Duane Model

This chapter includes the following sections:

Model History and Development

In 1962, J. T. Duane published a report in which he presented failure data of different systems during their development programs [8]. While analyzing the data, it was observed that the cumulative MTBF versus cumulative operating time followed a straight line when plotted on log-log paper (Figure 4.1).

duane.wmf

Figure 4.1: Cumulative MTBF vs. Cumulative Test Time postulated by Duane

Based on that observation, Duane developed his model as follows. If N(T) is the number of failures by time T, the observed mean (average) time between failures, MTBFc at time T is: [Note]

The equation of the line can be expressed as:

Setting:

MATH

yields:

Then equating MTBFc to its expected value, and assuming an exact linear relationship, gives:

or:

(1)

And, if you assume a constant failure intensity, then the cumulative failure intensity, λc, is:

or:

(2)

Also, the expected number of failures up to time T is:

MATH (3)

where:

= the average estimate of the cumulative failure intensity, failures/hr.
T = the total accumulated unit hours of test and/or development time.
1/b = the cumulative failure intensity at T = 1, or at the beginning of the test, or the earliest time at which the first is predicted, or the for the equipment at the start of the design and development process.
α = the improvement rate in the , 0 £ α £ 1.

The corresponding MTBFc, or $\hat{m}_{c}$ , is equal to:

(4)

where b = cumulative MTBF at T = 1 or at the beginning of the test, or the earliest time at which the first $\hat{m}$ can be determined, or the $\hat{m}$ predicted at the start of the design and development process (b > 0).

The cumulative MTBF, $\hat{m}_{c}$ , and tell whether m is increasing or λ is decreasing with time, utilizing all data up to that time. You may want to know, however, the instantaneous $\hat{m}_{i}$ or to see what you are doing at a specific instant or after a specific test and development time. The instantaneous failure intensity, λi, is:

MATH (5)

Similarly, using Eqn. (3), this procedure yields:

MATH (6)

where α = 1 implies infinite MTBF growth.

It can be seen from Eqn. (5) that the instantaneous failure intensity improvement line is obtained by shifting the cumulative failure intensity line down, parallel to itself, by a distance of (1 - α). Similarly, it can be seen from Eqn. (6) that the current or instantaneous MTBF growth line is obtained by shifting the cumulative MTBF line up, parallel to itself, by a distance of , as illustrated in Figure 4.1.