IEC 60601-1 says… Estimate the probability per time pe of an electrical failure and of an oxygen leak po. Determine the accepted probability of dangerous failures [fire] per time r. Calculate the inspection time interval tc = r/(0.5*pe*po).

A friend asked, “What’s the 0.5 for? It doesn’t account for the fire event sequence: leak before spark.” I posted correction tc = r/((po/(po+pe))*pe*po) and notified the IEC committee which acknowledged, “We’ll consider your suggestion for edition 4.”

[An earlier, shorter version of this article on www.LinkedIn.com, July 5, 2018. This version describes an inspection-time and risk-analysis template.]

When nuclear power plants were built, companies had quality assurance programs and US Nuclear Regulatory Commission risk standards. Now the nuclear industry faces obsolescence. Qualifying replacement parts and replacing analog instrumentation and controls with digital systems generates some reliability testing work. NASA solicits unmanned nuclear power plants on the moon and Mars. Nevertheless, the demand for nuclear engineers is decreasing. Fortunately, the nuclear industry spawned risk analyses useful in other industries. 

Isn’t it enough to estimate the age-specific field reliability functions for each of our products and their service parts? Of course we quantify uncertainties in estimates: sample uncertainties and population uncertainties due to changes or evolution. That’s information to forecast service requirements, recommend spares, optimize diagnostics, plan maintenance, warranty reserves, recalls, etc. What else could we possibly need or do?

What can we do without reliability function estimates? FMEA? FTA? RCA? RCM? Argue about MTBFs and availability? Weibull? Keep a low profile? Run Admirals’ tests? Look for a new, well-funded project far from the deliverable stage? 

Ask for field data; there should be enough to estimate reliability and make reliability-based decisions, even if some data are missing. Field data might even be population data!

(This is chapter 5 of User Manual for Credible Reliability Prediction – Field Reliability (google.com), cleaned up and typeset for AccendoReliaiblity Weekly Update.) 

The nonparametric maximum likelihood estimator for an M/G/∞ self-service time distribution function G(t) extends to nonstationary, time-dependent, Poisson arrival process M(t)/G/∞ systems, under a condition. A linearly increasing Poisson rate function satisfies the condition. The estimator of 1-G(t) is a reliability function estimate, from population ships and returns data required by generally accepted accounting principles. 

At my job interview, the new product development director, an econometrician, explained that he tried to forecast auto parts’ sales using regression. His model was 

sales forecast = Σ[b(s)*n(t-s)] + noise; s=1,2,…,t,

where b(s) are regression coefficients to be estimated, n(t-s) are counts of vehicles of age t-s in the neighborhood of auto parts stores. The director admitted to regression analysis problems, because of autocorrelation among the n(t-s) vehicle counts, no pun intended. 

A computer company tiger team held a meeting to decide how to fix their laser printer ghosting problem. Bearings seized in the squirrel-cage cooling fan for the fuser bar. The fan bearing was above fuser bar, which baked the bearing. A fix  decision was made, voted on, and accepted. Party time. I asked, “How do you verify the fix?” Boo!

This an example of using current status life data. I checked status every laser printer laser-printer fan in company headquarters: operating or failed? Date of manufacture was encoded in the printer serial number, so I estimated the fan’s age-specific failure rate function, before the fix. Premature wearout was evident. Could I observe repaired or new printers at a later time and test the hypothesis that the problem had been fixed? Yes. 

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The ASQ Reliability Division (RD), copyrighted the 2003 monograph “Credible Reliability Prediction” (CRP) but lost all copies circa 2014. I pestered the RD to let me republish CRP, because people asked “How do I make credible reliability predictions?” Copyright reversion to authors is accepted practice when a publisher no longer supports a document. 

ASQ Reliability Division published “Credible Reliability Prediction” (CRP) in 2003. Harold Williams, Reliability Division monograph series editor, wrote, “[CRP] …delineates statistical methods that effectively extend MTBF prediction to complex, redundant, dependent, standby, and life-limited systems… This is the first text that describes a credible method of making age-specific reliability predictions…. This monograph presents insights and information inspired by real applications and [still] not covered in contemporary reliability textbooks.” 

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“Component D” had some failures in its first 12 months. How many more would fail in 36-month warranty? ASQ’s Quality Progress Statistics Roundtable published the data and Weibull analysis. The data included left-censored failure counts collected at one calendar time. The Weibull analysis included actuarial failure forecasts. This article describes nonparametric alternatives to Weibull and quantifies extrapolation uncertainty. The nonparametric forecasts are larger than the Weibull forecasts. Alternative extrapolations of nonparametric failure rates from data subsets quantify uncertainty. [Read more…]

I learned actuarial methods working for the USAF Logistics Command. We used actuarial rates to forecast demands and recommend stock levels for expensive engines tracked by serial number, hours, and cycles. I had a hunch that actuarial methods could be applied to all service parts, without life data. [Read more…]

Ergodicity means that cross-section probabilities equal longitudinal lifetime probabilities. (“Ergos” is Greek for “work.” Think of “ergonomics”.) Ergodicity means that we can estimate age-specific field reliability functions from cross-section data: ships (installed base) and returns (complaints, failures, service parts’ sales, etc.). Ships and returns provide information about lifetimes. Returns are the superpositions of failures of products or their parts started at different times. What does ergodicity have to do with toilet paper? [Read more…]

Andrew Vázsonyi led an interesting life. He collaborated with mathematician Paul Erdös, he was co-founder of The Institute of Management Sciences, and he wrote “Which Door has the Cadillac: Adventures of a Real-Life Mathematician”. Around 1970, Andrew Vázsonyi interviewed for a teaching job in Sauder School of Business, University of British Columbia. During the job interview, he taught us Gozinto Theory. [Read more…]

Myron Tribus’ UCLA Statistical Thermodynamics class introduced me to entropy, -SUM[p(t)ln(p(t))]. (p(t) is the probability of state t of a system.) Professor Tribus later advocated maximum-entropy reliability estimation, because that “…best represents the current state of knowledge about a system…” [Principle of maximum entropy – Wikipedia] Caution! This article contains statistical neurohazards.

Claude Shannon wrote that entropy (log base 2) represents information bits, “…an absolute mathematical limit on how well data from the source can be losslessly compressed onto a perfectly noiseless channel.”  [Beirlant et al.]

Maximum likelihood estimation is one way to estimate reliability from data. It maximizes the probability density function of observed data, PRODUCT[p(t)], e.g., for observed failures at ages t. It is equivalent to maximize -SUM[ln(p(t)]. Maximum entropy reliability estimation maximizes entropy -SUM[p(t)ln(p(t)]. That’s same as maximizing the expected value, -SUM[p(t)ln(p(t)], of the log likelihood -ln(p(t). Fine, if you have life data, ages at failures t censored or not. [Read more…]

Ralph Evans was editor of the IEEE Transactions on Reliability from 1969 until 2004. He was a very good editor for my 1977 article, and he used me as a reviewer, because I was critical of BS and academic exercises. Ralph moved to University Retirement Community, Davis, CA. He died in 2013, https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=6587564. I wish I’d known he lived nearby so I could have visited and argued with him.

Ralph’s editorials [1 and 2] pled, “Data, Data, Oh Where Art Thou Data?” He wrote, “Field-data are largely garbage. I believe they deserve all the negative thinking possible.” “True field-data are wonderful-much better than fancy equations. Unfortunately, they are very difficult to get. Thus data from the field are largely garbage because they do not represent what really happened.” [Read more…]