Reliability Growth Analysis

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:

library(ReliaGrowR)

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 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)

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)

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)