MTBF Prediction With Variable Operating Hours Per Calendar Hour

”What if some subsystems don’t operate full time? How does that change MTBF prediction?”

### I replied…

“Good question! Funny, none of the MTBF prediction guides deal with this. There are at least three cases, of increasing complexity:

1.      Some subsystem works a known, fixed proportion of calendar time; e.g., 8 hours per day, 7 days per week, 52 weeks per year, so scale failure rates by the proportion

2.      Some customers’ subsystem works 8 hours per day and other customers’ subsystem work 16 hours per day; say ~50% of each

3.      Some subsystems work a random proportion of calendar time between zero and one

I modified the MIL-HDBK-217F MTBF prediction workbook to handle these alternatives and post in on the Internet.”

### Theory

Case 1 is how some people deal with operating hours less than calendar hours; e.g., http://quanterion.com/RIAC/Publications/RIACJournal/PDFFiles/3RD_Q1999.pdf and http://www.galaxyscientific.com/agingaircraft2002/SESSIONS/POSTERS/Dylis_PTR_DOC.pdf.

These web sites explain how a customer can scale his or her MTBF prediction for different operating time ratios to calendar hours. They do not explain how a supplier should modify an MTBF prediction for customers with random operating time ratios per calendar hour. Case 2 covers that.

Case 2 is almost as easy; multiply subsystem failure rates by ratios and their probabilities. I.e., let r(i) and p(i) denote possible operating/calendar-hour ratios and their probabilities. Then

Subsystem-failure-rate(calendar time) = Subsystem-failure-rate(operating time)*Sr(i)p(i)

where the sum is over all customer subsets and Sp(i) = 1.0. This scaling is a consequence of the constant failure rate assumption in traditional MTBF prediction. This is the P-factor method of scaling subsystem failure rates. With a discrete probability distribution of ratios; the P-factor is Sr(i)*p(i). Cases 1 and 2 are implemented in

The unknown complicates Case 3. In predictions, the probability distribution of the ratio of new system operating hours to calendar hours is unknown. So use operating data from prior, similar systems to estimate that probability distribution. Assume a beta probability distribution,

(1-r)r(a-1)(1-r)(b-1)/B(a,b),

where B(a,b) is the beta function, a ratio of gamma functions, http://en.wikipedia.org/wiki/Beta_distribution. This assumption was good enough for PERT, and it’s convenient to estimate the parameters a and b from three values, http://www.visionarytools.com/decision-making/3-point-estimating.htm

The reliability function of a subsystem with constant failure rates and random, beta-distributed proportion of operating hours to calendar hours is

òexp[-lt/r](1-r)r(a-1)(1-r)(b-1)/B(a,b)dr

where l is the subsystem failure rate and integration is from zero to 1. This reliability function is not exponential and its failure rate is not constant. Don’t add subsystem failure rates to get system failure rate and invert the sum for an MTBF prediction! Subsystem reliability function involves gamma, beta, and confluent hypergeometric functions. Don’t panic, they’re computable. Subsystem MTBF has a rather simple formula,

G(1+a)G(b)/(lB(a,b)G(1+a+b)).

The resulting the system MTBF of a system of multiple subsystems can be simulated, although perhaps it is not possible to not solve for MTBF symbolically.

### It later occurred to me that...

MTBF prediction guides have a reason for not dealing with part-time operation. Adjusting MTBF predictions loses comparability, perhaps the only value to MTBF predictions. Vendors should predict MTBF for full time operation and let each customer adjust for his or her own operating hours per calendar hour.

#### What about operating cycles and hours?

Part lives may be the minimum of operating cycles or hours, assuming they haven’t failed before their life limits. Let’s deal with this in two steps: life limits in terms of hours and life limits as the Min[cycles, hours], with random cycles per operating hour and perhaps random operating hours per calendar hour.

Typically, MTBF predictions are in hours alone, assuming no life limits whatsoever. Imposing life limits in terms on operating hours is equivalent to finite mission time. Making MTBF predictions for a finite-time mission is explained in “How Many Spares are Needed for a Finite Mission with Repair?” Fernau.htm. The simulated MTBF prediction is much greater than mission time for reasonably reliable systems, and the ratio, mission time/MTBF prediction is the probability of failure during a mission, under the usual assumption of constant failure rates.

If the ratio of cycles to operating hours is random, then simulate TBFs, in hours, even though the cycle limit may terminate system lives. Input the joint distribution of cycles and operating hours (and operating hours per calendar if random). Alternatively, input the distribution of cycles per operating hour, if that ratio is independent of age. (Old aircraft may to be used on shorter hauls thereby increasing cycles, takeoffs and landings, per flying hour.) In each simulation, save the operating hours at which the cycle limit is reached, if no prior failure occurs. Average the simulated operating hours to get an MTBF prediction and its standard deviation, representing sample uncertainty. Convert that to a calendar hour MTBF prediction using the methods of case 1 or 2.

#### Recommendations

If Cases 1 or 2 suit you, use the workbook, MH217F3.xls. Please let me know if you have criticism, corrections, or additions. If Case 3 suits you, send field data on operating hours per calendar hour to pstlarry@yahoo.com and I will estimate the beta distribution parameters and simulate your system MTBF prediction and its standard deviation, free of charge. If you have cycle and hour life limits, I will modify Fernau.xls to suit your joint distribution of cycles and hours.

Question the constant failure rate assumption! Test that assumption. Send field data on parts and I will send back nonparametric estimates of age-specific field failure rates, and modify the Case 3 MTBF prediction simulation to suit. It’s easy to program the simulation of any discrete distribution, because it simulates ages at failures from failure rate function estimates.