Reliability – lesson 2
Identification of the system reliability
Testing huge number of system products with reports about failures
Reports:
 time of failure of systems
 number of failures for all systems in short time period (short time with respect to the
total test time)
Statistical data processing
 approximate detected values
 compare with some standard probability distribution and set the parameters of
distribution
Selected distribution enable to compute others parameters of system (including all other
parameters like time to wait for repair, failure of reserve)
Distribution of continuous variable
1. Exponential distribution
has one parameter , can be easy express as analytic functions
R(t)= e
 t
Q(t)= 1  e

f(t)=   e
(t)=
R(t)
Q(t)
Q(t)
 t
t
1

1
D=
2
TS=
 It is the most used distribution
 MTTF is 1/, the distribution is
defined by mean value
 Very good approximation of systems
in time of normal operation
 Do not match the early stage and
end-of-life stage
 Some systems with flexible failure
rate are not convenient for
exponential distribution
R(t)
0
1
2
3
t
3
t
(t)
f(t)
(t)
=1
f(t)
0
1
2
Průběhy veličin R(t), Q(t), f(t) a (t) pro exponenciální rozdělení. =1.
2. Rayleight distribution
is defined by one parameter k and the failure rate is increasing linearly
suppose k>=0
R(t)
Q(t)
k
 t
R(t)= e 2

R(t)
k
 t
Q(t)= 1  e 2
k
 t2
f(t)= k  t  e 2
Q(t)
(t)= k t
TS=


1.253
k
2k

2
2  0.429
D=
k
k
t
1
0
(t)
f(t)
(t)
=1
f(t)
t
1
0
Průběhy veličin R(t), Q(t), f(t) a (t) pro Rayleightovo rozdělení. K=1.
3. Weibull distribution
has two parameters, m and t0
R(t)
1
m=1,5
R(t)= e
tm

t0
Q(t)= 1  e
m=1
m=0,5

tm
t0
m m -1
t .e
t0
m
(t)= t m -1
t0
f(t)=

f(t)
1
1

 1
m 
2


1
 1   2   1 
 m 
 m 


1
tm
t0
1

TS= t0m 
2
 
D= t 0m  
0
t
m=1,5
m=1
m=0,5
0
1
t
(t)
2
m=1,5
m=1
1
m=0,5
0
1
t
Průběhy veličin R(t), f(t) a (t) pro Weibullovo rozdělení.
Wiebull distribution contains exponential and Rayleight distribution as a special cases.
for m=1  it is exponential distribution for m=2  Rayleight distribution
Wiebull distribution for m<1 is very good approximation of the product the early stage
Wiebull distribution is the second most used distribution (after exponential)
4.
Gama distribution
R(t)
Q(t)
1
two parameters m>0, c>0, system with backup
f(t)=
t m -1
c m 
m
e

R(t)
t
c
Q(t)
0
t
c
m -1
m - 1!
e

t
f(t)
t
c
0
1
2
3
t
Průběhy veličin R(t), Q(t), f(t) a (t) pro rozdělení gama. c=1, m=2.
m…change the shape of function f(t)
c…change the scale factor of time
where:

3
(t)
if m is a natural number the definition of
m  m - 1! so:
m
2
f(t)
(t) 1
TS=m.c
D=m.c2
f(t)=
1
gama distribution is for m=1 equal to exponential distribution with =1/c
5 Truncated normal distribution:
two parameters > 0
f(t)=
c
e
 2

t -  2
2 2
1
   
where c     
   
 reliability theory use only t  0 it means that for  t<0 the probability of failure is zero
 characterization of reliability with truncated normal distribution t>0
 …is distribution function of normal distribution n=0 n2=1
x  :
2
x  
1
2
x t
e 2 dt


probability density of normal distribution is:
 x  
dx 
1

e
dx
2

x2
2
-t


 
R(t)  

 
 
Q(t)  1 - R(t)
R(t)
Q(t)
R(t)
 t

 

f(t) 

   
 
t
 

 

 (t) 
 t
  

  
 t
 

 

TS    
 
2  1


 
Q(t)
0
1
t
0
1
t
0
1
t
f(t)
(t)


for  > 2 is TS very similar to 
truncated normal distribution is suitable
for approximation of reliability of product
in end-of-life stage (see figure)
6.
Logarithmic – normal distribution
Průběhy veličin R(t), Q(t), f(t), (t) pro useknuté normální rozdělení.
t0=1, =0,3
 is defined by logarithm of random variable with normal distribution:
random variable x has normal distribution with parameters ,  and x=ln(t), so x  - ;
and t > 0
The normal distribution need not to be truncated
probability density.
x -  2

2
1
f x  
e 2 where: x  - ;
 2
with respect to variable t:
f t  
M
e
t 2

log(t) -log(t 0 ) 2
2 2
and
  log( t0 )
where constant M  0,4343 is for transformation ln  log
2
Mean time to failure:
MTTF  TS  t 0  e 2M
can be evaluated log(T S )  t 0  1.1513 2
2
Dispertion:
D  t S2
t S2
t 02
1

for small size of  < 0.1 is similar to normal distribution and can be used for time of
system back up

previous simple distribution cannot approximate the whole system for the life time of
product  combination of distributions
Superimposing of two exponential distribution

approximation of reliability in early stage and normal working product
R(t)= c1e
 1t
f(t)= 1c1e
 c2e2t
1t
 2 c2 e 2t

as the probability distribution is probability it need to satisfies
 f t dt  c
1
 c1 1
0
from this we can calculate  t  
c11e 1t  c22e 2 t
c1e 1t  c2e 2 t
start point =1c1+2c2
Mean time to failure = TS 
c1
1

c2
2
Typically we have 1<2 for big t is
e2t significantlly smaller than
e 1t , so for big time t this
distribution is transforming into
exponential distribution with t)  
(t).10-3
2
1
0
50
100
150
t
Průběhy intenzity poruch pro superpozici dvou exp. rozdělení. =0,001,
=0,04, c1=0,95, c2=0,05.
Combination of exponential and truncated normal distribution

is suitable for approximation of reliability in normal working product and end-of-life stage
of product
failure rate is combination: t)=t)+t)

probability of working without failure is
R(t)  R1 (t). R 2 (t)
 t


 
- 1t

R(t)  e 

 
 
-t
 




 t   1 
-t
  

  
(t)
1
0
5
t
Průběhy intenzity poruch pro kombinaci exp. a useknutého norm. rozdělení.
=0,2, t0=10, =1.
Partially Linear Distribution

so called bathtub curve
t

-K1t
-K2t

0

t1
t2
t3
t
stages are approximated by linear functions:
constant value  exponential distribution
increasing  Reileight distribution
decreasing  small problem - new type distribution
Decreasing linear function has for t=0 value 0 and then is linearly going to 0, so the function
is defined as (t)=0-K1t. But we know, that (t)  0 for all t  0. Therefore this definition is

valid only in interval 0, 0 . But this function does not satisfies the condition that
K1
t
if t      t dt   . The solution is that we define (t)=constant_value for t 
0
0
K1

probability of working without failure is defined as integration of (t). The integration is
t
divided into three intervals (see the bathtub curve) R(t)  e
-   (t)dt
0
This partially linear distribution satisfies:
 t; 0  t  t1:
(t)=0-K1t
R(t)= e
 0 t 
K1 2
t
2
=R1(t)
the end of the first interval is t 1 
 t; t1  t  t2:
1
0   
K1
(t)=
 0 t1 
K1 2
t1   t  t1 
2

R(t)= e
using the previous equation we receive:
 t t1 
R(t)=R1(t1) e
=R2(t)
 t; t2  t  t3:
(t)=+K2(t)
R(t)= R2(t2) e
Remarks: 1)
2)
  (t t2 ) 
K2
(t t2 ) 2
2
the distribution is defined by parameters 0, K1, , t2, K2, t3
the end of the interval t3 is possible to shift till   .
Discrete distributions
 The most used distributions are:
 binomial (Bernoulli)
 Poisson
1. Binomial distribution
Used: probability of some event in selection from defined set or in repeating tests. The event
is present or not present.
Suppose that we make n test, in m tests the event is present and the probability of this event is
p
Probability distribution of binomial distribution is:
n
nm
f m   f m; n, p     p m 1  p 
m; m  0,1,..., n
 m
m – tests satisfy event
n-m – test don’t satisfy event
n
n!
number of combinations is:   
 m  m!n  m !
 all combinations are independent  the product defines the probability of all components
in one time
 probability that the event is in all n tests: f n; n, p   p n
probability that the event is not in any test: f 0; n, p   1  p 
n
Distribution function of random variable M with binomial distribution is:
F m  PM  m 
m
 f k ; n, p 
k 0
Remark:
for big n the F(m) is similar to normal distribution
Mean value:
Dispersion:
EM   n. p
DM   n. p.1  p 
M
1
we receive: E  X   E M   p
n
n
1
p.1  p 
D  X   2 D M  
n
n
Remark: with substitution to random variable X 
f(m)
f(m)
0,6
n=10
p=0,05
0,5
0,25
0,4
0,2
0,3
0,15
0,2
0,1
0,1
0,05
0
1
2
3
n=10
p=0,5
4
m
0
2
4
6
8
10 m
Discrete binomial distribution
2. Poisson distribution
used:
to describe presence of isolated events isolated in time, duration. The number of
events in one unite has to be defined
Example: number of failures in time unit, number of failures in distance unit (for a car)
Remark: Poisson distribution is limit case of binomial distribution when
p  0 … probability of event
n   … number of tests
Probability density of Poisson distribution is:
am
f m; a  
exp  a 
m!
m=0,1,2,… is number of event occurrence
a………… mean number of event occurrence in one time unit
Mean value:
Dispertion:
EM   n. p  a
DM   a
f(m)
f(m)
0,6
a=5
a=0,5
0,5
0,25
0,4
0,2
0,3
0,15
0,2
0,1
0,1
0,05
0
1
2
3
4
m
0
2
4
6
8
10 m
Value of discrete probability density for Pisson distribution

The shape of Poisson distribution for different process in time:
 suppose the events are independent
 the occurrence of event in interval  t with probability p that is proportional to
size of time interval  t
 mark  number of event occurrence in time unit t 
 p  .t

t m  t
f m;  , t  
e
m!
number of event
and … mean value of event in time unit
 t
Remark: f 0;  , t   e
probability that in time t was no event  expression for working without failure:
 t
Rt   e