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Reliability Growth Analysis
Introduction
Reliability Growth Analysis (RGA) is a vital
methodology for evaluating how the reliability of systems or products
improves over time, particularly during development and testing phases.
By monitoring failures and implementing corrective actions, engineers
can track the rate of reliability improvement and forecast future
performance.
RGA is especially useful in scenarios where iterative enhancements
are made—such as design updates or process optimizations—with the
ultimate goal of achieving a mature, dependable product.
The Crow-AMSAA Model
The Crow-AMSAA introduced by Larry Crow in 1975, is
a widely adopted model in reliability engineering. The model takes
failure behavior as a Non-Homogeneous Poisson Process (NHPP) governed by
a power law, making it particularly effective for systems undergoing
reliability growth due to continuous improvements.
Non-Homogeneous Poisson Process:
In an NHPP, the rate of failures is not constant over time—it changes
as improvements are made. While the individual failure events still
follow a Poisson distribution, the intensity (or rate) of failures is
time-dependent. This characteristic allows the Crow-AMSAA model to
capture both reliability growth (decreasing failure rate) and
degradation (increasing failure rate).
The Power Law Model:
The failure intensity (rate of failures per time unit) is modeled as
a power function of time:
\[
\lambda(t) = \beta \cdot t^{\beta - 1}
\]
where:
- \(\lambda(t)\) is the failure
intensity at time \(t\),
- \(\beta\) is the shape
parameter,
- \(t\) is the time.
The cumulative number of failures up to time \(t\) is given by:
\[
N(t) = \lambda_0 \cdot t^{\beta}
\]
Where:
- \(N(t)\) is the cumulative number
of failures,
- \(\lambda_0\) is the scale
parameter.
Parameters:
The Shape Parameter (\(\beta\))
indicates whether the system is improving or deteriorating:
If \(\beta\) > 1, failures
are increasing over time, indicating that reliability is
worsening.
If \(\beta\) < 1, failures
are decreasing, indicating that reliability is improving over
time.
\(\beta\) = 1 implies a constant
failure rate (no growth or degradation).
The Scale Parameter (\(\lambda_0\)) is related to the initial
failure rate.
Example
First, load the package:
Next, set up some cumulative time and failure data:
times <- c(100, 200, 300, 400, 500)
failures <- c(1, 2, 1, 3, 2)
Then run the rga and plot the results:
result <- rga(times, failures)
plot_rga(result)
The Piecewise Weibull NHPP Model
The Piecewise Weibull NHPP model, developed by Guo
et al. (2010), extends the traditional NHPP by allowing different time
segments to follow their own Weibull failure distributions. This method
is especially useful when failure behavior shifts across phases of
development — such as early, middle, and late stages — where the
underlying reliability dynamics may differ.
The Weibull Distribution
The Weibull intensity function for each time segment \(i\) is modeled as:
\[
\lambda_i(t) = \frac{\beta_i}{\eta_i} \left( \frac{t - t_{i-1}}{\eta_i}
\right)^{\beta_i - 1}
\]
Where:
- \(\lambda_i(t)\) is the failure
intensity in time interval \(i\),
- \(\beta_i\) is the shape parameter
for interval \(i\),
- \(\eta_i\) is the scale parameter
(characteristic life) for interval \(i\),
- \(t_{i-1}\) is the start time of
the interval.
Example
First, set up some cumulative time/failure data and specify the
breakpoint:
times <- c(25, 55, 97, 146, 201, 268, 341, 423, 513, 609, 710, 820, 940, 1072, 1217)
failures <- c(1, 1, 2, 4, 4, 1, 1, 2, 1, 4, 1, 1, 3, 3, 4)
breakpoints <- 500
Then run the rga and plot the results:
result <- rga(times, failures, model_type = "Piecewise Weibull NHPP", breakpoints = breakpoints)
plot_rga(result)
The Piecewise Weibull NHPP with Change Point Detection
This enhanced version of the Piecewise Weibull NHPP model
incorporates change point detection, which identifies
time points where the underlying failure process shifts. Rather than
manually specifying breakpoints, this model automatically segments the
data based on significant changes in reliability behavior using
statistical techniques.
Example
First, set up some cumulative time and failure data:
times <- c(25, 55, 97, 146, 201, 268, 341, 423, 513, 609, 710, 820, 940, 1072, 1217)
failures <- c(1, 1, 2, 4, 4, 1, 1, 2, 1, 4, 1, 1, 3, 3, 4)
Then run the rga and plot the results:
result <- rga(times, failures, model_type = "Piecewise Weibull NHPP")
plot_rga(result)
The Duane Model
The Duane Model, introduced by J.T. Duane in 1964,
is a graphical technique for evaluating reliability growth. The model is
based on the assumption that reliability improves over time and
visualizes this via a log-log plot of cumulative failure rate (or mean
time between failures, MTBF) versus time.
Example
First, set up some cumulative time and failure data:
times <- c(100, 200, 300, 400, 500)
failures <- c(1, 2, 1, 3, 2)
Then plot the results:
fit <- duane_plot(times, failures)
Summary
Reliability Growth Analysis (RGA) provides a powerful statistical
framework for tracking and predicting how a system’s reliability evolves
throughout development. Models like Crow-AMSAA, Piecewise Weibull NHPP,
and the Duane Model allow engineers and analysts to quantify
improvements (or regressions) in reliability, understand system behavior
across development phases, detect significant shifts in failure
patterns, and guide design changes and process improvements.
These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.
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