The Anomaly of Calculating MTBF from Reliability

[itKiwi]

I would like some help in practical, non-numeric, terms to explain the following “anomaly”.

We all know that Reliability, A = MTBF /(MTBF + MTTR)

So, the higher(longer) the MTBF and the lower(shorter) the MTTR, the better (more 9's) our Reliability.

At the end of our spreadsheet, we have calculated the reliability of our complex system including all the series and parallel elements, redundant elements, etc. A = 0.9999999, etc. Great.

Now, the client wants to know the MTBF of his system. OK. We rearrange the above formula, and we get,

MTBF = A x MTTR/(1-A)

What's this ? MTBF is directly proportional to MTTR, which means the longer my MTTR, the better my MTBF !!! This is contrary to the original basic concept, that a short MTTR is best.

How do we explain this to a non-numeric client ?

It also raises the question; “Which MTTR do I use in this system calculation ? In practical terms, one would think it ethical to use the longest MTTR, (gives us the lowest Reliability). However, this gives us our best system MTBF.

Any answers or discussions on these two points would be appreciated.

Thanks.

[Tim Folkerts]

itKiwi,

The variable A is typically called the Availability. Reliability is the probability of working at a particular time under specific circumstances. Availability typically considered a constant (a long-term average of how much of the time the system is available); reliability decreases with time.

The two are obviously related, but not the same. A common function to express reliability is
R = exp(-t/MTTF)
but the actual function to use depends on the situation.

MTBF = A x MTTR/(1-A)

What's this ? MTBF is directly proportional to MTTR, which means the longer my MTTR, the better my MTBF !!! This is contrary to the original basic concept, that a short MTTR is best.

Ah, but MTBF is not directly proportional to MTTR. MTBF also depends on A, which in turn depends on MTTR. So when you change, MTTR, you also change A.

With a little algebra, it is easy to show that the equation above reduces to
MTBF = MTFB

In other words, MTBF in independent of MTTR, as would be expected.

If you did want to use this equation to calculate to MTBF, you would need the MTTR of the system. This would be some sort of average of the MTTR for each element combined with the probability of each component actually failing.


To calculate the MTBF, the best answer is probably just to go back to the original numbers. If you know the MTBF for each element and the appropriate series & parallel combinations, you should be able to find the overall MTBF directly.


Tim F