Issue 52, June 2005

[Editor's Note: This article has been updated since its original publication to reflect a more recent version of the software interface.]

Lloyd and Lipow (1962) considered a situation in which a test program is conducted in N stages, each stage consisting of a certain number of trials of an item undergoing testing, and the data are recorded as successes or failures. All tests in a given stage of testing involve similar items. The results of each stage of testing are used to improve the item for further testing in the next stage. For the k^th group of data, taken in chronological order, there are n_k tests with S_k observed successes. The reliability growth is then:

(1)

such that:

    R_k= the actual reliability during the stage of the testing

    R   = the ultimate reliability that would be attained if k

    α > 0 = modifies the rate of growth

Note that, essentially, R_k =S_k /n_k. If the data given are reliability data, then S_k is assumed to be the observed reliability given and n_k is considered 1.

In ReliaSoft's RGA software, this model is available for the Success/Failure (Sequential, Sequential with Mode, Grouped per Configuration) data type and the Reliability data type.

Maximum Likelihood Parameters Estimations

For the k^th stage:

and assuming that the results are independent between stages:

Then taking the natural log gives:

Differentiating with respect to R and α yields:

(2)

(3)

Rearranging Eqns. 2 and 3 and setting equal to zero gives:

Eqns. 4 and 5 can both now simultaneously be solved for R  and α. It should be noted that there is no closed form solution for either of the parameters, thus they must be estimated numerically.

Least Squares Parameter Estimation

To obtain least squares estimators for R and α,the sum of squares, Q, of the deviations of the observed success-ratio, S_k/n_k, is minimized from its expected value R - α /k with respect to the parameters R  and α. Thus, Q is expressed as:

Taking the derivatives with respect to R and α and setting equal to zero yields:

(6)

(7)

Solving Eqns. 6 and 7 simultaneously, the least squares estimates of R and α are:

or:

and:

or:

Example

The following table presents results of a 20-stage reliability development test program for a high volume rotary/vortex downhole gas separator designed to maximize production from wells with significant gas content. Improvements were implemented between these developmental testing stages. The table lists 20 groups that were tested sequentially and indicates the number that were tested in each group and the number of failures obtained from each group.

The previous figure also shows the MLE estimates of Lloyd-Lipow R and α parameters, which are found to be 0.8690 and 0.7582 respectively. The reliability growth throughout the 20 developmental stages is shown next.

The estimated reliability, based on the calculated parameters, at the end of the 20th stage is 83.11%. The next figure shows how this value can be obtained in RGA.

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