Renewal vs. Generalized Renewal Process?

How to distinguish a renewal process from a “generalized” renewal process? Compare observed monthly returns vs. actuarial returns forecasts using actuarial return rate estimates of TTFF and TBF (Time To First Failure and Time Between Failures). A geophysicist masquerading as an Apple reliability engineer said, “It’s too hard to figure out the probability that a return came from a computer made in an earlier year.”  It’s harder if returns could be second, third, or???

Renewal processes have independent and identically distributed times between returns. Generalized renewal processes have different distributions for TTFF and subsequent TBFs. I made nonparametric estimates of generalized renewal process TTFF and TBF distributions from M88-A1 unit starts and engine rebuilds. I did the same from published periodic Ford sales and warranty returns or repair counts [George 2001 and 2021, Salzman and Liddy (1988 Ford V-8 460 CID engines)]. Automotive publications gave 1988 me Ford sales counts. I sent the nonparametric estimates to Ron Salzman. He sent back the real sales counts.

Could Test Data be a renewal process?

Recently, someone on LinkedIn asked a question about distinguishing first, second, third, etc. returns from test data. I didn’t understand the question so I looked for their data. Table 2 shows first and subsequent multiple returns from the 245 units that were observed. The column “Cumulative Multiple returns” could include 3^rd, 4^th, etc. returns as well as 2^ndreturns.   

Table 1. First return and multiple cumulative returns data from retech-mtbf.com

Mo-Yr	Cumulative Returns	Cumulative Field Returns	Cumulative  Multiple  Returns	Returns per Month	Multiple Returns per Month
Mar-14	4	4	0	4	0
Apr-14	12	12	0	8	0
May-14	17	18	1	6	1
Jun-14	34	36	2	18	1
Jul-14	75	80	5	44	3
Aug-14	86	91	5	11	0
Sep-14	101	107	6	16	1
Oct-14	108	115	7	8	1
Nov-14	117	125	8	10	1
Dec-14	159	174	15	49	7
Jan-15	188	210	22	36	7
Feb-15	201	226	25	16	3
Mar-15	220	253	33	27	8
Apr-15	228	261	33	8	0
May-15	239	278	39	17	6
June-15	245	284	39	6	0

It’s easy to estimate the reliability R(t) of TTFF from the Cumulative Returns column r(t) of table 1, if you know when each units started operating. If I assume units started operating In February of March of 2014, R(t) = r(t)/245, t=1,2,…,16, and MTFF = 8.7 months. If units were started operating randomly before and during 2014 and 2015, then it is still possible to compare TTFF and TBF [George, 2008].

I used the spreadsheet for the M88-A1 engine generalized renewal process to estimate the distribution of TBFs from the “Cumulative Returns” and “Cumulative Multiple Returns” columns of table 1 [George, Oct. 2001]. The spreadsheet computes actuarial return rates, a(t)=(R(t)-R(t+1))/R(t), to see whether they produced accurate actuarial forecasts of multiple return counts. The actuarial multiple returns forecast for age t months is ∑a(s)n(t-s), s=1,2,…,t, where n(t-s) is the number of units of age t-s that have already returned once (differences in successive entries in column “Cumulative Multiple Returns”.  

Compare with observed multiple returns at each age, SSE=SUMXMY2(E[multiple Rets], observed). Excel Solver finds the TBF probability density function for the actuarial return rates a(s) that minimized SSE. The actuarial return rates in table 2. 

Table 2. Actuarial return rate estimates for TTFF and TBFs

Age, Months	TTFF	TBFs
1	0.016327	0.055695
2	0.033195	0
3	0.021459	0
4	0.074561	0
5	0.194313	0
6	0.064706	0.004975
7	0.09434	0.097545
8	0.048611	0.024306
9	0.065693	0.084243
10	0.328125	0
11	0.337209	0
12	0.22807	0
13	0.431818	0
14	0.32	0
15	0.647059	0
16	1.	0

Conclusions and Recommendations?

This article shows that renewal process estimation can be done, without TBF lifetime data. It would have been nice to have the data on when units started operating so actual TTFF reliability could be estimated, instead of assuming the came from February-March 2024. I asked for that data. You don’t need the actual unit start dates;  monthly unit counts would have been statistically sufficient.  

This article analyzed multiple returns as if they could include third and perhaps fourth, etc. returns [George, 2001 and 2008]. The TTFF and TBF reliability function estimates are clearly different so returns could be a generalized renewal process if TBF2, TBF3, etc. had same or similar distributions to TBF. 

1988 Ford 460 CID engines did have different TTFF and TBF, because TTFF included time for shipment from factory to dealer plus time from dealer receipt to sale and the first time the new owner took it back to the dealer to fix drivability complaints. Subsequent returns were because dealers couldn’t fix the problem(s). The 1988 Ford V-8 460 engine was the last Ford with carburetors.

The TTFF and TBF for the test units in table 1 could differ because somebody fixed the cause of the first failures. I recommend computing the reliability function estimates for TTFF, TBF1, TBF2, etc. and using Statistical Reliability Control to quantify reliability growth [George 2024], not just MTBF growth.  

References

L. L. George, “Statistical Reliability Control,” Weekly Update, https://fred-schenkelberg-project.prev01.rmkr.net/statistical-reliability-control/#more-522710/, Aug. 2023

L. L. George, “Renewal Process Estimation Without Life Data,” Weekly Update, https://fred-schenkelberg-project.prev01.rmkr.net/renewal-process-estimation-without-life-data/, Sept. 2021 

L. L. George, “Estimate Renewal Process Reliability without Renewal Counts,” ASQ Tech Briefs, Vol. 2, 2008

L. L. George, “User Guide for M88 A1 Field Reliability Estimation.” Oct. 22, 2001, revised Oct. 25, 2001

Salzman, Ronald H. and Richard G. Liddy. “Product Life Predictions from Warranty Data.” SAE Transactions, vol. 105, pp. 908–11. JSTOR, http://www.jstor.org/stable/44734119, 1996