Estimate Field Reliability Without Life Data!

Fred wrote, “I would like to suggest that you continue writing articles – make them more tutorial in nature as if teaching someone the stats from scratch that you use in your articles. Instead of loads of references to papers and procedures, explain the concepts and math involved.” OK, I’ll try.

Nonparametric field reliability estimators require no unwarranted distribution assumptions and they preserve all information in data. Here’s how to compute them, without life data, while preserving all relevant information in ships and returns counts. 

 “…explain the concepts and math involved” in Queuing Theory and the PAV algorithm

Several publications prove that the cumulative outputs of an M/G/Infinity self-service system have nonstationary Poisson distributions with expected values λG(t) [George, Oct. 2021]. (“M” denotes that λ is the Poisson arrival rate, G(t) represents the service time distribution function, P[Service time ≤ t], and Infinity means infinite number of service stations or self-service.) Estimates of G(t) (λG(t) divided by arrival rate λ) have to be nondecreasing (increasing or constant as a function of age t) to be distribution functions. I found the maximum likelihood of a nondecreasing G(t) in [Marshall and Proschan]. It is called the PAV, “Penalize Adjacent Violators”, algorithm. This was the gist of my first thesis, estimating service time distributions without observing each customer’s service time. It was rejected by my advisor. he said, “It was unpublishable”, [George and Agrawal 1973]. We didn’t realize its potential for estimating reliability without life data.

In 1980, Mike Johanns of Apple Computer told me, “Imagine the lifetime of a product or part as the service time in an M/G/Infinity self-service system.” Reliability is 1‑G(t). Mike told me to do a literature search on statistics and self-service systems. The only hit was my first thesis [George and Agrawal 1973]. I programmed it and used it to estimate the reliability of Apple products and their service parts. This article describes the spreadsheet version that computer program.

Statistically Sufficient Data are Free!

Data are periodic ships, aka “cohorts” (products’ or parts’ installed base by age: sales, deliveries, production, etc.), and returns (complaints, repairs, failures, spares sales, etc.). Such data are required by GAAP (Generally Accepted Accounting Principles). You may have to work to dig ships out of revenue = sales*price and returns out of warranty costs = returns*service cost or similar sources, such as spare parts’ sales. I have even used industry publications for ships counts by calendar period. What if you want parts’ ships by age from product ships counts by age?

Use “gozinto” theory to convert product bills of materials and installed base into parts’ installed base by age [George, May 2021]. The link to the gozinto.xls workbook in Gozinto Theory and Parts’ Installed Base – Accendo Reliability/ is broken. Here is a link to a new version, Gozinto3.xlsm: https://docs.google.com/spreadsheets/d/1Z2vQncs8CuzMFHKtpBtCIrR3eO_QnoMJVGdAoBFGwT8/edit?usp=sharing/. I will try to explain the distribution of parts installed base induced by variability in product ships in my next “Weekly Update” article.

Table 1 is a spreadsheet example of reliability estimation from ships and returns counts. Column A is date or time presumably equally spaced. If your data are not equally spaced, send your data and explain spacing; this is important for computing the failure rate function per unit of time. I put the complete workbook into a Google sheet named “npmle”, https://docs.google.com/spreadsheets/d/1fb_pjwygp_2CUeayNEV4PAX6wyR8sdWqPr76qeJ_n4I/edit?usp=sharing/.

Table 1. Ships and returns data. Returns are not identified by which ships cohort they came from. There is at most one return from each unit shipped in current or earlier time periods.

	A	B	C
1	Period or  Age t	Ships	Returns
2	1	47	1
3	2	41	7
4	3	45	12
5	4	39	17

Table 2 shows the computation of ratios of cumulative returns divided by cumulative ships. Each of these ratios is an estimate of G(t), t=1,2,3,4. They may not yield a legitimate, non-decreasing distribution function G(t).

Table 2. Ratios of cumulative returns and cumulative ships numerical values followed by formulas

	A	B	C	D	E	F	G
1	Age t	Ships	Returns	1	2	3	4
2	1	47	1	0.0213
3	2	41	7	0.0909	0.1707
4	3	45	12	0.1504	0.2209	0.2667
5	4	39	17	0.2151	0.2880	0.3452	0.4359

Table 3. Excel formulas for table 2

	A	B	C	D	E	F	G
1	Age t	Ships	Returns	1	2	3	4
2	1	47	1	$C$2/$B$2
3	2	41	7	SUM($C$2:$C3)/ SUM($B$2:$B3)	$C$3/$B$3
4	3	45	12	SUM($C$2:$C4)/ SUM($B$2:$B4)	SUM($C$3:$C4)/ SUM($B$3:$B4)	$C$4/$B$4
5	4	39	17	SUM($C$2:$C5)/ SUM($B$2:$B5)	SUM($C$3:$C5)/ SUM($B$3:$B5)	SUM($C$4:$C5)/ SUM($B$4:$B5)	$C$5/$B$5

The PAVA (Penalize Adjacent Violators algorithm [Marshall and Proschan]) smooths the estimates of G(t) = λ(t)G(t)/λ(t) so that G(t) is a non-decreasing distribution function. Table 3 shows the PAV algorithm computations.

Table 4. PAVA maximum likelihood estimates of G(t) and reliability.

	A	B	C	D	E	F
6	G(t)	R(t)	1	2	3	4
7	0.0213	0.9787	0.0213
8	0.1707	0.8293	0.0909	0.1707
9	0.2667	0.7333	0.1504	0.2209	0.2667
10	0.4359	0.5641	0.2151	0.2880	0.3452	0.4359

Table 5. Excel PAVA formulas for table 4. Spreadsheet users can see I’ve written formulas to copy down. 

	A	B	C	D	E	F
6	G(t)	R(t)	1	2	3	4
7	=MIN(C7:F7)	=1-A6	=MAX(C2:C$5)
8	=MIN(C8:F8)	=1-A7	=MAX(C3:C$5)	=MAX(D3:D$5)
9	=MIN(C9:F9)	=1-A8	=MAX(C4:C$5)	=MAX(D4:D$5)	=MAX(E4:E$5)
10	=MIN(C10:F10)	=1-A9	=MAX(C5:C$5)	=MAX(D4:D$5)	=MAX(E5:E$5)	=MAX(F5:F$5)

This maximum likelihood method extends to nonstationary Poisson input processes. It’s hard to disprove the nonstationary Poisson hypothesis from one observation for each age t [Nelson and Leemis]. However there’s a stationarity condition; “If E[λ(t–S)] is nondecreasing, then the mle [maximum likelihood estimator] of a nondecreasing failure rate function may be used to estimate E[λ(t–S)] and G(t)”, the distribution function of service time S [Eick, Massey and Whitt]. I program this test into some estimator workbooks [George, Oct. 2021]. 

Recommendations?

This method produces the maximum likelihood estimator for products or parts returns that are dead forever, not for renewals, “relevations”, or other recurrent process counts. I use least squares to estimate the reliability from ships and returns counts for renewal processes when the returns counts might be the first, second, or ??? return [George, 1993]. 

Warning, this article’s spreadsheet in the tables is oversimplified. It may produce #NA() values or negative numbers if returns exceed ships in some time period intervals. The Google spreadsheet may be more robust. If you want help, send some ships and returns counts, and I will send back the workbook that contains your field reliability estimates.

References

S. G. Eick, W. A. Massey, and W. Whitt, “The Physics of the Mt/G/Infinity Queue,” Ops. Res., Vol. 41, pp. 731-742, 1993 

L. L. George, “Estimate Reliability Functions Without Life Data”, ASQ Reliability Review, Vol. 13, No. 1, March 1993

L. L. George, “Gozinto Theory and Parts’ Installed Base”, Weekly Update, Gozinto Theory and Parts’ Installed Base – Accendo Reliability, May 2021

L. L. George, “Estimation of a Hidden Service-Time Distribution of an M(t)/G/∞ Self-Service System”, Weekly Update, Estimation of a Hidden Service-Time Distribution of an M(t)/G/∞ Self-Service System – Accendo Reliability, Oct. 2021

L. L. George and A. Agrawal, “Estimation of a Hidden Service Distribution of an M/G/∞ Service System,” Naval Research Logistics Quarterly, vol. 20, pp. 549-555, https://doi.org/10.1002%2Fnav.3800200314, 1973

Albert W. Marshall and Frank Proschan, “Maximum likelihood estimation for distributions with monotone failure rate,” Annals of Math. Statist., Vol. 36, pp. 69-77, https://doi.org/10.1214%2Faoms%2F1177700271, 1965

Barry L. Nelson and Lawrence M. Leemis, “The Ease of Fitting but Futility of Testing a Nonstationary Poisson Processes From One Sample Path,” Proceedings of the 2020 Winter Simulation Conference, IEEE, 2020