Bayesian Reliability Analysis of Certain Types

of Systems

With Discrete Failure Time

By

, . .Ketan AGajjar M N Patel

1.IntroductIonThe main concern of a maintenance engineer has been the estimation of reliability using operational data on a system .

Estimates based on operational data can be updated by incorporating past environmental experiences on random variations in the life-time parameters of

the system.

A component in a system which is capable of just two modes of performance, represented by a Bernoulli random variable“X” “X” assumes values 1 and 0 for the two modes - functioning and nonfunctioning respectively, having a probability distribution ,

10,)0(1)1( <<==−== θθXPXP

To investigate the ability of electronic tubes to withstand successive voltage overloads and the performance of electric switches which are repeatedly turned on and off, the geometric distribution can be used as a discrete failure model. In each of these cases failure can occur at the xth trial ( x=1,2,3,………) with the probability of failure at any trial as 1 - θ.

The probability mass function of the failure at the xth trial is

f (x, θ ) = (1.1)

10,.....;3,2,1,1)1( <<=−− θθθ xx

In the present study we have considered Bayesian reliability analysis of series, parallel, k out of n and standby systems when lifetime of a component is a geometric random variable with p.m.f given in (1.1).

2. AssumptIons i) Let x be the number of completed cycles / trials of an item denoting discrete failure time,following geometric distribution with Mean Time Between Failures (MTBF) as ii) The prior distribution of parameter θ in (1.1) is

(2.1)

0,;10,),()1(

11)( ><<−

−−= ml

ml

mlθβ

θθθπ

θ−1

1

iii)If u represents the number of failures during total testing time of T-1 trials ( i.e. the number of failures up to T-1) and follows the Binomial Distribution having p.m.f.(See: Patel and Patel (2006))

(2.2) );10;,1,...2,1,0(

,)1(11

)|(

θφθ

φφθ

nNTTu

uuTu

TuP

=<<∈−=

−−−−=

iv) The posterior distribution of θ given that n failures have been observed up to trial

T-1 can be obtained as

(2.3)

∫

=1

0)()|(

)()|()|(1

θθπθ

θπθθduP

uPug

)0),1(;10(

;

),)1(0

()1(

)1()1(11)1(

>+−−<<

++−−∑=

−

−−−−+−−

=

mjuTn

mljuTnu

j

jj

u

n umluTn

θ

β

θθθ

3. BAyesIAn relIABIlIty AnAlysIs of A k out of n system

In view of (1.1), reliability of an item for a mission time, t is given by

(3.1)

10;,.......3,2,1;)( <<== θθ tttR

Hence, reliability of a k out of n system (kns) becomes

Cases k=1 and k=n correspond respectively, to a parallel and a series system.

))(1())(()( tRin

tRin

ki i

ntR kns

−−

∑=

=

Upon using the posterior distribution of θ given in (2.3) and assuming the squared error loss function, the Bayes estimator of the reliability of a k- out-of- n system, can be obtained as,

=E ( |u) =

)(* tRkns )(tR kns

),)1(0

()1(

1

0)1()1(

11)1()1()(

mljuTnu

j

jj

u

dn umluTnt initn

ki i

n

++−−∑=

−

∫ −−−−+−−

−−

∑=

β

θθθθθθ

After some algebraic manipulations, we get as

=

(3.2)

Estimates for series and parallel systems follow as particular cases.

Bayes risk of estimator is defined as

(3.3)

)(* tRkns

)(* tRkns),)1(

0()1(

),)1(0

)(()1(0

}]{[mljuTn

u

j

r

r

u

mlruTnu

rtji

jr

r

uin

j j

inn

ki i

n

++−−∑=

−

++−−∑=

++−+

∑−

=

−∑=

β

β

)(*

tRkns

][ ))()(*(2

),*( tRtR knsEuERRr −= θ

A few algebraic manipulations, give as

(3.4)

),*( RRr

=),*( RRr

]

[

)(

)(

)()(

12,)1

(1

122,)1

2()(

21

1,)1

()(*21

0)(

* 2

+−−++∑≠

∑=

++−+∑=

++−+∑=

−∑

−

=

jint

jin

ji

n

kj j

n

i

n

t

int

in

ki i

nt

int

in

ki i

ntRknst

T

utRkns

β

β

β

4. BAyesIAn relIABIlIty AnAlysIs of A cold stAndBy system (css) We, now consider an n- component system in which only one component is operating and the remaining (n-1) components are kept standby to take over the operation in succession when the component in operation fails, assuming that components cannot fail in the standby state and that the switching and sensing device is 100 % reliable.

If R(t) denotes the reliability of each component as given in (3.1), then the reliability of a css is given by (See Govil 1983, page 68)

(4.1)

Finally, upon using the posterior distribution of θ in (2.3) and assuming the squared error loss function, the Bayes estimator of can be derived as

∑−

=−=

1

0 !))(log(

)()(n

r rtRr

tRtRcss

)(tRcss

=

After some algebraic manipulations, we get =

(4.2)

Bayes risk of estimator of can be defined as stated in (3.3).

),)1(0

()1(

1

0)1()1(

11)1(1

0 !))(log(

mljuTnu

j

jj

u

dn umluTnn

r rtRr

t

++−−∑=

−

∫ −−−−+−−∑

−

=−

β

θθθθθ)(tRcss

)(* tR ssc

),)1(0

()1(

1

0 0 ))1((

)1(11

0}][{

mljuTnu

j

jj

u

m

i

u

j iljuTnt r

ijj

u

i

mtrn

r

++−−∑=

−

∑−

=∑= +++−−+

−+

−∑−

=

β

)(* tR ssc

5. sImulAtIon study And dIscussIon 5.1 sImulAtIon studIes In this section, we have studied reliability of different systems discussed in sections 3 and 4 at mission time t for different values of number of trials performed (T), number of failures (u) during trial and hyper parameters l and m.

5.2 conclusIons (i)For fixed l, m, k, t and T Bayes risk increases with n for all the system.

(ii)For k out of n system (k≠1, n): For l = m Bayes risk is smaller than that of under l > m for fixed t, T and n.

(iii)For series system : When l = m , Bayes risk is smaller then that of under l < m for fixed t, T and n.

conclusIons contd…

(iv)For parallel system : For fixed l, as m increases, Bayes risk decreases fixed t,T and n.

(v)For standby system : For l < m , Bayes risk is smaller than that of under l = m for fixed t, T and n.

Some recent studies by *Sharma and Bhutani (1992, 1993),

*Sharma and Krishna (1994) *Martz and Waller (1982) deal with these aspects in detail.

references

1. Govil, A.K. (1983): Reliability Engineering, Tata McGraw Hill.2. Kalbfeisch, J.D. and Prentice R.L. (1980): The Statistical Analysis of Failure Time Data, John Wiley, New York.3. Martz, H.F. and Waller, R.A. (1982): Bayesian Reliability Analysis, John Wiley, New York.4. Nelson, W.B. (1970): Hazard Plotting Methods for Analysis of Life Data with Different Failure Models, Journal of Quality Technology.Vol.2, pp. 126-149.5. Patel, M.N. and Gajjar, A.V. (1990): Progressively Censored Samples from Geometric Distribution, The Aligarh Journal of Statistics, Vol.10, pp 1-8.

6. Patel, N.W. and Patel, M.N. (2003):Estimation of Parameters of Mixed Geometric Failure Models From Type –I Progressively Group Censored Sample, IAPQR Transactions, Vol.28, pp. 33-417.Patel, N.W. and Patel, M.N. (2006): Some Probabilistic Properties of Geometric Lifetime Model, IJOMAS, Vol.22, pp. 1-3.8. Sharma, K.K. and Bhutani, R.K. (1992):Bayesian Reliability Analysis of a Parallel System, Reliability Engineering System Safety. Vol. 37, pp. 227-230.

9. Sharma, K.K. and Bhutani, R.K. (1993): Bayesian Analysis of System Availability, Microelectronics and Reliability, Vol. 33, pp. 809-811.10. Sharma, K.K. and Krishna, Hare (1994): Bayesian Reliability Analysis of A k - out of n - System and the Estimation of a Sample Size and Censoring Time, Reliability Engineering System Safety. Vol. 44, pp. 11-15.11. Yaqub, M and Khan, A.H (1981): Geometric Failure Law in Life Testing, Pure and Applied Mathematika Sciences, Vol. XLV, No.1-2, pp. 69-76.