Repairable Systems Analysis • ReliaPlotR

To run a repairable systems analysis, start by loading ReliaGrowR and ReliaPlotR:

library(ReliaGrowR)
library(ReliaPlotR)

The examples below use a fleet of five field units. Each row records which unit failed and when:

id    <- c("U1","U1","U1","U2","U2","U3","U3","U3","U4","U5","U5")
time  <- c( 120, 310, 560, 200, 480, 95, 280, 430, 370, 150, 490)

Mean Cumulative Function

The Mean Cumulative Function (MCF) is a non-parametric estimate of the expected cumulative number of failures per system by time t. It is computed using the Nelson-Aalen estimator, which accounts for systems that are still operating (censored) at the end of the observation window.

end_times <- c(U1 = 600, U2 = 600, U3 = 600, U4 = 600, U5 = 600)
fit_mcf   <- mcf(id = id, time = time, end_time = end_times)
plotly_mcf(fit_mcf)

The shaded band shows the pointwise confidence interval. Hover over the step function to read exact MCF values at each event time.

Exposure Analysis

The exposure plot tracks the cumulative event rate — total failures divided by total system-time — over the observation period. A rising rate indicates a deteriorating fleet; a flat or falling rate indicates stability or improvement.

fit_exp <- exposure(id = id, time = time)
plotly_exposure(fit_exp)

NHPP — Power Law Model

The Non-Homogeneous Poisson Process (NHPP) fits a parametric model to the recurrent failure process. The Power Law model (also called the Crow-AMSAA model for repairable systems) describes how the failure intensity changes over time:

$\lambda(t) = \lambda \beta \, t^{\beta - 1}$

A shape parameter $\beta < 1$ indicates reliability improvement; $\beta > 1$ indicates deterioration; $\beta = 1$ reduces to a homogeneous Poisson process.

times  <- c(120, 200, 310, 370, 430, 480, 490, 560)
events <- c(  1,   1,   1,   1,   1,   1,   1,   1)
fit_nhpp <- nhpp(time = times, event = events, model_type = "Power Law")
plotly_nhpp(fit_nhpp)

NHPP — Piecewise Model

When a corrective action or design change is introduced mid-program, the failure process may shift. A piecewise NHPP fits separate Power Law segments on either side of a specified breakpoint, shown as a vertical dotted line:

# Dense failures before the design change (t < 500), sparse afterwards
times2  <- c(30, 65, 105, 150, 200, 255, 315, 380, 450, 800, 1300, 1950, 2800)
events2 <- rep(1, 13)
fit_nhpp2 <- nhpp(time = times2, event = events2, breaks = 500, method = "LS")
plotly_nhpp(fit_nhpp2, fitCol = "steelblue", confCol = "steelblue", breakCol = "red")

Overlay

Multiple MCF or NHPP estimates can be overlaid on the same plot by passing a list of objects. This is useful for comparing two fleets, two time periods, or a before/after scenario.

id2   <- c("A","A","B","B","B","C")
time2 <- c(180, 420, 90, 310, 530, 260)
end2  <- c(A = 600, B = 600, C = 600)
fit_mcf2 <- mcf(id = id2, time = time2, end_time = end2)

plotly_mcf(list(fit_mcf, fit_mcf2), cols = c("steelblue", "tomato"))