Chapter 6 Time dependent
reliability of components and
system
6.2 Failure rate time curve
6.3 Reliability and Hazard functions
Reliability:
N f (t )
N s (t ) N  N f (t )
R(t ) 

 1
N
N
N
dR(t )
1 dN f (t )


dt
N dt
dN f (t )
dR(t )

 N
dt
dt
(6.1)
(6.2)
(6.3)
Failure rate (instantaneous rate of failure, hazard
function or hazard rate):
1 dN f (t ) N dR(t )
1 dR(t )
(6.4,6.5)
h(t ) 


N s (t )
dt
Ns
dt
R(t ) dt
dR(t )
0 h( x)dx  0 R(t )   ln R(t )
t
t
reliability at time t:
t
R(t )  exp[ h( x)dx]
0
The distribution function:
t
FT (t )   fT ( x)dx]
0
The reliability of the component:
R(t )  P(T  t )  1  P(T  t )  1  FT (t )
t
 exp[ h( x)dx]
0
t
 ln[1  Ft (t )]   h( x)dx
0
Differentiation, yield
fT (t )

 h(t )
1  FT (t )
i.e.
h(t ) 
fT (t )
f (t )
 T
R(t ) 1  FT (t )
6.4 Modeling of failure Rates
h( x)  
failure
per
unit time
f (t )  h(t )[1  F (t )]
t
 h(t ) exp[ h( )d ]  e t
0
and
t
R(t )  exp[  h( )d ]  e t
0
Assuming linear variation:
h(t )  c1t  c2
6.5 Estimation of failure rate from emperical data
The estimate reliability function at time t:
N s (t )
R(t ) 
N
The failure rate can be computed as
N s (t )  N s (t  t )
h(t ) 
N s (t )t
example:6.1
(6.23)
(6.24)
6.6 Mean time before failure (MTBF)


0
0
MTBF   tf t (t )dt   

dR (t )

t
dt   tR(t ) 0   R(t )dt
0
dt
Since all system fail after a finite time, we have
R(t )  0
as
and tR(t )  0
t 
at t  0

 MTBF   R(t )dt
0
(6.26)
6.7 Series system
n
Rs  R1 R2 R3 ......Rn   Ri
i 1
Failure time of the series system is
t s  min ti
1i  n
The failure time distribution :
n
n
i 1
i 1
Fs (t )  P(t s  t )  1  P(t s  t )  1   P(ti  t )  1 1  Fi (t )
The probability function of the failure time:
n
n
dFs (t ) n Fs Fj
f s (t ) 

  f j (t ) [1  Fi (t )]
dt
j 1 Fj t
j 1
i 1,i  j
6.7.1 Failure rate of the system
Ri (t )  eit
n
Rs (t )   Ri (t ) e
(
n
 i )t
i 1
 e  s t
i 1
Where the failure rate of the system:
n
s   i
j 1
The reliability can be expressed as:
t
Ri (t )  exp[   hi ( )d ]
0
t
Rs (t )   Ri (t )   e 0
e

n
 0
t
n
n
i 1
i 1
e
i 1
n
hi ( x )) dx
0 
i 1
t
hi ( x ) dx
 hi ( x ) dx
 (
n
 hs (t )   hi (t )
i 1
n
or hs (t )   hi
if
i 1
hi
is cons
6.7.2 MTBF of the system


0
0
n
MTBF   R (t )dt   ( Ri (t ))dt

 MTBF   e
(
i 1
n
 i )t
i 1
0
dt




 ( n  )t
1   i1 i 

  n
e
 

 

  i  
 i 1 
0
1
n

i 1
i
6.8 Parallel System
Reliability of parallel system, first seven events
6.8.1 Failure Rate of the system
The system failure rate is given by:
n
n
f (t )
h p (t ) 

1  F (t )
f
j
j 1
(t )  Fi (t )
i 1,i  j
n
1   Fi (t )
i 1
6-8-2 MTBF of the system


0
0
MTBF   R (t )dt  


0


n


1   1  Ri (t ))dt
 i 1

n

 i t
1

1

e
 
 i 1
 



 dt

 

  1  1  e 1t 1  e 2t ....1  e nt dt
0
For special case for n=2

 


MTBF   1  1  e 1t 1  e 2t dt
0
1
1
1
  
1 2 1  2
Where relation is used:


0

1
 1

e t dt   e t  
 
0 
6-9 (k,n) systems
 n
i
n i
Rk (t )    1  F (t ) F (t )
i k  i 
n
Probability distribution function of the system
 n
i
n i
Fk (t )  1  Rk (t )    1  F (t ) F (t )
i 0  i 
k 1
Failure time of the system
dFk (t )
n!
nk
k 1
F (t ) 1  F (t ) f (t )
f k (t ) 

dt
(n  k )!(k  1)!
6.9.1 MTBF of the system


0
0
MTBF   Rk (t )dt  
 n  n  t i
t


e
1

e
  
 i k  i 
 

n i

 dt

6.10 Mixed series and parallel system
The system failure rate is given by:
The reliability R0 shown in fig. 6-8
6.11 complex systems
A
C
D
B
E
6.11.1 Enumeration method
 RB RD  RB RE
No component fail
1. ABCDE
0
6.11.2 Conditional Probability method
 Re liability of the
 Re liability of the



of  
 Re liability

   system with com ponent Ccri  com ponent
Ccri 
 the system
 
 in operating condition
 in operating condition


 Re liability of the
 Re liability of the 



  system with com ponent Ccri  com ponent
Ccri 
 in failed condition
 in failed condition 



A
C
D
A
B
AB
6.11.3 Cut set method
1. Identify the minimal cut sets of the system
2. Model the components of each minimal cut set to be in
parallel .
3. Assume that the various cut sets are in series
4. Find the reliabilityof the system using the parallel-series
model.
6.12 Reliability Enhancement
6.12.1 Series system
R0  dR0  R1 R2 .......Ri 1 ( Ri  dRi ) Ri 1.......Rn
 n

  R j    R j dRj ; i  1,2.....n
j 1
 j 1, j i 
n
Constant constrain:
 n

 n

dR0  dRi   Rk   dRj   Rk 
 k 1,k i 
 k 1,k  j 
That is
R0
R0
dRi
 dRj
Ri
Rj
 dRi  dRj
Ri
Rj
The cost involved in achieving the new system
 ci Ri
R0
R0
dRi
 dRj
 ci dRi  c j dRj 
c R
Ri
Rj
 j j
ci Ri 
min
j 1, 2 ,.....,n
c j Rj




6.12.2 parallel system
(6.73)
dRi 
dR0


(
1

R
)

k 
k 1,k i

n
; i  1,2......,n
For minimum cost:
ci (1  Ri )  min c j (1  R j )
j 1, 2 , 3.....,n
6.13 reliability allocation- agree method
 t  t
1  e ti / mi  1  1  i   i
 mi  mi
wi ti
 c  constant
Ti
6.87