ELSEVIER

Reliability Engineering and System Safety 44 (1994) 11-15 © 1994 Elsevier Science Limited

Printed in Northern Ireland. All fights reserved 0951-8320/94/$7.00

Bayesian reliability analysis of a k-out-of-m system and the estimation of sample size and

censoring time

K. K. Sharma & Hare Krishna Department of Statistics, Meerut University, Meerut, Pin-250004, (UP)India

(Received 5 April 1993; accepted 4 August 1993)

In the absence of lifetime data on a complete system, it is desirable to use operational experience on its components in the Bayesian analysis of the system. The present study deals with Bayesian reliability analysis of a k-out-of-m system using two types of censored failure information. In another development, Bayesian confidence intervals for unreliability have been used for estimating the sample size and the censoring time needed to get sufficient failure information.

1 INTRODUCTION

Bayesian reliability analysis combines the prior knowledge of the random variations in the parameters of the lifetime distribution with observed lifetime data. Some recent studies 1-3 deal with this aspect in detail and provide the conceptual framework and methodology for such analyses. In order to overcome a difficult situation when the operational data on the complete system is non-existent or expensive, it is suggested to make use of the operational data on its components. Reference 4 presented the Bayesian reliability analysis of a parallel system using time censored failure information and a prior belief about the failure rate of the components of the system. In this study, however, failure information is qualitative, i.e. the number of failures in a stipulated time interval has been used in the analysis.

Pursuing these concepts, in the present study, Bayes estimates of reliability and MTSF have been derived for a more general system configuration, namely k-out-of-m. These estimates have been obtained with two types of censored failure information which are both qualitative and quantitative in nature. The posterior estimates of system reliability for a parallel system in Ref. 4 suffer from some mathematical lacunae. The corresponding corrected estimates are also mentioned as a special case.

In life-testing experiments censored samples are used to get sufficient failure information within a

11

reasonable time limit. Time censored data (type I censoring) is used when one wishes to limit the period of experimentation where the cost increases with time. On the other hand, failure censored data (type II censoring) is helpful in experimenting with costly items. However, in a life-testing experiment, the size of the sample and the censoring time are two conflicting variables. For example, placing 30 items on test and waiting until 10 of them fail will involve, on average, a longer testing time than if one places 50 items on test and waits for 10 failures.

Therefore, depending on the nature of the experiments a problem of considerable interest is to perform an optimisation between censoring time and sample size so that the experiment results in a pre-assigned amount of failures. In section 5 of the present study, Bayes confidence limits for unreliability of the system have been used for defining confidence limits for the sample size with a pre-specified confidence coefficient and varying censoring time. The results are useful for making an economic trade-off between the sample size and the censoring time.

2 BACKGROUND

(a) k-out-of-m system: A k-out-of-m system con- sists of m independent and identical components such that the system operates as long as at least k of its components operate. In particular,

(i) for k = m, a k-out-ofom system reduces to a

12 K. K. Sharma, Hare Krishna

series system which fails as soon as any one of its components fails;

(ii) for k = 1, the system reduces to a parallel system which functions as long as any one of its components operates.

(b) Let the lifetime distribution of each component be exponential with probability density function (pdf)

f(x, #) = #e-W; #, x > 0 (1)

Here, # is the failure rate of the components and 1/# is the mean time to system failure (MTSF). The component reliability for a mission time t is

Re(t) = P[X > t] = e - ' ; t > 0 (2)

(c) The prior distribution of the failure rate # is assumed to be gamma with p.d.f.

0 #e- ou g(#) = _ _ #t~-,; O, fl ,#>O (3)

r ( ~ )

(d) The reliability Rs(t) of a k-out-of-m system, with each component having lifetime distribution given in eqn (1) is

R~(t)= ~ (7)(e-~)i( l -e-Ut) m-i (4) i=k

and its MTSF is

tv r r sF = - (5) l

(i) for k = m, i.e. a series system,

R, (t) = e-m~" (6)

1 MTSF = - - (7)

m#

(ii) for k = 1, i.e. a parallel system,

Rs(t) = 1 - [1 - e-~t] m

MTSF = - - #i=1 i

(8)

(9)

3 POSTERIOR ANALYSIS OF A k-OUT-OF-m SYSTEM WITH TIME CENSORED DATA

As assumed in Ref. 4, an experiment is conducted with a large number of components and the number of failures, r say, at time T is recorded. Obviously, r has a Poisson distribution with probability mass function (pmf)

e-"r(#T)" P(r [/~) = r! , r = 0, 1, 2 . . . . . (10)

Then, the posterior pdf of # given r failures upto time T, with prior distribution of # given in eqn (3)

becomes p(r I t~)g(t~)

fo-~P(r #)g(#) d# i

= (T + O) r+O #,+t~_le_Utr+o); r ( r +/~)

N, ( T + 0), (r + f l ) > 0 (11)

which is a gamma distribution with parameters (T + 0) and (r + fl). Therefore, the Bayes estimator of # becomes

#* = E(# I r) = #a'(# I r) d#

r + fl (12) =T+O

Further, the Bayes estimates of component reliability

i=k i e-/~( 1 - e-u')m-'~r(# I r) dlt

( r + o)r+" f~ r+._~

X e-u(r+°+i')(1 - e-Ut) "-i d#

= ~ (m)(r__+_ o_y+o~ i (m 7 i ) (_ l ) j ,=, r ( r +/~)

X fo ® ll~r+13-1e-U(T+O+it+#) d#

m m , 1 =

1 . i + j ]r+t3 (15)

× [ l + , ~ - ~ j

Similarly, the Bayes estimate of MTSF can be put as

1 m MTSF* = E l - Y , 1 I r ]

l #i=k [

-- T"~-O ( i ~ k ~ ) r + r - - 1 _ ( 1 6 )

and MTSF are, respectively, given by

g*(t) = E(e -~ I r) 1

= r 1 t I r+~ (13) [

and

MTSF* = E r - (14) r + f l - 1

Now, the Bayes estimate of Rs(t), say R*~(t) is obtained as follows:

m

E Z - e-m) m-' R*( / )= [ i=k(7)e- iUt(1 " r]

Bayesian reliability analysis of a k-out-of-m system 13

Particular cases (i) Putting k = m in eqns (15) and (16) the

corresponding estimates for a series system can be obtained as

1 mt 1 ~+0 (17) R*(t) = [1 + ~ - - ~ j

T + O MTSF* - (18)

(r + fl - 1)m

(ii) Similarly, we put k = 1 in eqns (15) and (16) to get the estimates for a parallel system.

R,( t) = 1 _ ~ ( 7 ) ( _ 1 ) ~ 1 [ "r i=0 1 + T + 0J

(19)

MTSF* (20) r + f l - 1 .=

Here, it is notable that the estimates of Rs(t) and MTSF for the parallel system with fl = 1 as given in eqns (19) and (20) differ from the respective estimates mentioned in eqns (9) and (11) of Ref. 4. This is because of incorrect formulation of integrals there. However, the correct formulation of eqns (9) and (11) in Ref. 4 should be

Rs(t) = E[1 - (1 - e-Ut) m I rl

MTSF=E[I(m.~=I~)Ir ]

4 POSTERIOR ANALYSIS OF A k-OUT-OF-m SYSTEM WITH FAILURE CENSORED DATA

In section 3, the censored information used in the analysis is qualitative in nature. Now, let us consider an experiment in which n components are put on test and the experiment is terminated as soon as a pre-assigned number r(<_n) of them fail resulting in quantitative observed lifetimes x~ <- x2 <-" • • <- Xr. Assuming the component's life-time distribution as exponential in eqn (1), the likelihood of x = (xl, x2 . . . . . Xr) is given by

where

n~ L(x, It) = (n - r)! l'tre-USr

Sr = ~ xi + (n - r)xr i = 1

is the total testing time. Taking the prior of tt as in eqn (3), the posterior

distribution of/~ given x becomes

L(x, Ix)gOt ) Ix ) =

fo~L(x, It)g(l~) d#

( S r -J¢- o)r+fl r+0_le_~,(S+o). -ff; (St + 0), (r + fl), # > 0 (21)

which is a gamma distribution with parameters (Sr + O) and (r + fl). Here, it is interesting to note that the distribution in eqn (21) is same as the one in eqn (11) with T replaced by S, Therefore, in type II censoring, the desired Bayes estimates of Rs(t), MTSF etc. remain the same as obtained in type I censoring except that T will be replaced by Sr.

Further, T and xr stand for the time of experimentation in type I and type II censoring, respectively. Obviously, T>xr . Therefore, for the same failure information of r components, the time of experimentation as well as the sample size n to be put on test is larger in type I censoring. In view of this, while testing with costly items like electronic and electrical components, it is more desirable to use type II censoring with pre-assigned sample size and resulting in a quantitative failure information.

5 BAYES ESTIMATION OF SAMPLE SIZE AND CENSORING TIME

While using censored sample information a problem of considerable interest is to know how large the sample size should be in order to realise a reasonable failure information in a stipulated time. In type I censoring the truncation point t is arbitrarily chosen where as in type II censoring, the sample size is at the discretion of the experimenter. For example, in section 3, the sample size is taken to be large enough and in section 4, it is some arbitrarily chosen number n(>-r). Here, in failure censoring case the sample size has to be restricted while experimenting with high-cost items and, in a time censored sample, one has to limit the time of experimentation when the cost increases with time.

In the present section, Bayes confidence intervals for unreliability have been used to develop confidence limits for t and n. In view of cost constraints these limits provide an economic trade-off between the testing time and the sample size with desired confidence. The Bayes confidence interval for # with confidence coefficient (1 - tr), is given by

e[a --</~ --< b] = 1 - a~ (22)

where a and b are the lower and upper limits,

14 K. K. Sharma, Hare Krishna

respectively, and are such that

~ ( . I r ) d/~ = :z(/~ I r ) d/~ = oc/2

or

fa(T+O) xr+fl--I f • r+fl--1 X -x Og 3o F-~ + [3 ) e -x dx = Jb e dx = -

<r+o, F -~+f l ) 2

(23)

The values of a and b in eqn (23) can be obtained for pre-assigned /3, 0, T, r and a~ using the tables for incomplete gamma function. 5 Now on using eqn (22), Bayes confidence interval for unreliability for a mission time t, i.e., for/~(t) = 1 - e -m can be p u t a s

P[1 - e -at <<. R( t ) = 1 - e -~t <_ 1 - e - b t ] - - 1 - a~ (24)

By definition, /~(t) gives the proportion or the probability of the component failing till time t. Thus, R ( t ) = r / n . From eqn (24), the Bayes confidence limits for r /n become

P [ 1 - - e - " ' < r < l - - e - b ' ] = l - t r n (25)

Using eqn (25), the confidence limits for r /n for varying t can be obtained. In the following section, we consider an example clarifying the procedure of getting Bayes estimates of censoring time and sample size.

6 E X A M P L E S

(a) The Bayesian estimates of the MTSF and reliability of a k-out-of-m system has been evaluated with m = 3 and k = 1, 2, 3. Assuming fl = 1, 0 = 5 and the testing time T = 500 units (or total testing time Sr = 500 units), the posterior estimates of MTSF for r = 0, 1, 2, 3 are presented in Table 1. Using the same data in eqn (15) the estimates of reliability for varying t have been plotted in Figs 1-3 in case of k = l (parallel system), k = 2 (2-out-of-3 system) and k = 3 (series system), respectively.

Curves in Figs 1-3 show that the posterior reliability decreases uniformly as the failure informa- tion r increases in testing time T -- 500 units. Further,

Table 1. Bayes estimates of MTSF for a k-out-of-3 system

Failure Parallel 2-out-of-3 Series information system system system

r k = l k = 2 k = 3

0 (30 oo oo

1 926 421 168 2 463 210 84 3 309 140 56

R;(t)

1-0

0.8

r . 0

0-6

0'4 r =l

r=2

O" - . . . . . . . . . . . ~t

0 100 200 300 400 500 600

Fig. l . Bayes estimates of reliability of a parallel system.

R~(t )

1.0

0.8

0.6

r = O 0"4.

02 t r 1

0 . . . . . . . . . . . ~ r t 2 0 IO0 200 300 400 SO0 6OO

Fig. 2. Bayes estimates of reliability of a 2-out-of-3 system.

~t F~(t)

1.2

1.0,

(>8'

0'6

0"4"

0'2- r ,0

o . . . . .

100 200 300 400 500 600

Fig. 3. Bayes estimates of reliability of a series system.

Bayesian reliability analysis of a k-out-of-m system 15

Table 2. Bayes estimates of n and t for r = 2

Mission /~(t) = r Sample size n time, t n

Lower Upper Lower Upper limit limit limit limit

and cost constraints, one can easily make an economic trade-off between the sample size and the censoring time by analysing the information in Table 2.

50 0.0595 0.5111 4 34 75 0-0879 0.6581 3 23

100 0-1155 0.7609 3 17 200 0.2177 0.9429 2 9 300 0-3081 0.9863 2 7

a comparison of the three figures reveals that the Bayes estimate of system reliability increases uni- formly as the number of redundancies, i.e. (m-k) increases in a k-out-of-m system.

(b) As an example for clarifying the procedure developed in section 5, let cr = 0.05, T = 500 units and f l = l , 0 = 5 in eqn (23). This set-up gives a=0-00122787 and b =0.01431067. On using eqn (25), the 95% confidence limits for unreliability/~(t) and hence for n, with r = 2 and varying t have been summarised in Table 2. From the table it is observed that for getting failure information of r = 2, the upper limit for the sample size n is 34 when the experiment is conducted for time t = 50 units. On the other hand, at the most nine sample observations are needed to get r - -2 in time t = 200 units. Depending on the nature of experiment, system or item and other time

A C K N O W L E D G E M E N T

The authors thankfully acknowledge the financial assistance provided by the University Grants Com- mission, India for this research work. Thanks are also due to the learned reviewer for his valuable comments.

R E F E R E N C E S

1. Kaplan, S., Liming, J. K., Dykes, A. A., Pritchett, T. C., Shaver, D., Budo, E. P., Cunha, D. M. & Mihalkanin, P. A., A methodology for assessing the reliability of boxes. Reliability Engng System Safety, 26 (1989) 249-69.

2. Apostolakis, G., The concept of probability in safety assessment of technological systems. Science, 250 (1990) 139-64.

3. Martz, H. F. & Walher, R. A., Bayesian Reliability Analysis. John Wiley and Sons, New York, USA, 1982.

4. Sharma, K. K. & Bhutani, R. K., Bayesian reliability analysis of a parallel system. Reliability Engng System Safety, 37 (1992) 227-30.

5. Pearson, K., Tables of the Incomplete Gamma Function. The University Press, Cambridge, UK, 1965.