SRC Forum  Message Replies
Forum: Reliability & Maintainability Questions and AnswersTopic: Reliability & Maintainability Questions and Answers
Topic Posted by: Reliability & Maintainability Forum
(src_forum@alionscience.com
)
Organization: System Reliability Center
Date Posted: Mon Aug 31 12:47:36 US/Eastern 1998
Original Message:
Posted by: Geoffrey HampdenSmith
(geoff.hampdensmith@shell.com
)
Organization:Shell UK
Date posted: Fri Jun 24 5:40:18 US/Eastern 2005
Subject: Component failure rate and availability of unrevealed failure data
Message: We carry out valve testing at irregular periods ranging
from 12 to 364 days. (This arises as we take advantage of
opportunities duringprocess shut downs).
A series of successful tests (valve passes)takes place, and then the
valve is found to have failed (once the failure is revealed it is
repaired immediately in about 1 hour). Can the average availability
calculated as:
1)
[SUM TEST PERIODS (days) VALVE PASSED + 1/2*(SUM OF TEST PERIODS
(days)
IN WHICH VALVE FAILURES ARE REVEALED]/
[SUM TEST PERIODS VALVES PASSED (days) + TEST PERIODS (days) IN WHICH
VALVES FAILED]
(The Denominator equals the TOTAL TIME)
2)
Is the valve failure rate calculated as:
[NUMBER OF VALVE FAILURES]/[TOTAL of TEST PERIODS (days) VALVES
PASSED
+ 1/2*(SUM OF TEST PERIODS (days) IN WHICH VALVE FAILURES ARE
REVEALED)]
Reply:
Subject: Lengthbiased sampling and left censored failure rate
Reply Posted by: Larry George
(pstlarry@yahoo.com
)
Organization: Problem Solving Tools
Date Posted: Tue Jun 28 17:37:54 US/Eastern 2005
Message: Interesting questions!
1. Your attempt to adjust the asymptotic availability formula may be biased by the random inspections. Assuming the inpections constitute a renewal process, the interval in which a failure occurs is longer than that in which a failure doesn't occur. See renewal theory and length biased sampling.
Are you sure you want to use the asymptotic availability formula for a process with an annual cycle? Even assuming constant failure rate?
2. I did a quick and dirty derivation of the mle of the assumed constant failure rate (lamdba) assuming equal intervals between inspections. The likelihood is
PRODUCT[(RiRi1)^ri]*PRODUCT[Ri^(niri)]
where Ri is the reliability function, P[Life > i],
ri is the number of failures observed in (i1,i], and
niri is the number of survivors t age i.
With one little approximation (1exp(lambda)) ~ lambda, the max. likelihood estimator (mle) for lambda is
SUM[ri]/SUM[i*ni] = number of failures/total observation time.
In other words, this approximate mle uses the whole test period in the denominator.
The mle of lambda can be found numerically, without the approximation. Test the assumption of constant failure rate. Make the nonparametric mle of the failure rate function and compare it with the mle of constant lambda failure rate, incorporating random inspection intervals in the likelihood. Use age specific availability.
Send data, and I will send back and attempt to do those suggestions.
