CHAPTER 12
Reliability &
Maintainability
McGraw-Hill/Irwin
Copyright © 2009 by The McGraw-Hill Companies, Inc. All rights reserved.
12-2
Reliability vs. Quality
While product reliability is certainly an important
part of quality, the fields treat different problems.
Quality: process monitoring, statistical testing of
manufactured goods, and product design issues.
Reliability: the statistical aspects of the failure of
items in the field.
12-3
Reliability of a Single Component
Suppose that T is a random variable representing the lifetime
of a single component. Then
F ( t )  P (T  t ) is the cum ulative distribution func tion of T .
f (t ) 
dF ( t )
is the probability density function o f T .
dt
r (t ) 
f (t )
1  F (t )
is the failure rate function of T .
12-4
The Exponential Failure Law
The exponential cumulative distribution function
and density function are represented
mathematically as:
F (t )  1  e
f (t )   e
 t
t
(The exponential distribution and density function are pictured on the next
slide).
12-5
The Exponential Density
and Distribution Functions
12-6
Memoryless Property of the
Exponential Distribution
The exponential distribution is the only continuous
distribution possessing this property. Let T be the lifetime
of an item. Then it can be shown that if T has the
exponential distribution:
P (T  t  s | T  t )  P (T  s )
In words: the probability that an item will not fail in the next s
units of time, given it has not failed up until now (time t),
is the same as the probability that a new item will not fail
in s units of time. The item “forgets” the t units of time it
has been operating. This means that failures occur
completely at random.
12-7
Increasing and Decreasing
Failure Rate Functions
A common probability law for describing the failure
characteristics of items in the field is the Weibull
distribution:
F (t )  1  e
 t

The Weibull distribution is useful for describing both
increasing and decreasing failure rate functions.
(See Figure on the next slide).
12-8
Failure Rate Functions for the
Weibull Lifetime Distribution
12-9
The Poisson Process in
Reliability
Suppose that a single item is instantly replaced upon failure, and
that successive times between failures are given by
independent random variables T1 , T2 , . . . having the
exponential distribution. Successive failure times are
W 1  T1
W 2  T1  T 2
W 3  T1  T 2  T3
(Refer to the figure on the next slide).
12-10
Realization of a
Poisson Process
12-11
Poisson Process (continued)
 From the figure, we define N(t) as the number of failures
that occur up until time t. Then it can be shown that Wn has
the Erlang distribution with parameters n and , and N(t)
has the Poisson distribution with parameter t.
n 1
P (W n  t ) 

e
 t
k!
k 0
P ( N (t )  n ) 
( t )
e
t
( t )
n!
n
k
12-12
Failures of Components in Series
Suppose that N identical components each with failure
distribution F(t) are arranged in series (that is, the
system fails as soon as a single component fails). Then
the cumulative distribution function of the time until
failure of the series system, FS(t) is given by:
FS ( t )  1  [1  F ( t )]
N
12-13
Failures of Components in
Parallel
When N identical components are arranged in parallel
(where the system fails only when all of the
components in the system fail), the cumulative
distribution function of the time until failure of the
parallel system, FP (t), is:
F P ( t )  [ F ( t )]
N
12-14
K Out of N systems
A K out of N system is one which functions when at
least K of the components function. Suppose that R(t)
is the probability that a single component functions at
time t. Then the probability that the system functions
at time t is given by:


N!
j
Nj
R K (t )   
R
(
t
)
F
(
t
)

j  K  j !( N  j ) ! 
N
12-15
Maintenance Terminology
MTBF: Mean Time Between Failures
MTTR: Mean Time to Repair
Availability =
MTBF
____________
MTBF + MTTR
12-16
Deterministic Age Replacement
Assumptions:
Equipment is operating continuously
Ignore downtime for repair and maintenance
Planning horizon is infinite
All new equipment is identical
Only maintenance and replacement costs are considered
Objective is to minimize long run costs of replacement
and maintenance.
The cost of maintaining an item of age u is cu and the
replacement cost of a failed item is K.
12-17
Deterministic Age Replacement
(continued)
The optimal policy is to replace the item when it reaches
age t* given by:
t* 
2K
a
Extensions: this basic model can be extended to more
complex maintenance and salvage value functions, but
the optimal solution is often difficult to find.
12-18
Planned Replacement
Preventive Maintenance: replace items before they
fail to avoid failure during operation.
Note: there is no advantage to replacing operating
items that fail according to an exponential
distribution, because of the memoryless property
of the exponential. Planned replacement only
makes sense for items with IFR – Increasing
Failure Rate - distributions. (See Figure 12-14)
12-19
Successive Cycles for Planned
Replacement of a Single Item
12-20
Analysis of Warranties
Warranties are a big business for most consumer
items such as home electronics, computers,
appliances, and automobiles.
Two common types of warranties are:
Free Replacement Warranty: items that fail are
replaced with new items.
Pro-rata warranty: A rebate is given
proportional to the remaining life of the failed
item.
12-21
Free Replacement Warranty
Define:
T = Lifetime of an item picked at random.
 = Failure rate of each item.
F(t) = Cumulative distribution function of T.
C1 = Cost of purchasing a new item with warranty.
K = Cost of purchasing a new item with no warranty.
W1 = Time that the free replacement warranty is in effect
after purchase.
(Refer to Figure 12-16 on the next slide).
12-22
Replacement Cycles
for Free Replacement Warranty
12-23
Optimal policy for free
replacement warranty
Assuming that items fail completely at random,
(according to an exponential distribution with rate
), the free replacement warranty should be
purchased if the cost is less than C1* given by
C1* = (  W1 + 1) K
12-24
Optimal Policy for the ProRata Warranty
Let W2 be the term of the pro-rata warranty. Then the
value at which the consumer is indifferent between
purchasing the item with a pro-rata warranty and
an item with no pro-rata warranty, say C2* ,
solves:
C2 
*
 KW2
1 e
 W 2