I couldn't find any where to report this mistake so I'll report it
here. The version of this book published in 1998 appears to have the
wrong formula for the 1oo2 (1 out of 2) case on page 87, section 8.7.1
formula Set 2:. The error jumps out at you because P(a,b)=p(a)*p(b) if
a and b are independent. Thus the probability of failure for the 1oo2
case should be roughly equal to the square of the 1oo1 case. Thus we
suspect that the coefficient in front of the 2 out of 2 case should be
1 and not 2.

The formula for the 1oo1 case is very intuitive. That is the availability is equal to roughly the MTTR (mean time to repair) divided by the MTBF (mean time before failure). The book says these equations are simplifications of results derived from Markov models in the book: Reliability, Maintainability, and Risk by David J. Smith. http://www.knovel.com/knovel2/Toc.jsp?BookIDQ9

The exact formulas given by Gruhn are: 1oo2 lambda_d*(MTTR+(TI_a/2)) 1oo2 2*(lambda_d)^2*(MTTR+(TI_a/2))^2 2oo2 2

If we look at section 8.1 of Smith, table 8.6 we see that all of the formulas agree with Gruhn's formula's except for the 1oo2 case. We also see that our intuition is correct for the 1oo2 case. The other difference we notice is Smith does not include a term for the automatic diagnostic time in his book. However, later on Smith addresses the case in section 8.1.4 where the time to start repairs is not instantaneous but occurs at some periodic manual test interval T. This is very analogous to the automatic diagnostic time.

Grunth also provides formulas for this case in formula set 3 but instead calls the automatic diagnostic time the Manual test interval and misses the fact that as the MTTR approaches zero the formulas for the availability given above should approach the formulas for the availability where the Manual test interval is much greater then the MTTR but with the automatic diagnostic time replaced by the manual test interval.

The formulas' by Gruhn for the case where the manual test interval TI dominates the mean time to repair are:

1oo1 lambda_d*(TI/2) 1oo2 ((lambda_d)^2*(TI)^2)/3 2oo2 lambda_d*TI 2oo3 (lambda_d)^2*(TI)^2

These formula's agree with the formula's given by Smith in table 8.8 in section 8.1. Thus without derivation a reasonable conjecture of a general equation set is: 1oo2 lambda_d*(MTTR+(TI_a/2)) 1oo2 1*(lambda_d)^2*(MTTR+(TI_a/3))^2 2oo2 2

This of course ignores the nuances that some failures could be detected automatically well other types of error would need manual testing. For these cases proper expressions or simulated results would be needed using Markov models.

The formula for the 1oo1 case is very intuitive. That is the availability is equal to roughly the MTTR (mean time to repair) divided by the MTBF (mean time before failure). The book says these equations are simplifications of results derived from Markov models in the book: Reliability, Maintainability, and Risk by David J. Smith. http://www.knovel.com/knovel2/Toc.jsp?BookIDQ9

The exact formulas given by Gruhn are: 1oo2 lambda_d*(MTTR+(TI_a/2)) 1oo2 2*(lambda_d)^2*(MTTR+(TI_a/2))^2 2oo2 2

***(lamgda_d)***(MTTR+(TI_a/2)) 2oo3 6*(lambda_d)^2*(MTTR+(TI_a/2))^2 TI=Automatic diagnostic interval MTTR=mean time to repair Lambdaúilure rate (1/MTBF) dÚngerous (inhibiting) failureIf we look at section 8.1 of Smith, table 8.6 we see that all of the formulas agree with Gruhn's formula's except for the 1oo2 case. We also see that our intuition is correct for the 1oo2 case. The other difference we notice is Smith does not include a term for the automatic diagnostic time in his book. However, later on Smith addresses the case in section 8.1.4 where the time to start repairs is not instantaneous but occurs at some periodic manual test interval T. This is very analogous to the automatic diagnostic time.

Grunth also provides formulas for this case in formula set 3 but instead calls the automatic diagnostic time the Manual test interval and misses the fact that as the MTTR approaches zero the formulas for the availability given above should approach the formulas for the availability where the Manual test interval is much greater then the MTTR but with the automatic diagnostic time replaced by the manual test interval.

The formulas' by Gruhn for the case where the manual test interval TI dominates the mean time to repair are:

1oo1 lambda_d*(TI/2) 1oo2 ((lambda_d)^2*(TI)^2)/3 2oo2 lambda_d*TI 2oo3 (lambda_d)^2*(TI)^2

These formula's agree with the formula's given by Smith in table 8.8 in section 8.1. Thus without derivation a reasonable conjecture of a general equation set is: 1oo2 lambda_d*(MTTR+(TI_a/2)) 1oo2 1*(lambda_d)^2*(MTTR+(TI_a/3))^2 2oo2 2

***(lamgda_d)***(MTTR+(TI_a/2)) 2oo3 6*(lambda_d)^2*(MTTR+(TI_a/2))^2This of course ignores the nuances that some failures could be detected automatically well other types of error would need manual testing. For these cases proper expressions or simulated results would be needed using Markov models.