this paper was selected at the national level seminar and presented

1. Bayesian Reliability Analysis of Certain
Types
of Systems
W Discrete Failure Time
ith
By
Ketan A Gajjar, M.N .Patel

2. 1.IntroductIon
The 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.

3. 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 ,
P ( X = 1) = 1 − P ( X = 0) = θ , 0 <θ <1

4. 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.

5. The probability mass function
of the failure at the xth trial is
f (x, θ ) = (1 − θ )θ x − 1, x = 1,2,3,.....; 0 < θ < 1
(1.1)

6. 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).

7. 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 1
1−θ
ii) The prior distribution of parameter θ in
(1.1) is
l −1 m−1
(1−θ )
π (θ ) = θ ,0 < θ < 1 ; l , m > 0
β (l , m)

8. 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))
 T − 1 u
P (u |θ ) =   φ T −1−u
 u  (1−φ ) ,
 
n
( u = 0,1,2,...T − 1, T ∈ N ; 0 < θ <1;φ = θ )
(2.2)

9. iv) The posterior distribution of θ given that
n failures have been observed up to trial
T-1 can be obtained as
P(u | θ )π (θ )
g1(θ | u ) = 1
∫ P(u | θ )π (θ ) dθ
0
n(T −1−u )+l −1 m−1 nu
θ (1−θ ) (1−θ )
= ;
u  u j (2.3)
∑   (−1) β ( n(T − 1 − u + j ) + l, m)
j =0  j 
 
(0 < θ <1 ; n(T − 1 − u + j ), m > 0 )

10. 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
R(t ) = θ t ; t =1,2,3,.......; 0 <θ <1
(3.1)

11. Hence, reliability of a k out of n
system (kns) becomes
n  n i n−i
 
R (t ) = ∑  i  ( R(t )) (1− R(t ))
kns i=k  
Cases k=1 and k=n correspond
respectively, to a parallel and a series
system.

12. 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,
* (t ) (t )
R kns R
=E ( kns |u)
1 n  n  it
  t n−i n(T −1−u )+l −1 m−1 n u dθ
∫ ∑  (θ ) (1−θ ) θ (1−θ ) (1−θ )
0 i=k  i
= u  u
  j
∑  j (−1) β ( n(T − 1 − u + j ) + l , m)
j =0  

13. After some algebraic manipulations, we get
*
R kns (t ) as
*
n
∑
i=k




[
 ∑ 
i  j =0  j 
 
{
n  n−i  n − i 


u  u
∑   (−1)
 r
r =0  
r+ j
}]
β ((i + j )t + n(T − 1 − u + r ) + l, m)
R kns (t )= u  u r
∑   (−1) β ( n(T − 1 − u + j ) + l, m)
 r
j =0  
(3.2)
Estimates for series and parallel systems
follow as particular cases.
*
Bayes risk of estimator R kns (t ) is defined as
r(R *, R ) = E [ E ( * (t ) − R ( t ) )2 ] (3.3)
θ u R kns

14. A few algebraic manipulations, give
as
r ( R*, R )
r(R *, R ) = ∑ [
T −1
u =0
* 2 2 *
t
n
(R kns (t )) − Rkns (t ) ∑  n  β ( (i + 1) , n − i + 1) +
 
 
i =k  i  t
2
1 n  n 1
∑ ( ) β ( (2i + ) ,2 n − 2 i + 1) +
t i =k  i  t
1 n n  n  n  1
∑ ∑    β ( (i + j + ) ,2 n − i − j + 1)
t i≠ j j =k  i  j 
   t
]
(3.4)

15. 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.

16. 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)
r
n−1 (− log R(t ))
R css(t ) = R(t ) ∑ r!
r =0 (4.1)
Finally, upon using the posterior distribution
of θ in (2.3) and assuming the squared error
loss function, the Bayes estimator of R css(t )
can be derived as

17. 1 t n−1 (− log R(t )) r n(T −1−u )+l −1 m−1 nu
∫ θ ∑ r! θ (1−θ ) (1−θ ) dθ
0 r =0
R css(t ) u  u
= j
∑   (−1) β ( n(T − 1 − u + j ) + l, m)
j =0  j 
 
After some algebraic manipulations, we get
 u j +i
 
 j  (−1)
*
Rc ss (t )
n−1 r
∑ t
r =0
{ m−1
∑
i =0
 m − 1

 i 



[ ∑
u  
j =0 ( t + n(T −1− u + j ) + l + i) r
]}
= u  u j
∑   (−1) β ( n(T − 1 − u + j ) + l, m)
j =0  j 
 
(4.2)
*
Bayes risk of estimator of Rc ss (t ) can be defined
as stated in (3.3).

18. 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.

19. 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.

20. 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.

21. Some recent studies by
*Sharma and Bhutani (1992, 1993),
*Sharma and Krishna (1994)
*Martz and Waller (1982)
deal with these aspects in detail.

22. 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.

23. 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. 33-41
7.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.

24. 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.