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Reliability Growth Analysis
Reliability Growth Analysis (RGA) is a method to assess the
reliability of systems or products during the development and testing
phases. RGA tracks the rate at which reliability improves over time,
typically as design changes, fixes, or process improvements are
implemented. The primary objective is to measure the system or product’s
failure behavior and predict how it will perform once the product is
fully mature or operational.
The Crow-AMSAA Model
The Crow-AMSAA or Non-Homogeneous Poisson
Process (NHPP) model is a statistical model used to track
reliability improvements over time. This model is particularly useful
for analyzing failure data in systems where the failure rate changes due
to improvements or corrective actions during the development or
operational phase.
The Non-Homogeneous Poisson Process (NHPP):
In a NHPP, the failure rate changes over time, increasing or
decreasing depending on actions taken to improve reliability. Failures
follow a Poisson process, but the rate at which they occur changes with
time, hence “non-homogeneous.”
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 is an extension of
the standard NHPP model that includes different segments or phases of
time that follow separate Weibull failure distributions. This model is
particularly useful when a system experiences changes in failure
behavior over different development phases, such as the initial, interim
and final phases of a development process.
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(100, 200, 300, 400, 500, 600, 700, 800, 900, 1000)
failures <- c(1, 2, 1, 1, 1, 2, 3, 1, 2, 4)
breakpoints <- 400
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
The Piecewise Weibull NHPP with Change Point
Detection is an advanced model to identify changes in failure
behavior and model system reliability. This method builds on the
Piecewise Weibull NHPP model by introducing the concept of change
points, which represent the time when the underlying failure behavior
changes. Detection of change points involves statistical techniques that
analyze failure data to automatically identify when the behavior
changes, allowing for a more precise segmentation of the model into
different Weibull distributions.
Example
First, set up some cumulative time and failure data:
times <- c(100, 200, 300, 400, 500, 600, 700, 800, 900, 1000)
failures <- c(1, 2, 1, 1, 1, 2, 3, 1, 2, 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 is another graphical method to track
and visualize the reliability improvement of a product or system over
time. The Duane Plot is a log-log plot that shows the cumulative failure
rate or Mean Time Between Failures (MTBF) versus time, helping to assess
whether reliability is improving, stabilizing, or worsening over
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
RGA provides a quantitative framework for understanding how a
system’s reliability changes over time and guides decisions for further
improvements based on observed failure patterns.
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.