Reliability Growth⚠ unverified

Understanding the Parameters

• Growth exponent $\beta$ — the key parameter. In the Duane framework the cumulative failure rate falls as $t^{\beta - 1}$, so the growth rate is often written as $1 - \beta$ (or $-m$ in slope terms). A larger growth exponent means faster improvement:

• Effective TAAF programmes achieve growth slopes around $0.3$–$0.5$ (steady debugging).

• A slope near $0$ means no growth — failures accumulate linearly, the design is not improving.

• Values above $\sim0.6$ are rarely sustained (they imply near-heroic fix effectiveness). The exponent is read directly as the slope of the log–log Duane plot.

• Scale coefficient $\alpha$ — sets the vertical position of the line: the early failure level before growth takes effect. It reflects the initial (immature) reliability and the test conditions.

• Cumulative test time $t$ — total accrued test hours, the horizontal axis. Reliability growth is charged against cumulative testing: the more you test and fix, the more the design matures.

• The output $N(t)$ — the cumulative failures expected by test time $t$ (or a derived intensity). Its slope on log–log axes, not its absolute value, is what tells you whether and how fast reliability is growing.

Derivation (Approaching a Proof)

J. T. Duane (1964) observed empirically that when the cumulative failure rate of a system under development is plotted against cumulative operating time on log–log paper, the data fall on a straight line. A straight line on log–log axes is a power law: if $\ln(\text{cumulative failure rate}) = \ln k - m\ln t$, then the cumulative failure rate is $C(t)/t \propto t^{-m}$, so the cumulative number of failures is

$$N(t) = C(t) = k\,t^{1 - m} = \alpha\,t^{\beta}, \qquad \beta = 1 - m. \qquad\blacksquare$$

Here $m$ is the growth slope (larger = faster improvement) and $\beta = 1 - m$ is the exponent in this calculator's form. The instantaneous failure rate is the derivative $dN/dt = \alpha\beta\,t^{\beta - 1}$, which declines when $\beta < 1$ ($m > 0$) — the signature of a design whose reliability is improving. The continuous statistical version of this idea is the Crow–AMSAA (NHPP power-law) model, which puts the Duane observation on a rigorous footing for confidence bounds and projections.

Dimensional check. With $\alpha$ absorbing the units, $N(t) = \alpha t^\beta$ is a count (dimensionless) when $\alpha$ has units of (failures)/(time$^\beta$); the physically meaningful output is the slope $\beta$, which is dimensionless. $\checkmark$

History and Development

• Duane, 1964. J. T. Duane of General Electric published the empirical log–log linearity of cumulative failure rate versus time across several development programmes — the first quantitative reliability-growth model, giving managers a way to track and project maturing reliability.

• Crow–AMSAA. Larry Crow (US Army Materiel Systems Analysis Activity) recast Duane's observation as a non-homogeneous Poisson process with a power-law intensity, providing statistical rigour, goodness-of-fit tests, and confidence intervals — now the standard reliability-growth analysis (MIL-HDBK-189).

• Test–analyse–fix. Reliability growth formalises the TAAF development philosophy: test to failure, find root causes, fix them, and repeat, driving the failure intensity down along the growth curve. It connects development testing to the in-service reliability models (Exponential Reliability, Weibull Reliability) that describe the matured design.

Related Concepts: Exponential Reliability, Weibull Reliability, Bathtub Curve, Failure Rate From Mtbf, Mtbf From Failure Rate, Availability

Notes: Registry calculator reliability-growth (unverified). Duane model $N(t) = \alpha t^\beta$ — correct as shipped; models a design improving through test-fix (not a fixed design). Growth slope $m = 1 - \beta$; instantaneous rate $\propto t^{\beta-1}$ declines for $\beta<1$. Rigorous form = Crow–AMSAA (NHPP power law). The slope is the meaningful output. Registry labels $t$ dimensionless (is a time).