Theory of recovery


Till now: component is working only to time of failure
Usually failed component can be repaired  recovery process
Sequence of sate without failure
& states of failure of random process
1. Process with direct recovery:
 Time of recovery  0 (~
system
or
component
and infinite number of backups)
 Suppose failure: 
Time of working without failure: random variable time 
1
0
2
1
3
2
with
switch
4
3
4
n
t
n
 n   i
i 1


Time of working without failure for n-1 recovery is equal to time of n-th failure
If the components after recovery are identical, it holds distribution function, created by
multi convolution integral:
t
Fn t   Pn  t    Fn1t   dF  
0
where: F1 t   F t  is distribution function of failure for single component
t
The derivation of distribution function is density:
f n t    f n 1 t    f  d
0
where: f1 t   f t  is probability density of failure for single component.
 t
Suppose components with constant failure rate (t)=then f t     e
(Poisson process)
and we can express the previous probability density of working time for 1,2,…,n recovery:
Lf n (t )  Lf (t )n   s   


n
  
t
f1 t   L1 
   e
s   
 2 
f 2 t   L1 
 2  t  e  t
2
 ( s   ) 
n1
n






.

t



1
. f n t   L 

 e t
n
n  1!

(s   ) 

distribution of time working without failure

From previous equation it is ´possible to determine:
Erlang
1.
2.
So:
it holds:
and:
probability of n-th recovery before time t
how many recovery will be till t
suppose N(t) is number of recovery till time t
N (t )  n   n  t  PN t   n  P n  t 
P n  t   1  P n  t   1  Fn t 
integration of probability density
PN t   n  1  Fn t 
we want to determine: PN t   n that holds:
PN t   n  PN t   n  1  PN t   n  Fn (t )  Fn1 (t )
(1)
using substitution from equation fn(t) – Erlang distribution and integrating receive:
t
PN t   n  
0
 t n1
n  1!
e
 t
t
dt  
 t n
n!
0
e t dt
distribution function:
t
Fn t   
 t n1
( n  1)!
e
 t
n 1
dt  1  
t k e t
k 0
0
using substitution equation (2). into equation (1) receive:
n
t k  t n1 t k  t
PN t   n 

 
k!
k 0


e

 
k 0
k!
e

t n  t

e
n!
Number of recoveries has Poisson distribution
Mean number of recovery (here mean number of failure) that arrive till t is recovery
function H(t)

H t   E N t    n.PN t   n
n 0
mean value

H t  

 nFn t   Fn1t    Fn t 
n0
It can be changed for :
n 1
 t
H t   F t     Fn t   dF  
n 1 0
t
or:
(2)
k!
H t   F t    H t   dF  
0
t
H t    1  H t   dF  
or:
0

Mean umber of failures in time interval (t1,t2) is H(t1)- H(t2)
For exponential distribution
LH (t )  Lf (t )  2 , appropriate function of recovery
s1  L f (t ) s
in time t H t   t
With Poisson process the number of recovery increase linearly with time
Density of recovery: ht  
dH t 
dt
number of failures in single time interval (  0)
 ht dt ~ probability that receive (t,t+  t)
Derivation of expression for H(t):
ht   f t    ht    f  d

or:
ht    f n t 
n 1
H(t)
f n t  …n-th convolution of probability density
function of
recovery
h(t) density of recovery
norm.

norm.
exp
E(x)-1
exp
t
t
Process with finite time of recovery

Time of recovery is comparable to time of working without failure
Process run:
1
01
0
1
Without failure
2
01
02
2
n
02
…
n
0n
n
0n
t
failure
Time of n-th failure begin:
 n   1   01   2   02  ...   n
Time of n-th failure end:
0n   1   01   2   02  ...   n   0n
Total time working without failure:
T p   1   2  ...   n
Total time of recovery:
T0   01   02  ...   0 n
 From previous equations we can express:
Time of n-th failure end is:  0 n  T p  T0
We suppose that working without failure and recovery has the sane distribution functions F(t)
and G(t)

Then the function of recovery:
H t    F0 n t  mean number of recovery in time
n 1
interval (0,t)
where F0 n t   P0 n  t 
t
F0 n t    Fn t   dGn  
0
failures + recovery =distribution. function
failures:
Fn t   PT p  t    Fn1 t   dF  
recoveries:
Gn t   PT0  t    Fn1 t   dG 
F1 t   F t 
G1 t   Gt 
dF t 
dH t  
  f 0 n t  ; where f 0 n t   0 n
density of recovery: ht  
dt
dt
n 1
for exponential distribution failures and recoveries
 t  ; f t  .et and  t  ; g



t   .et
Lf n (t ) Lf (t )n  Lg(t )n1 and Lf on (t )  Lf (t ) Lg(t )n
LH (t ) 

s 2 s     
function of recovery: H t  



t
1  e (    ) t
2
      
mean number of failures in time t:


2
t
1  e (    ) t
2
      
Remark:


availability factor ~ system with finite time of recovery has probability kˇp(t),
that in time t is in state without failure:

k p t    P 0 n  t   n 1 
n 0
can be expressed as:
t
k p t   1  F t    1  F t   h d
0
Availability factor ~ ratio of whole time without failure till time t to whole time of operation
(failures + recovery) till time t:
k p t  
T p t 
T p t   T0 t 
; where Tp(t) – working without failure
T0(t) – recovery
for t   :
k p  lim k p t  
t 
TSp
TSp  TS 0
; where TSp is mean time of working without failure
TS0 is mean time of recovery
Remark:
If recovery time () and working without failure () with exponential distribution
TTs 
1

TS 0 
1

is
k p t  


Remark: more detailed approaches divide recovery to wait time for repairs and time of repair
with different distributions
Another characterization of system with repair
Remark:
1)
system with repair
working without failure
working without failure, out of order
Each combination has different failure
distribution
2) time of repair
find failure
repair failure
wait for failure repair (bulk service)
3) Safety critical systems
-depends on reliability of operations
-some components has essentially worse reliability then others
replacement before failure
replace has sense  if the failure rate is increasing (components becomes old)
constant failure rate ~
exponential
the same probability of failure in time interval
(regardless of elapsed time)
a)
Using
distribution
the
early
replace
has
no
sense
b)
For real components (in the early-stage the components has increased failure rate) the
early replace of component can lead to decrease the system reliability!