Download MTBF

MAT 4830
Mathematical Modeling
05
Mean Time Between
Failures
http://myhome.spu.edu/lauw
Reading for Next Wednesday
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
MCM Contest Rules, Registration and
Instructions
http://www.comap.com/undergraduate/c
ontests/mcm/instructions.php
Preview
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Use Monte Carlo Method to estimate the
life of products
Emachine…
Mean Time Between Failures
Suppose that we make a product composed of
n components where the ith component is
guaranteed by its manufacturer to possess a
time before failure that is normally distributed
about the mean i with standard deviation i.
ith component
Ti ~ N ( i ,  i )
Mean Time Between Failures
Assume that our product will fail when any one
of its components fail.
Questions: What guarantee can we offer our
customers for our product? What is our
product's mean time between failures (MTBF)?
Analytic Solution
Ti ~ N ( i ,  i )
T  min(Ti )
1i  n
MTBF  

 tf (t )dt

Analytic Solution
Fi (t )  Cumulative Distribution Function of Ti


n
 tf (t )dt  



 t j 1 1  Fj (t )  Fi (t )dt
i 1 
n
j i
Monte Carlo Solution

3 components
T1 ~ N (11, 1)
T2 ~ N (12, 2)
T3 ~ N (13, 3)
Monte Carlo Solution

3 components
T1 ~ N (11, 1)
T2 ~ N (12, 2)
T3 ~ N (13, 3)

Let us watch this product fail over and
over by simulation
Idea
i T1
T2
T3 T  min(T1 , T2 , T3 )
1 11.5 11.4 13.0
t1  11.4
2 10.6 12.0 10.2
t2  10.2
n 11.0 10.9 15.5
tn  10.9
Idea
i T1
T2
T3 T  min(T1 , T2 , T3 )
1 11.5 11.4 13.0
t1  11.4
2 10.6 12.0 10.2
t2  10.2
n 11.0 10.9 15.5
tn  10.9
Idea
i T1
T2
T3 T  min(T1 , T2 , T3 )
1 11.5 11.4 13.0
t1  11.4
2 10.6 12.0 10.2
t2  10.2
n 11.0 10.9 15.5
tn  10.9
Idea
i T1
T2
T3 T  min(T1 , T2 , T3 )
1 11.5 11.4 13.0
t1  11.4
2 10.6 12.0 10.2
t2  10.2
n 11.0 10.9 15.5
tn  10.9
1 n
1 n 2
   ti , m   ti ,  2  m   2
n i 1
n i 1
Programming Outline
Within a loop:
1. Generate random numbers T1 , T2 , T3
(with the corresponding distribution)
2. Compute ti  min(T1 , T2 , T3 )
n
n
i 1
i 1
3. Keep track of  ti and  ti2
1 n
1 n 2
2
2
Then compute:    ti , m   ti ,   m  
n i 1
n i 1
Sample Output
> MTBF(1000);
The mean is approximately 10.138361
The standard deviation is approximately 1.355540
Classwork

Individual (Each of you need to think
through the process)
Problem 1
The example above
Problem 1
Maple Commands:
HW Problem 1
A manufactured product consists of three
components. Suppose the time before
failure for the parts are given by
Ti ~ N (10, 1), i  1, 2,3
HW Problem 1
The first component is a backup for the
second; i.e., either both must fail or the
third component must fail before the
construct fails.
HW Problem 1
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Write a program to estimate the mean
and standard deviation of the life of the
product.
Briefly explain the key underlying
methodology of your program.