Duane 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).

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, at time T is: (Note: The development or TAAF (Test-Analyze-And-Fix) time considered in reliability growth in terms of the MTBF, failure rate or reliability improvement for continuously operating units, is the unit hours accumulated by one or more units involved in the growth process. For one-shot items, the TAAF time considered in reliability growth is the accumulated number of missions, events, launches, or discrete periods of operation, e.g. month of TAAF or cycles of operations to a failure.)

The equation of the line can be expressed as:

Setting:

yields:

Then equating 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, , is:

or:

(2)

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

(3)

where:

The corresponding , or , 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 can be determined, or the predicted at the start of the design and development process (b > 0).

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

(5)

Similarly, using Eqn. 3, this procedure yields:

(6)

where 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 . 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.

This chapter includes the following sections:

Parameter Estimation for the Duane Model

Confidence Bounds for the Duane Model

General Examples Using the Duane Model

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