6 Repairable Systems Analysis

6.1 Introduction

Unlike life data analysis (Chapter 3), which focuses on the time to first failure of non-repairable components, Repairable Systems Analysis (Ascher and Feingold 1984; Meeker and Escobar 1998) models the entire sequence of recurring failures on a single system or fleet. This chapter covers the foundational theory and the ReliaGrowR tools for fitting models, computing fleet-level exposure, and estimating the Mean Cumulative Function.

6.2 Learning Objectives

By the end of this chapter, you will be able to:

Define the distinction between repairable and non-repairable systems.
Explain the Homogeneous Poisson Process (HPP) and Non-Homogeneous Poisson Process (NHPP).
Describe the Power Law Process and its parameters.
Fit and interpret the Crow-AMSAA model using nhpp() from ReliaGrowR.
Apply the Piecewise NHPP model to detect changes in failure behavior.
Compute fleet-level exposure using exposure().
Estimate the Mean Cumulative Function (MCF) using mcf().
Forecast future failure counts from a fitted Power Law model.

The ReliaGrowR package (Govan 2024) provides functions for repairable systems analysis in R.

library(ReliaGrowR)

6.3 Repairable vs Non-Repairable Systems

A non-repairable system is discarded after failure and replaced with a new, statistically identical unit. The unit of analysis is the time to first failure. Weibull analysis applies.

A repairable system is restored to service after each failure and continues accumulating operating time. The unit of analysis is the sequence of inter-failure times or cumulative failure counts over the system’s life.

Two repair assumptions are common:

Repair type	Description	Model
Perfect repair	As-good-as-new after each repair	Renewal process
Minimal repair	As-bad-as-old — failure fixed, nothing else changed	NHPP

Most field maintenance data best fits the minimal repair assumption, making NHPP the standard framework.

The recurrence rate (intensity function) $\rho(t)$ is the instantaneous rate of failure occurrence at time $t$:

\[\rho(t) = \frac{d\,E[N(t)]}{dt}\]

6.4 Failure Processes: HPP and NHPP

Homogeneous Poisson Process (HPP)

The HPP assumes a constant failure rate $\lambda$ throughout the system’s life:

\[E[N(t)] = \lambda t \qquad \rho(t) = \lambda \qquad \text{MTBF} = 1/\lambda\]

The HPP is appropriate when failures occur at random with no trend.

Non-Homogeneous Poisson Process (NHPP)

The NHPP allows the intensity function to change with time:

\[\rho(t) = \frac{d\Lambda(t)}{dt}\]

When $\rho(t)$ increases, the system is deteriorating; when it decreases, the system is improving.

t <- seq(1, 100, by = 1)

# HPP (constant rate)
hpp_intensity  <- rep(0.05, length(t))

# NHPP Power Law scenarios
nhpp_improving  <- 0.01 * 0.7 * t^(0.7 - 1)   # beta = 0.7
nhpp_stable     <- 0.05 * 1.0 * t^(1.0 - 1)   # beta = 1.0
nhpp_worsening  <- 0.002 * 1.5 * t^(1.5 - 1)  # beta = 1.5

ylim_max <- max(c(hpp_intensity, nhpp_improving, nhpp_stable, nhpp_worsening)) * 1.1

plot(t, nhpp_improving, type = "l", col = "blue", lwd = 2,
     ylim = c(0, ylim_max),
     xlab = "Time", ylab = "Intensity ρ(t)",
     main = "NHPP Power Law Intensity Functions")
lines(t, nhpp_stable,    col = "red",       lwd = 2)
lines(t, nhpp_worsening, col = "darkgreen", lwd = 2)
abline(h = 0.05, col = "gray50", lwd = 2, lty = 2)
legend("topright",
       legend = c("β = 0.7 (improving)", "β = 1.0 (stable / HPP)",
                  "β = 1.5 (worsening)", "HPP reference λ = 0.05"),
       col = c("blue", "red", "darkgreen", "gray50"),
       lwd = 2, lty = c(1, 1, 1, 2))

6.5 The Power Law Process

The Power Law Process is the most widely used NHPP for repairable systems:

\[E[N(t)] = \lambda t^{\beta} \qquad \rho(t) = \lambda \beta t^{\beta-1}\]

Parameter interpretation:

$\lambda > 0$ — scale parameter; overall magnitude of the failure rate.
$\beta < 1$: decreasing intensity (system improving).
$\beta = 1$: constant intensity = HPP.
$\beta > 1$: increasing intensity (system deteriorating).

The instantaneous MTBF at time $t$:

\[\text{MTBF}(t) = \frac{1}{\rho(t)} = \frac{1}{\lambda\beta\,t^{\beta-1}}\]

Maximum likelihood estimates given $n$ failures at times $t_1 < \cdots < t_n$ over total time $T$:

\[\hat{\beta} = \frac{n}{\displaystyle\sum_{i=1}^{n} \ln(T/t_i)} \qquad \hat{\lambda} = \frac{n}{T^{\hat{\beta}}}\]

Review

A repairable system is observed to have an increasing failure rate over time. Which model is most appropriate?

Answer

NHPP Power Law with $\beta > 1$. An increasing intensity function is characteristic of a deteriorating system. HPP (constant rate) cannot capture this trend.

6.6 NHPP Analysis with nhpp()

The nhpp() function fits the Power Law Process (Crow-AMSAA model) (Crow 1975) to cumulative failure data by maximum likelihood, returning estimates of $\lambda$ and $\beta$.

A curve that bends downward (concave) indicates $\hat{\beta} < 1$ — improving system. Upward bending (convex) indicates $\hat{\beta} > 1$ — deteriorating system.

times    <- c(500, 1200, 2100, 3300, 4800, 6500, 8400, 10500, 12800, 15000)
failures <- c(3, 4, 4, 5, 4, 3, 3, 3, 2, 2)

fit <- nhpp(time = times, event = failures)
plot(fit,
     main = "Crow-AMSAA Model: Industrial Pump Fleet",
     xlab = "Cumulative Operating Hours",
     ylab = "Cumulative Failures")

The concave curve and $\hat{\beta} < 1$ indicate the pump fleet’s failure rate is decreasing — a favorable operating regime.

Try It

A compressor fleet recorded these cumulative operating hours and failure counts. Fit a Power Law model using nhpp() with method = "LS".

times    <- c(1000, 2500, 4000, 5500, 8000)
failures <- c(2, 3, 4, 5, 5)
# fit <- nhpp(time = times, event = failures, method = "LS")
# plot(fit)

Solution

times    <- c(1000, 2500, 4000, 5500, 8000)
failures <- c(2, 3, 4, 5, 5)
fit <- nhpp(time = times, event = failures, method = "LS")
plot(fit, main = "Crow-AMSAA: Compressor Fleet",
     xlab = "Cumulative Hours", ylab = "Cumulative Failures")

6.7 Piecewise NHPP

The Piecewise NHPP (Guo et al. 2010) fits a separate Power Law to each segment of the time axis, allowing for structural changes in failure behavior (overhauls, design modifications, operational changes):

\[E[N(t)] = \begin{cases} \lambda_1 t^{\beta_1} & 0 < t \leq \tau \\ E[N(\tau)] + \lambda_2(t-\tau)^{\beta_2} & t > \tau \end{cases}\]

Pass a breakpoint estimate via the breaks argument. The segmented algorithm then optimizes the breakpoint location from the data.

times2    <- c(500, 1200, 2100, 3300, 4800, 6500, 8400, 9200, 10100, 11300,
               12800, 14500, 16400, 18500, 20800)
failures2 <- c(3, 4, 4, 5, 6, 6, 5, 3, 3, 2, 2, 2, 2, 1, 1)

fit_pw <- nhpp(time = times2, event = failures2, breaks = 8000, method = "LS")
plot(fit_pw,
     main = "Piecewise NHPP: Fleet Overhaul at Hour 8000",
     xlab = "Cumulative Operating Hours",
     ylab = "Cumulative Failures")

The first segment (pre-overhaul, steeper slope) gives way to a flatter or downward-bending second segment, confirming the overhaul reduced the failure rate. The fitted breakpoint may differ slightly from the input value — here it is estimated near hour 9,820.

Review

What is the primary reason to use a Piecewise NHPP instead of a single Crow-AMSAA model?

Answer

To account for a known or suspected structural change in the failure process (overhaul, design change, operational change). A piecewise model fits each segment independently, capturing the change in failure behavior.

6.8 Exposure Analysis

When analyzing a fleet, individual units enter and leave observation at different times. Exposure is the total operating time accumulated across all units currently at risk.

The exposure() function computes: - Cumulative exposure — total system-time at risk up to each event time. - Number at risk — how many units were under observation at each time. - Event rate — cumulative events divided by cumulative exposure.

pump_data <- data.frame(
  id    = c(1, 1, 1,  2, 2, 2,  3, 3,  4, 4, 4, 4,  5, 5),
  time  = c(310, 850, 1620,  420, 1050, 2100,  580, 1890,  240, 710, 1380, 2400,  530, 1740),
  event = c(1, 1, 1,   1, 1, 1,   1, 1,   1, 1, 1, 1,   1, 1)
)

exp_result <- exposure(id = pump_data$id, time = pump_data$time,
                       event = pump_data$event)
plot(exp_result)

exp_result is a data frame containing three columns: cum_exposure (total system-time accumulated across all units at risk), n_risk (number of units under observation at each time), and event_rate (cumulative events divided by cumulative exposure). The exposure plot shows how total observed system-time accumulates and how the fleet-wide event rate behaves over time.

Review

Why is it important to account for exposure when comparing failure rates across systems with different observation periods?

Answer

Without normalizing by exposure, a system observed for twice as long will appear to have twice as many failures even if its underlying rate is identical. Exposure provides the denominator — failures per unit time — so rates can be compared fairly.

6.9 Mean Cumulative Function

The Mean Cumulative Function (MCF) is a non-parametric estimate of the expected cumulative number of failures per system as a function of time. It makes no distributional assumptions and handles systems with different observation periods.

The Nelson-Aalen estimator is a non-parametric method that requires no distributional assumption and naturally handles systems with different observation lengths (administrative censoring):

\[\hat{M}(t) = \sum_{t_j \leq t} \frac{d_j}{n_j}\]

where $d_j$ is the number of failures at time $t_j$ and $n_j$ is the number of systems at risk at $t_j$.

Pass end_time to mcf() when systems were observed beyond their last recorded failure — otherwise the risk set is deflated too early, inflating the rate estimate.

end_times  <- c("1" = 3000, "2" = 3000, "3" = 3000, "4" = 3000, "5" = 3000)

mcf_result <- mcf(id = pump_data$id, time = pump_data$time,
                  event = pump_data$event,
                  end_time = end_times)
plot(mcf_result,
     main = "Mean Cumulative Function: Industrial Pump Fleet",
     xlab = "Operating Hours",
     ylab = "Expected Cumulative Failures per System")

The slope of the MCF reveals the recurrence trend: steepening = increasing failure rate; flattening = improving.

Try It

Three systems were observed with the following failure times. Each was observed until hour 2000. Compute and plot the MCF.

id    <- c(rep("1", 3), rep("2", 4), rep("3", 2))
time  <- c(400, 900, 1500,   300, 700, 1200, 1800,   600, 1400)
event <- rep(1, 9)
end_times2 <- c("1" = 2000, "2" = 2000, "3" = 2000)
# fit <- mcf(id, time, event, end_time = end_times2)
# plot(fit)

Solution

id    <- c(rep("1", 3), rep("2", 4), rep("3", 2))
time  <- c(400, 900, 1500,   300, 700, 1200, 1800,   600, 1400)
event <- rep(1, 9)
end_times2 <- c("1" = 2000, "2" = 2000, "3" = 2000)
fit <- mcf(id = id, time = time, event = event, end_time = end_times2)
plot(fit, main = "MCF Example", xlab = "Time (hours)", ylab = "MCF")

6.10 Interpreting Results and Predicting Future Failures

Reading $\hat{\beta}$

$\hat{\beta}$	Interpretation	Typical cause
$< 1$	Decreasing failure rate	Effective maintenance, infant mortality resolved
$= 1$	Constant failure rate	Random external shocks
$> 1$	Increasing failure rate	Wear-out, accumulating damage, aging

Cumulative vs Instantaneous MTBF

Cumulative MTBF = $T / N(T)$ — historical average over all observed failures.
Instantaneous MTBF at time $t$ = $1/\rho(t) = 1/(\hat{\lambda}\hat{\beta}\,t^{\hat{\beta}-1})$ — expected time to the next failure.

For $\hat{\beta} > 1$ (aging system), the instantaneous MTBF falls below the cumulative average at late times — the system’s near-term reliability is worse than its history suggests.

Predicting Future Failures

Expected additional failures from current time $T$ to a future time $T + \Delta t$:

\[E[\Delta N] = \hat{\lambda}(T + \Delta t)^{\hat{\beta}} - \hat{\lambda}\, T^{\hat{\beta}}\]

lambda    <- 0.0003
beta      <- 0.85
T_current <- 15000
T_future  <- 17000

expected_add <- lambda * T_future^beta - lambda * T_current^beta
cat("Expected additional failures:", round(expected_add, 2), "\n")

Expected additional failures: 0.12

Try It

A system has been operating for 5,000 hours. A Power Law fit yielded $\lambda = 0.0005$ and $\beta = 1.2$. Calculate the expected additional failures in the next 1,000 hours.

lambda    <- 0.0005
beta      <- 1.2
T_current <- 5000
T_future  <- 6000
# E[ΔN] = lambda * T_future^beta - lambda * T_current^beta

Solution

lambda    <- 0.0005
beta      <- 1.2
T_current <- 5000
T_future  <- 6000
expected_failures <- lambda * T_future^beta - lambda * T_current^beta
cat("Expected additional failures:", round(expected_failures, 2), "\n")

Expected additional failures: 3.36

Review

A system has a fitted Power Law model with $\beta = 1.35$. What is the operational implication?

Answer

The failure rate is increasing — wear-out or aging is occurring. The system may be approaching a major overhaul threshold. The instantaneous MTBF is below the cumulative historical average.

6.11 Summary

Key takeaways:

Repairable systems analysis models the sequence of recurring failures, not just the first failure.
The Power Law Process ($E[N(t)] = \lambda t^\beta$) is the standard NHPP for repairable systems.
$\hat{\beta}$ determines whether the failure rate is improving ($< 1$), stable ($= 1$), or worsening ($> 1$).
nhpp() fits the Crow-AMSAA model; add breaks for piecewise fitting.
exposure() computes fleet-level operating time at risk.
mcf() estimates cumulative failures per system non-parametrically.
Future failure counts: $E[\Delta N] = \hat{\lambda}(T + \Delta t)^{\hat{\beta}} - \hat{\lambda}\,T^{\hat{\beta}}$.

Ascher, Harold, and Harry Feingold. 1984. Repairable Systems Reliability. Marcel Dekker.

Crow, Larry H. 1975. Reliability Analysis for Complex, Repairable Systems. Army Material Systems Analysis Activity. https://apps.dtic.mil/sti/citations/ADA020296.

Govan, Paul. 2024. ReliaGrowR: Reliability Growth Analysis. https://doi.org/10.32614/CRAN.package.ReliaGrowR.

Guo, H., A. Mettas, G. Sarakakis, and P. Niu. 2010. “Piecewise NHPP Models with Maximum Likelihood Estimation for Repairable Systems.” 2010 Proceedings – Annual Reliability and Maintainability Symposium (RAMS), 1–7. https://doi.org/10.1109/RAMS.2010.5448029.

Meeker, William Q., and Luis A. Escobar. 1998. Statistical Methods for Reliability Data. Wiley-Interscience.

# Repairable Systems Analysis {#sec-rs} ## Introduction Unlike life data analysis (@sec-lda), which focuses on the *time to first failure* of non-repairable components, **Repairable Systems Analysis** [@Ascher1984; @Meeker1998] models the entire sequence of recurring failures on a single system or fleet. This chapter covers the foundational theory and the `ReliaGrowR` tools for fitting models, computing fleet-level exposure, and estimating the Mean Cumulative Function. ## Learning Objectives By the end of this chapter, you will be able to: - Define the distinction between repairable and non-repairable systems. - Explain the Homogeneous Poisson Process (HPP) and Non-Homogeneous Poisson Process (NHPP). - Describe the Power Law Process and its parameters. - Fit and interpret the Crow-AMSAA model using `nhpp()` from ReliaGrowR. - Apply the Piecewise NHPP model to detect changes in failure behavior. - Compute fleet-level exposure using `exposure()`. - Estimate the Mean Cumulative Function (MCF) using `mcf()`. - Forecast future failure counts from a fitted Power Law model. The `ReliaGrowR` package [@ReliaGrowR] provides functions for repairable systems analysis in R. ```{r} #| message: false library(ReliaGrowR) ``` ## Repairable vs Non-Repairable Systems A **non-repairable system** is discarded after failure and replaced with a new, statistically identical unit. The unit of analysis is the *time to first failure*. Weibull analysis applies. A **repairable system** is restored to service after each failure and continues accumulating operating time. The unit of analysis is the *sequence of inter-failure times* or cumulative failure counts over the system's life. Two repair assumptions are common: | Repair type | Description | Model | |---|---|---| | Perfect repair | As-good-as-new after each repair | Renewal process | | Minimal repair | As-bad-as-old — failure fixed, nothing else changed | NHPP | Most field maintenance data best fits the **minimal repair** assumption, making NHPP the standard framework. The **recurrence rate** (intensity function) $\rho(t)$ is the instantaneous rate of failure occurrence at time $t$: $$\rho(t) = \frac{d\,E[N(t)]}{dt}$$ ## Failure Processes: HPP and NHPP ### Homogeneous Poisson Process (HPP) The **HPP** assumes a *constant* failure rate $\lambda$ throughout the system's life: $$E[N(t)] = \lambda t \qquad \rho(t) = \lambda \qquad \text{MTBF} = 1/\lambda$$ The HPP is appropriate when failures occur at random with no trend. ### Non-Homogeneous Poisson Process (NHPP) The **NHPP** allows the intensity function to change with time: $$\rho(t) = \frac{d\Lambda(t)}{dt}$$ When $\rho(t)$ increases, the system is deteriorating; when it decreases, the system is improving. ```{r} t <- seq(1, 100, by = 1) # HPP (constant rate) hpp_intensity <- rep(0.05, length(t)) # NHPP Power Law scenarios nhpp_improving <- 0.01 * 0.7 * t^(0.7 - 1) # beta = 0.7 nhpp_stable <- 0.05 * 1.0 * t^(1.0 - 1) # beta = 1.0 nhpp_worsening <- 0.002 * 1.5 * t^(1.5 - 1) # beta = 1.5 ylim_max <- max(c(hpp_intensity, nhpp_improving, nhpp_stable, nhpp_worsening)) * 1.1 plot(t, nhpp_improving, type = "l", col = "blue", lwd = 2, ylim = c(0, ylim_max), xlab = "Time", ylab = "Intensity ρ(t)", main = "NHPP Power Law Intensity Functions") lines(t, nhpp_stable, col = "red", lwd = 2) lines(t, nhpp_worsening, col = "darkgreen", lwd = 2) abline(h = 0.05, col = "gray50", lwd = 2, lty = 2) legend("topright", legend = c("β = 0.7 (improving)", "β = 1.0 (stable / HPP)", "β = 1.5 (worsening)", "HPP reference λ = 0.05"), col = c("blue", "red", "darkgreen", "gray50"), lwd = 2, lty = c(1, 1, 1, 2)) ``` ## The Power Law Process The **Power Law Process** is the most widely used NHPP for repairable systems: $$E[N(t)] = \lambda t^{\beta} \qquad \rho(t) = \lambda \beta t^{\beta-1}$$ **Parameter interpretation:** - $\lambda > 0$ — scale parameter; overall magnitude of the failure rate. - $\beta < 1$: decreasing intensity (system improving). - $\beta = 1$: constant intensity = HPP. - $\beta > 1$: increasing intensity (system deteriorating). The **instantaneous MTBF** at time $t$: $$\text{MTBF}(t) = \frac{1}{\rho(t)} = \frac{1}{\lambda\beta\,t^{\beta-1}}$$ Maximum likelihood estimates given $n$ failures at times $t_1 < \cdots < t_n$ over total time $T$: $$\hat{\beta} = \frac{n}{\displaystyle\sum_{i=1}^{n} \ln(T/t_i)} \qquad \hat{\lambda} = \frac{n}{T^{\hat{\beta}}}$$ ::: {.callout-tip} ## Review A repairable system is observed to have an increasing failure rate over time. Which model is most appropriate? <details><summary>Answer</summary> **NHPP Power Law with $\beta > 1$**. An increasing intensity function is characteristic of a deteriorating system. HPP (constant rate) cannot capture this trend. </details> ::: ## NHPP Analysis with nhpp() The `nhpp()` function fits the Power Law Process (Crow-AMSAA model) [@Crow1975] to cumulative failure data by maximum likelihood, returning estimates of $\lambda$ and $\beta$. A curve that bends downward (concave) indicates $\hat{\beta} < 1$ — improving system. Upward bending (convex) indicates $\hat{\beta} > 1$ — deteriorating system. ```{r} times <- c(500, 1200, 2100, 3300, 4800, 6500, 8400, 10500, 12800, 15000) failures <- c(3, 4, 4, 5, 4, 3, 3, 3, 2, 2) fit <- nhpp(time = times, event = failures) plot(fit, main = "Crow-AMSAA Model: Industrial Pump Fleet", xlab = "Cumulative Operating Hours", ylab = "Cumulative Failures") ``` The concave curve and $\hat{\beta} < 1$ indicate the pump fleet's failure rate is decreasing — a favorable operating regime. ::: {.callout-note} ## Try It A compressor fleet recorded these cumulative operating hours and failure counts. Fit a Power Law model using `nhpp()` with `method = "LS"`. ```{r} times <- c(1000, 2500, 4000, 5500, 8000) failures <- c(2, 3, 4, 5, 5) # fit <- nhpp(time = times, event = failures, method = "LS") # plot(fit) ``` <details><summary>Solution</summary> ```{r} times <- c(1000, 2500, 4000, 5500, 8000) failures <- c(2, 3, 4, 5, 5) fit <- nhpp(time = times, event = failures, method = "LS") plot(fit, main = "Crow-AMSAA: Compressor Fleet", xlab = "Cumulative Hours", ylab = "Cumulative Failures") ``` </details> ::: ## Piecewise NHPP The **Piecewise NHPP** [@Guo2010] fits a separate Power Law to each segment of the time axis, allowing for structural changes in failure behavior (overhauls, design modifications, operational changes): $$E[N(t)] = \begin{cases} \lambda_1 t^{\beta_1} & 0 < t \leq \tau \\ E[N(\tau)] + \lambda_2(t-\tau)^{\beta_2} & t > \tau \end{cases}$$ Pass a breakpoint estimate via the `breaks` argument. The `segmented` algorithm then optimizes the breakpoint location from the data. ```{r} times2 <- c(500, 1200, 2100, 3300, 4800, 6500, 8400, 9200, 10100, 11300, 12800, 14500, 16400, 18500, 20800) failures2 <- c(3, 4, 4, 5, 6, 6, 5, 3, 3, 2, 2, 2, 2, 1, 1) fit_pw <- nhpp(time = times2, event = failures2, breaks = 8000, method = "LS") plot(fit_pw, main = "Piecewise NHPP: Fleet Overhaul at Hour 8000", xlab = "Cumulative Operating Hours", ylab = "Cumulative Failures") ``` The first segment (pre-overhaul, steeper slope) gives way to a flatter or downward-bending second segment, confirming the overhaul reduced the failure rate. The fitted breakpoint may differ slightly from the input value — here it is estimated near hour 9,820. ::: {.callout-tip} ## Review What is the primary reason to use a Piecewise NHPP instead of a single Crow-AMSAA model? <details><summary>Answer</summary> To account for a **known or suspected structural change** in the failure process (overhaul, design change, operational change). A piecewise model fits each segment independently, capturing the change in failure behavior. </details> ::: ## Exposure Analysis When analyzing a *fleet*, individual units enter and leave observation at different times. **Exposure** is the total operating time accumulated across all units currently at risk. The `exposure()` function computes: - **Cumulative exposure** — total system-time at risk up to each event time. - **Number at risk** — how many units were under observation at each time. - **Event rate** — cumulative events divided by cumulative exposure. ```{r} pump_data <- data.frame( id = c(1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5), time = c(310, 850, 1620, 420, 1050, 2100, 580, 1890, 240, 710, 1380, 2400, 530, 1740), event = c(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1) ) exp_result <- exposure(id = pump_data$id, time = pump_data$time, event = pump_data$event) plot(exp_result) ``` `exp_result` is a data frame containing three columns: `cum_exposure` (total system-time accumulated across all units at risk), `n_risk` (number of units under observation at each time), and `event_rate` (cumulative events divided by cumulative exposure). The exposure plot shows how total observed system-time accumulates and how the fleet-wide event rate behaves over time. ::: {.callout-tip} ## Review Why is it important to account for exposure when comparing failure rates across systems with different observation periods? <details><summary>Answer</summary> Without normalizing by exposure, a system observed for twice as long will appear to have twice as many failures even if its underlying rate is identical. Exposure provides the denominator — failures per unit time — so rates can be compared fairly. </details> ::: ## Mean Cumulative Function The **Mean Cumulative Function (MCF)** is a non-parametric estimate of the expected cumulative number of failures *per system* as a function of time. It makes no distributional assumptions and handles systems with different observation periods. The **Nelson-Aalen estimator** is a non-parametric method that requires no distributional assumption and naturally handles systems with different observation lengths (administrative censoring): $$\hat{M}(t) = \sum_{t_j \leq t} \frac{d_j}{n_j}$$ where $d_j$ is the number of failures at time $t_j$ and $n_j$ is the number of systems at risk at $t_j$. Pass `end_time` to `mcf()` when systems were observed beyond their last recorded failure — otherwise the risk set is deflated too early, inflating the rate estimate. ```{r} end_times <- c("1" = 3000, "2" = 3000, "3" = 3000, "4" = 3000, "5" = 3000) mcf_result <- mcf(id = pump_data$id, time = pump_data$time, event = pump_data$event, end_time = end_times) plot(mcf_result, main = "Mean Cumulative Function: Industrial Pump Fleet", xlab = "Operating Hours", ylab = "Expected Cumulative Failures per System") ``` The slope of the MCF reveals the recurrence trend: steepening = increasing failure rate; flattening = improving. ::: {.callout-note} ## Try It Three systems were observed with the following failure times. Each was observed until hour 2000. Compute and plot the MCF. ```{r} id <- c(rep("1", 3), rep("2", 4), rep("3", 2)) time <- c(400, 900, 1500, 300, 700, 1200, 1800, 600, 1400) event <- rep(1, 9) end_times2 <- c("1" = 2000, "2" = 2000, "3" = 2000) # fit <- mcf(id, time, event, end_time = end_times2) # plot(fit) ``` <details><summary>Solution</summary> ```{r} id <- c(rep("1", 3), rep("2", 4), rep("3", 2)) time <- c(400, 900, 1500, 300, 700, 1200, 1800, 600, 1400) event <- rep(1, 9) end_times2 <- c("1" = 2000, "2" = 2000, "3" = 2000) fit <- mcf(id = id, time = time, event = event, end_time = end_times2) plot(fit, main = "MCF Example", xlab = "Time (hours)", ylab = "MCF") ``` </details> ::: ## Interpreting Results and Predicting Future Failures ### Reading $\hat{\beta}$ | $\hat{\beta}$ | Interpretation | Typical cause | |:---:|---|---| | $< 1$ | Decreasing failure rate | Effective maintenance, infant mortality resolved | | $= 1$ | Constant failure rate | Random external shocks | | $> 1$ | Increasing failure rate | Wear-out, accumulating damage, aging | ### Cumulative vs Instantaneous MTBF - **Cumulative MTBF** = $T / N(T)$ — historical average over all observed failures. - **Instantaneous MTBF** at time $t$ = $1/\rho(t) = 1/(\hat{\lambda}\hat{\beta}\,t^{\hat{\beta}-1})$ — expected time to the *next* failure. For $\hat{\beta} > 1$ (aging system), the instantaneous MTBF falls *below* the cumulative average at late times — the system's near-term reliability is worse than its history suggests. ### Predicting Future Failures Expected additional failures from current time $T$ to a future time $T + \Delta t$: $$E[\Delta N] = \hat{\lambda}(T + \Delta t)^{\hat{\beta}} - \hat{\lambda}\, T^{\hat{\beta}}$$ ```{r} lambda <- 0.0003 beta <- 0.85 T_current <- 15000 T_future <- 17000 expected_add <- lambda * T_future^beta - lambda * T_current^beta cat("Expected additional failures:", round(expected_add, 2), "\n") ``` ::: {.callout-note} ## Try It A system has been operating for 5,000 hours. A Power Law fit yielded $\lambda = 0.0005$ and $\beta = 1.2$. Calculate the expected additional failures in the next 1,000 hours. ```{r} lambda <- 0.0005 beta <- 1.2 T_current <- 5000 T_future <- 6000 # E[ΔN] = lambda * T_future^beta - lambda * T_current^beta ``` <details><summary>Solution</summary> ```{r} lambda <- 0.0005 beta <- 1.2 T_current <- 5000 T_future <- 6000 expected_failures <- lambda * T_future^beta - lambda * T_current^beta cat("Expected additional failures:", round(expected_failures, 2), "\n") ``` </details> ::: ::: {.callout-tip} ## Review A system has a fitted Power Law model with $\beta = 1.35$. What is the operational implication? <details><summary>Answer</summary> The failure rate is **increasing** — wear-out or aging is occurring. The system may be approaching a major overhaul threshold. The instantaneous MTBF is below the cumulative historical average. </details> ::: ## Summary **Key takeaways:** - Repairable systems analysis models the *sequence* of recurring failures, not just the first failure. - The **Power Law Process** ($E[N(t)] = \lambda t^\beta$) is the standard NHPP for repairable systems. - $\hat{\beta}$ determines whether the failure rate is improving ($< 1$), stable ($= 1$), or worsening ($> 1$). - `nhpp()` fits the Crow-AMSAA model; add `breaks` for piecewise fitting. - `exposure()` computes fleet-level operating time at risk. - `mcf()` estimates cumulative failures per system non-parametrically. - Future failure counts: $E[\Delta N] = \hat{\lambda}(T + \Delta t)^{\hat{\beta}} - \hat{\lambda}\,T^{\hat{\beta}}$.

6.1 Introduction

6.2 Learning Objectives

6.3 Repairable vs Non-Repairable Systems

6.4 Failure Processes: HPP and NHPP

Homogeneous Poisson Process (HPP)

Non-Homogeneous Poisson Process (NHPP)

6.5 The Power Law Process

6.6 NHPP Analysis with nhpp()

6.7 Piecewise NHPP

6.8 Exposure Analysis

6.9 Mean Cumulative Function

6.10 Interpreting Results and Predicting Future Failures

Reading \(\hat{\beta}\)

Cumulative vs Instantaneous MTBF

Predicting Future Failures

6.11 Summary