Key Symbols | Symbol | Name | Definition | First used | |:---:|---|---|:---:| | $\beta$ | Shape parameter | Controls whether the failure rate increases, decreases, or stays constant over time | @sec-ram | | $\eta$ | Scale parameter (characteristic life) | Time at which $F(t) = 63.2\%$ for any $\beta$ | @sec-lda | | $\lambda$ | Failure rate | Failures per unit time (constant rate — exponential model) | @sec-ram | | $\rho(t)$ | Intensity / recurrence rate | Time-varying failure rate for repairable systems | @sec-rs | | $R(t)$ | Reliability function | Probability of survival to time $t$ | @sec-ram | | $F(t)$ | Cumulative distribution function | Probability of failure by time $t$; $F(t) = 1 - R(t)$ | @sec-lda | | MTTF | Mean Time To Failure | Expected life for non-repairable items | @sec-ram | | MTBF | Mean Time Between Failures | Expected time between failures for repairable systems | @sec-ram | | MTTR | Mean Time To Repair | Expected time to restore a failed system | @sec-ram | | $AF$ | Acceleration Factor | Ratio of life at use conditions to life at elevated stress | @sec-alt | | $E_a$ | Activation energy | Energy barrier for thermally activated failure (eV) | @sec-alt | | $k$ | Boltzmann constant | $8.617 \times 10^{-5}$ eV/K | @sec-alt | | $n$ | Stress exponent | Sensitivity of life to non-thermal stress (Power Law) | @sec-alt | | $t_0$ | Failure-free period | Minimum time before any failure can occur (3-parameter Weibull) | @sec-lda | | AD | Anderson-Darling statistic | Goodness-of-fit measure; lower values indicate a better fit | @sec-lda | | MCF | Mean Cumulative Function | Expected cumulative failures per system over time (non-parametric) | @sec-rs | | $\beta_{\text{CCF}}$ | CCF beta-factor | Fraction of total failure rate attributable to common cause failures | @sec-rbd | ## Key Formulas ### RAM — @sec-ram $$R = 1 - \frac{\text{failed time}}{\text{total time}}$$ $$A = 1 - \frac{\text{unavailable time}}{\text{total time}} = \frac{\text{MTTF}}{\text{MTTF} + \text{MTTR}}$$ $$\lambda = \frac{\text{failures}}{\text{total time}}, \qquad \text{MTTF} = \frac{1}{\lambda}, \qquad \text{MTBF} = \frac{1}{\lambda}$$ $$R(t) = e^{-\lambda t} \qquad \text{(exponential model)}$$ $$B_n \text{ life}: \text{time } t \text{ such that } F(t) = n/100$$ --- ### Weibull Distribution — @sec-lda $$R(t) = \exp\!\left[-\left(\frac{t}{\eta}\right)^{\!\beta}\right], \qquad F(t) = 1 - R(t)$$ $$F(\eta) = 1 - e^{-1} \approx 63.2\% \quad \text{for any } \beta$$ 3-parameter Weibull with failure-free period $t_0$: $$R(t) = \exp\!\left[-\left(\frac{t - t_0}{\eta}\right)^{\!\beta}\right]$$ --- ### Reliability Block Diagrams — @sec-rbd $$R_{\text{series}} = \prod_{i=1}^{n} R_i$$ $$R_{\text{parallel}} = 1 - \prod_{i=1}^{n}(1 - R_i)$$ $$R_{\text{k-out-of-n}} = \sum_{i=k}^{n} \binom{n}{i} R^i (1-R)^{n-i}$$ Beta-factor CCF model: $$\lambda_{\text{CCF}} = \beta_{\text{CCF}} \cdot \lambda_{\text{total}}, \qquad \lambda_{\text{ind}} = (1 - \beta_{\text{CCF}}) \cdot \lambda_{\text{total}}$$ --- ### Accelerated Life Testing — @sec-alt Arrhenius (temperature): $$AF = \exp\!\left[\frac{E_a}{k}\left(\frac{1}{T_{\text{use}}} - \frac{1}{T_{\text{stress}}}\right)\right]$$ Power Law (non-thermal stress): $$AF = \left(\frac{S_{\text{stress}}}{S_{\text{use}}}\right)^{n}$$ $$\text{Life}_{\text{use}} = AF \times \text{Life}_{\text{stress}}$$ --- ### Reliability Growth Analysis — @sec-rt Duane cumulative MTBF ($K$ = scale parameter estimated from data): $$\text{CMTBF}(t) = K \cdot t^{\beta - 1}$$ Crow-AMSAA cumulative failures ($\lambda_0$ = scale parameter; $\beta$ = shape parameter): $$N(t) = \lambda_0 \cdot t^{\beta}$$ | $\beta$ (Crow-AMSAA) | Meaning | |:---:|---| | $< 1$ | Failures decreasing — reliability improving | | $= 1$ | Constant rate — stable | | $> 1$ | Failures increasing — reliability worsening | ::: {.callout-warning} **Note**: Crow-AMSAA $\beta$ and Duane $\beta$ have **opposite** interpretations — Duane $\beta > 1$ means *improving*, Crow-AMSAA $\beta > 1$ means *worsening*. See @sec-rt for details. ::: --- ### Repairable Systems — @sec-rs Power Law intensity (recurrence rate): $$\rho(t) = \lambda \beta t^{\beta - 1}$$ Instantaneous MTBF at time $t$: $$\text{MTBF}(t) = \frac{1}{\rho(t)}$$