Accommodating degradation results within a qualification procedure

Reliability Engineering and System Safety 72 (2001) 103±109

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Accommodating degradation results within a quali®cation procedure A.E. Gera* Department of Quality Assurance Ð 3495, ELTA Electronics Industry Ltd, P.O. Box 330, Ashdod 77102, Israel Received 14 October 2000; accepted 8 November 2000

Abstract A quali®cation procedure may comprise different tests which may either be successful or a failure. However, there are cases of degradation which should be taken into account. The reliability of a consecutive k-out-of-n:G set of tests is evaluated, along with the possibility of degradation. The problem is formulated as a difference equation that is transformed into a digital ®lter equation. Numerical results indicate improved chances of success for a set of tests that take into account the case of degradation. q 2001 Elsevier Science Ltd. All rights reserved. Keywords: Consecutive k-out-of-n; Filter equation; Quali®cation test

1. Introduction Quali®cation tests for a system may involve the requirement of passing a set of k consecutive trials within the general set of n tests. This is the so-called `consecutive kout-of-n:G' system, which attracted a lot of research during the recent decades [1±6]. It is complementary to the `consecutive k-out-of-n:F' system, where a system fails whenever k consecutive trials fail. The basic problem assumes independent and identical probabilities of success for every test. Some references include the possibility of non-identical reliabilities or the dependence of the chances of passing each test on the previous results. Some work on this topic has been carried out earlier by the author [3±5]. Normally, it is assumed that each test may either be successful or failure (binary reliability). However, sometimes a certain operational degradation may occur which yields an undetermined state-of-result of a test (multivalued reliability). This possibility should be taken into account although the model and mathematics become more complicated. In the following sections, the multivalued case will be introduced as a modi®cation of the regularly accepted binary reliability. The problem of the reliability of a consequential k-out-ofn:G will be formulated as a linear recursion relation which may be transformed into a linear difference equation. Such an equation will be recognized as a digital ®lter equation which may be solved using the familiar Z transform. This

* Fax: 1972-8-857-2330. E-mail address: [email protected] (A.E. Gera).

technique has been applied previously to the case of binary resulting tests, and will be generalized. 2. Assumptions 1. Each test may be successful (s), degraded (dg), or fails (f). 2. The set of tests passes successfully if and only if it includes a subset of consecutive k successful tests. 3. Tests showing limited functional degradation are counted neither as a success nor as a failure. They do not `break' a chain of successful tests.

3. The mathematical model It has been previously shown that the model equation for the binary reliability is [3±5]: R…p; k; n† ˆ pk u‰n 2 kŠ 1 q

k X

pt21 R…p; k; n 2 t†

…3:1†

tˆ1

with the following boundary conditions for any t , k R…p; k; t† ˆ 0

…3:2†

Eq. (3.1) was solved using the Z transform. Closed-form expressions have been derived for different ranges of k/n. Explicitly, for k # n # 2k : R…p; k; n† ˆ ‰1 1 …n 2 k†qŠpk

0951-8320/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S 0951-832 0(00)00107-1

…3:3†

104

A.E. Gera / Reliability Engineering and System Safety 72 (2001) 103±109

Nomenclature n Number of tests k Minimum number of consecutive passed tests for a successful set of tests m n2k u [´] Unit step function Xi, s, dg, f Result of test i: Xi ˆ s (success), dg (degradation), f (failure) p,d,q p ˆ Pr{Xi ˆ s}; d ˆ Pr{Xi ˆ dg}; q ˆ Pr{Xi ˆ f} R(p, k, n) Reliability of a consequential k-out-of-n:G system with binary results (s or f) R(p, d, k, n) Reliability of a consequential k-out-of-n:G system with multi-valued results (s, f, dg) n!=…r!…n 2 r†!† nCr and more cumbersome expressions were presented for other ranges. Generalization is now carried out on the case for which a chain of successful tests may include degraded ones that will neither be counted as successes nor as failures. The following summarizes all the possible combinations that may yield a successful set of tests:

what different from Eq. (3.1). However, it may be easily checked to see if their solutions are identical. Eq. (3.4) is a linear difference equation. De®ning new independent and dependent variables: mˆn2k

…3:6†

y…t† ˆ R…p; d; k; t†

…3:7†

(a) First test: any result; remaining tests: successive k results `s' within n 2 1 tests (b) For any k 2 2 , r , n : if n ˆ r 1 1

first test : s

r remaining ones : each `s' or `dg'; including exactly k 2 1 results `s'

if n . r 1 1

first test : s

r next ones : each `s' or `dg' including exactly k 2 1 results `s'

test no:…r 1 2† : f

remaining n 2 …r 1 2† tests : no consecutive k results `s' Eq. (3.4) accepts the form:

The model equation will therefore be: R…p; d; k; n† ˆ R…p; d; k; n 2 1† 1

nX 21 rˆk 2 1

y…n† ˆ y…n 2 1† 1 r Ck21 p

k r2…k21†

d

nX 21 rˆk 2 1

{u‰n 2 …r 1 1†Š

fr ‰1 2 y…n 2 …r 1 2††Š

…3:8†

where

1 …q 2 1†u‰n 2 …r 1 2†Š}  ‰1 2 R…p; d; k; n 2 …r 1 2††Š

(3.4)

The boundary conditions are for t , k : R…p; d; k; t† ˆ 0

…3:5†

The private case of d ˆ 0 yields an equation which is some-

fr ˆ r Ck21 pk dr2…k21† {u‰n 2 …r 1 1†Š 1 …q 2 1†u‰n 2 …r 1 2†Š}

The boundary conditions are: y…0†; y…1†; ¼; y…k 2 1† ˆ 0

Fig. 1. Transposed direct-form II structure.

…3:9†

A.E. Gera / Reliability Engineering and System Safety 72 (2001) 103±109

105

Fig. 2. Probability of success for p ˆ 0:5; k ˆ 2; for various values of n and d.

4. Method of solution

and

Difference equation (3.4) may be replaced by a ®rst-order matrix difference equation. Such a transformation has been performed by the author in a recent publication [5]. However, the equation is recognized as a digital ®lter equation of the type:

an12 ˆ

n21 Ck21 p

k n2k

bn12 ˆ

n21 Ck21 p

k n2k

nX 21

y…n† ˆ y…n 2 1† 2

rˆk 2 1

1

nX 21

Y…z† 2

N X rˆ1

…4:1†

d

Oppenheim and Schafer [7] presented an algorithm to solve the ®lter equation with M inputs and N outputs. They considered the Z-transformed equation:

ar13 y…n 2 …r 1 2††

br13 x…n 2 …r 1 2†† 1 bk11 x…n 2 k†

d

ar z2r Y…z† ˆ

M X rˆ1

br z2r

…4:2†

for which {y(n)} are the set of outputs and {x(n)} the set of inputs. The expressions for the coef®cients are:

which is implemented as the so-called `transposed directform II structure' as shown in Fig. 1. This structure requires the minimal number of memory cells for storing the states, represented in Fig. 1 as junctions in the central horizontal line.

bk11 ˆ pk

5. Numerical results

rˆk 2 1

and for 1 , s , n 2 k 1 2; ak1s ˆ

k1s23 Ck21 p

k s22

d

{u‰n 2 …k 1 s 2 2†Š

1 …q 2 1†u‰n 2 …k 1 s 2 1†Š}

bk1s ˆ

k1s23 Ck21 …q

2 1†pk d s22 1k1s22 Ck21 pk ds21

The practical calculation of various examples was found to be very ef®cient using the Matlab software package. Results are presented for various examples appearing in the literature. The reliability values of the k-out-of-n:G systems for the binary case (for which every test may either succeed or fail) are compared to systems involving the possibility of accommodating functional degradation of some tests. The following values of the reliability of the set of tests have been evaluated (Figs. 2±8):

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A.E. Gera / Reliability Engineering and System Safety 72 (2001) 103±109

A. p ˆ 0:5; k ˆ 2 (Fig. 2) d 0

0.1

0.2

0.3

0.4

Kuo et al. [6] (d ˆ 0)

n 2 3 4 5 6

0.25 0.4 0.5275 0.6265 0.7050

0.25 0.425 0.56 0.6633 0.7423

0.25 0.45 0.5975 0.7055 0.7845

0.25 0.475 0.64 0.7548 0.8333

0.25 0.375 0.5 0.59375 0.671875

B. k ˆ 3; n ˆ 5; p ˆ 0:25; 0:5; 0:75 (Figs. 3±5) d 0 0.1

0.2

Sfakianakis et al. [8] (d ˆ 0)

p 0.25 0.5 0.75

0.0511 0.3275 0.8311

0.039 0.25 0.633

0.2

0.3

0.4

Kuo et al. [6] (d ˆ 0)

0.0959

0.1689

0.3130

0.0352

0.2

0.3

0.4

Sfakianakis et al. [8] (d ˆ 0)

0.0296

0.0737

0.2005

0.007

0.2

0.3

0.4

Sfakianakis et al. [8] (d ˆ 0)

0.0089

0.0267

0.0829

0.001

0.25 0.375 0.5 0.5938 0.6719

0.0391 0.25 0.6328

0.0446 0.2850 0.7193

C. p ˆ 0:5; k ˆ 7; n ˆ 14 (Fig. 6) d 0 0.1 0.0352

0.0570

D. p ˆ 0:5; k ˆ 10; n ˆ 20 (Fig. 7) d 0 0.1 0.0059

0.0128

E. p ˆ 0:5; k ˆ 12; n ˆ 20 (Fig. 8) d 0 0.1 0.0012

0.0032

Fig. 3. Probability of success for p ˆ 0:75; k ˆ 3; for various values of n and d.

A.E. Gera / Reliability Engineering and System Safety 72 (2001) 103±109

Fig. 4. Probability of success for p ˆ 0:5; k ˆ 3; for various values of n and d.

Fig. 5. Probability of success for p ˆ 0:25; k ˆ 3; for various values of n and d.

107

108

A.E. Gera / Reliability Engineering and System Safety 72 (2001) 103±109

Fig. 6. Probability of success for p ˆ 0:5; k ˆ 7; for various values of n and d.

Fig. 7. Probability of success for p ˆ 0:5; k ˆ 10; for various values of n and d.

A.E. Gera / Reliability Engineering and System Safety 72 (2001) 103±109

109

Fig. 8. Probability of success for p ˆ 0:5; k ˆ 12; for various values of n and d.

Obviously, the chances of passing the set of experiments increase if the possibility of functional degradation is taken into account. Results for the binary reliability correlate with those in the literature.

6. Conclusions The problem of evaluating the reliability of a consecutive k-out-of-n:G set of tests has been generalized to include the possibility of tests with some functional degradation. Such tests are neither considered as successes nor as failures. It is thus assumed that we should `skip' over such tests when searching for a continuum of k successes. This seems to be a practical contribution since normally many tests exhibit some functional degradation. Further work should incorporate different types of severity together with their chances of occurrence.

References [1] Bollinger RC, Salvia AA. Consecutive-k-out-of-n:F networks. IEEE Trans Reliab 1982;R-31:53±6. [2] Chiang DT, Niu SC. Reliability of consecutive-k-out-of-n:F system. IEEE Trans Reliab 1981;R-30:87±9. [3] Gera AE. A generalized version of the MTBF assurance test. Proceedings of the Eleventh International Conference of the Israel Society for Quality, Jerusalem, November 1996. p. 406±11. [4] Gera AE. An MTBF assurance test involving reliability improvement. Proceedings of the Thirteenth International Conference of the Israel Society for Quality, Jerusalem, December 1998, Jerusalem. [5] Gera AE. A consecutive k-out-of-n:G system with dependent elements Ð a matrix formulation and solution. Reliab Engng Syst Safety 2000;RESS-68:61±7. [6] Kuo W, Zhang W, Zuo M. A consecutive k-out-of-n:G system: the mirror image of a consecutive-k-out-of-n:F system. IEEE Trans Reliab 1990;R-39:244±53. [7] Oppenheim AV, Schafer RW. Discrete-time signal processing. Englewood Cliffs, NJ: Prentice-Hall, 1989 (p. 311±12). [8] Sfakianakis M, Kounias S, Hillaris A. Reliability of a consecutive kout-of-r-from-n:F system. IEEE Trans Reliab 1992;R-41:442±6.