Reliability Growth during a Development Testing Program

Abstract

During i t s development t e s t i n g , a system may undergo modification t o remedy design weaknesses which t h e tests r e v e a l . The changes are made t o improve t h e r e l i a b i l i t y of t h e system, and i f such improvanent o c c u r s , we s a y t h a t r e l i a b i l i t y growth i s t a k i n g place.This RAND Memorandum d e r i v e s s t a t i s t i c a l methods f o r e s t i m a t i n g r e l i a b i l i t y under growth assumptions which merely r e q u i r e t h a t changes made t o t h e system which is being t e s t e d do not make i t worse. The methods d i f f e r from previous methods, which e i t h e r ignore growth o r assume t h a t growth i s t a k i n g place i n a n assigned f u n c t i o n a l manner. While t h i s Memorandum i s addressed p r i m a r i l y t o s t a t i s t i c i a n s , i t should a l s o be of i n t e r e s t t o test engineers and managers concernedw i t h a s s e s s i n g a system's r e l i a b i l i t y .taken as a p a r t of t h e Apollo Contingency Planning Study which RAND is conducting f o r Headquarters, NASA, under Contract NASr-21(09).of t h e a u t h o r s , Richard E. Barlow, i s a c o n s u l t a n t t o The RAND Corporation. The i n v e s t i g a t i o n w a s under- One -V- SUMMAKYThis study examines t h e problem of e s t i m a t i n g t h e r e l i a b i l i t y o f a system t h a t is undergoing development t e s t i n g . I n such a program, changes a r e made t o t h e system from time t o time i n o r d e r t o i n c r e a s e i t s r e l i a b i l i t y . n o t d e l e t e r i o u s , and, u n l i k e some previous work i n t h i s area, i t does n o t assume t h a t system modifications cause r e l i a b i l i t y growth according t o a p r e s c r i b e d f u n c t i o n a l form. The method described h e r e i n does, however, r e q u i r e t h a t each f a i l u r e be c l a s s i f i e d a s i n h e r e n t o r ref l e c t i n g a c o r r e c t a b l e cause. This study assumes t h a t t h e s e changes a r e a t l e a s tThe study proceeds on t h e supposition t h a t t h e test program is conducted i n K s t a g e s , with s i m i l a r items b e i n g t e s t e d w i t h i n each s t a g e . For each s t a g e , t h e number o f i n h e r e n t f a i l u r e s , o f a s s i g n a b l e cause f a i l u r e s , and o f successes is recorded.It is supposed t h a t t h e p r o b a b i l i t y of a n inherent f a i l u r e , t h e test program and t h a t t h e p r o b a b i l i t y of an a s s i g n a b l e cause f a i l u r e i n t h e i -t h s t a g e , qi, does n o t i n c r e a s e from s t a g e t o s t a g e of t e s t i n g . This Memorandum obtains maximum l i k e l i h o o d e s t i m a t e s o f q and a l s o o b t a i n s .a c o n s e r v a t i v e lower confidence i n t e r v a l f o r r t h e r e l i a b i l i t y o f t h e system i n i t s f i n a l c o n f i g u r a t i o n o f t h e t e s t progra remains t h e same throughout ,and o f t h e q 's s u b j e c t t o t h e c o n d i t i o n t h a t they be non-increasi...

A Class of General Reliability Growth Prediction Models

“…Whether corrective action takes place through design changes or development modifications, it is common to assume that the action never decreases reliability and, at least occasionally, actually increases reliability, hence the term reliability growth. Generally speaking, the mathematical models developed for this problem concern two objectives: (1) the prediction, before testing is initiated, of the reliability that should be attained at various stages of development; and (2) the estimation of reliability as a function of one or more parameters from test data. In general, the latter objective is attained by making statistical inferences from observed test data, while the former is gained through the use of probabilistic models using certain assumptions concerning the failure modes and the repair policy.…”

“…A testing program is carried out for the purpose of discovering and removing the failure mode. On each trial, the system fails with (known) probability q (i. e. , the system has reliability 1 Weiss [5] …”

“…Substituting this in (1) yields the predicted reliability of the system after n trials, is the distribution of states occupied by the system after n trials. The matrix form (3) of the reliability given in (1) is easily extended to the present case, so that the reliability after n trials is given by…”

“…The matrix form (3) of the reliability given in (1) is easily extended to the present case, so that the reliability after n trials is given by…”

“…(n-1) C n ,(s) P: (1) i be obtained using the one-step transition probabilities given in (5) and (6) together with equation (8) . Thus,…”