Imagine observing inputs and outputs of a self-service system, without individual service times. How would you estimate the distribution of service time without following individuals from input to output? The maximum likelihood estimator for an M/G/Infinity self-service-time distribution function from ships and returns counts works for nonstationary arrival process M(t)/G/Infinity self-service systems, under a condition. A constant or linearly increasing arrival (ships) rate satisfies the condition. If you identify outputs by failure mode then you could estimate reliability by failure mode or quantify reliability growth, without life data. [Read more…]

# Progress in Field Reliability?

## Why Isn’t It Working Like You Said?

Nonparametric, age-specific field reliability estimates helped deal with a Customer’s bad experience using a Hewlett-Packard part in the Customer’s product: 110 part failures out of 3001 shipped in the first five months. Comparison of HP population vs. Customer reliability estimates showed the Customer’s infant mortality was not typical. Using population ships and failures or returns data eliminated sample uncertainty from the HP population field reliability estimate.

[Read more…]## Sample vs. Population Estimates?

Rupert Miller said, “Surprisingly, no efficiency comparison of the sample distribution function with the mles (maximum likelihood estimators) appears to have been reported in the literature.” (Statistical “efficiency” measures how close an estimator’s sample variance is to its Cramer-Rao lower bound.) In “What Price Kaplan-Meier?” Miller compares the nonparametric Kaplan-Meier reliability estimator with mles for exponential, Weibull, and gamma distributions.

This report compares the bias, efficiency, and robustness of the Kaplan-Meier reliability estimator from grouped failure counts (grouped life data) with the nonparametric maximum likelihood reliability estimator from ships (periodic sales, installed base, cohorts, etc.) and returns (periodic complaints, failures, repairs, replacement, spares sales, etc.) counts, estimator vs. estimator and population vs. sample.

[Read more…]## Uncertainty in Population Estimates?

Dick Mensing said, “Larry, you can’t give an estimate without some measure of its uncertainty!” For seismic risk analysis of nuclear power plants, we had plenty of multivariate earthquake stress data but paltry strength-at-failure data on safety-system components. So we surveyed “experts” for their opinions on strengths-at-failures distribution parameters and for the correlations between pairs of components’ strengths at failures.

If you make estimates from *population* field reliability data, do the estimates have uncertainty? If all the data were *population* lifetimes or ages-at-failures, estimates would have no sample uncertainty, perhaps measurement error. Estimates from population field reliability data have uncertainty because typically some population members haven’t failed. If field reliability data are from renewal or replacement processes, some replacements haven’t failed and earlier renewal or replacement counts may be unknown. Regardless, estimates from population data are better than estimates from a sample, even if the population data is ships and returns counts!

## Progress in LED Reliability Analysis?

ANSI-IES TM-21 standard method may predict negative L70 LED lives. (L70 is the age at which LED lumens output has deteriorated to less than 70% of initial lumens.) Philips-Lumileds deserves credit for publishing the data that inspired an alternative L70 reliability estimation method based on geometric Brownian motion of stock prices in the Black-Scholes-Merton options price model. This gives the inverse Gauss distribution of L70 for LEDs.

[Read more…]## What’s Wrong Now? Intermittent Failures?

“Aircraft LRUs test NFF (No-Failure-Found) approximately 50% of the time” {Anderson] Wabash Magnetics claimed returned crankshaft position sensors had 89-90% NTF (No-Trouble-Found), Uniphase had 20%, Apple computer had 50% [George].

[Read more…]## What’s Wrong Now? Multiple Failures?

How is failure testing done on the Space Station? Could FTA (Fault Tree Analysis) be used in reverse to detect multiple failures given symptoms? That’s what NASA was programming in the 1990s. I proposed that the ratios P[part failure]/(part test time) be used to optimally sequence tests. Those ratios work if there are multiple failures, as long as failure rates are constant and failure times are statistically independent.

[Read more…]## What’s Wrong Now? Shotgun Repair

Shotgun repair is trying to fix a system problem by replacing parts until the problem goes away. It is often done without regard to parts’ age-specific reliability information. Should you test before replacement? Which test(s) should you do? In which order? How long? Which part should you replace next if the test gave no indication of what’s wrong? What if test indication is imperfect or the fault is intermittent? What if there are more than one part failure?

[Read more…]## Failure Rate Classification for RCM

Which of these six failure rate functions do your products and their service parts have? You don’t know? You don’t have field reliability lifetime data by product name or part serial number? That’s OK. Lifetime data are not required to estimate and classify failure-rate functions, including attrition and retirement. GAAP requires statistically sufficient field reliability data to classify failure rate functions for RCM.

[Read more…]## Transient Markov Model of Multiple Failure Modes

COVID-19 Case Fatality Rate (CFR) is easy to estimate: CFR=deaths/cases. Regression forecasts of COVID-19 cases and deaths are easy but complicated by variants and nonlinearity. Epidemiologists use SIR models (Susceptible, Infectious, and “Removed”) to estimate Ro. These are baseball statistics. Reliability people need age-specific reliability and failure-rate function estimates, by failure mode, to diagnose problems, recommend spares, plan maintenance, do risk analyses, etc. Markov models use actuarial transition rates.

[Read more…]## Markov Approximation to Standby-System Reliability

Age-specific reliability of a standby system depends on components’ failure rates. Reliability computation is interesting when part failure rates depend on age, which is what motivates having a standby system. A Markov chain, approximates the *age-specific* reliability and availability, which are complicated to compute exactly, unless you assume constant failure rates. Why not use age-specific (actuarial) rates? They are Markov chain transition rates.

## Assuming Stationarity could be as bad as Assuming Constant Failure Rate!

Suppose installed base or cohorts in successive periods have different reliabilities due to nonstationarity? What does that do to forecasts, estimates, reliability predictions, diagnostics, spares stock levels, maintenance plans, etc.? Assuming stationarity is equivalent to assuming all installed base, cohorts, or ships have the same reliability functions. At what cost? Assuming a constant failure rate is equivalent to assuming everything has exponentially distributed time to failure or constant failure rate. At what cost?

## Forecast Parts’ Demands, Without Life Data, for a Nonstationary Process

In the 1960s, my ex-wife’s father set safety stock levels and order quantities for Pep Boys. He used part sales rates and the Wilson square-root formula to set order quantities.

Why not use the ages of the cars into which those parts go, to forecast part sales and recommend stock levels? Imagine you had vehicle counts (year, make, model, and engine) in the neighborhoods of parts stores, catalogs of which parts and how many go into which cars, and store sales by part number.

[Read more…]## Dependence in Production Lines

As I rode, I thought, how could I use reliability statistics to optimize a solar-tube production line? Then I noticed a brass glint in the scrub brush. It didn’t look like trash, so I stopped and found an old brass oil lamp like Aladdin’s. Naturally, I rubbed it. There was a flash and a puff of smoke, and out popped the genie who said, “Yes master, by the powers vested in me, I grant you three wishes.”

[Read more…]## Reliability Estimation With Life Limits but Without Life Data

Would you like age-specific field reliability of your products and their service parts? Age-specific field reliability is useful for reliability prediction, diagnoses, forecasting, warranty reserves, spares stock levels, warranty extensions, and recalls. Nonparametric estimation of age-specific field reliability is easy, *if you track parts or products by name and serial number for life data*. What if there are life limits? What if there’s no life data?