Module III Product Quality Improvement
Lecture – 2 What are component and system reliability and how it can be improved?
Reliability is a measure of the quality of the product over the long run. The concept of reliability
is an extended time period over which the expected operation of the product is considered and
we expect the product will function according to certain expectations over a stipulated period of
time. With the customer and warranty costs in mind, we must know the chances of successful
operation of the product for at least a certain stipulated period of time. Such information helps
the manufacturer to select the parameters of a warranty policy.
Technically, reliability is the probability of a product performing its intended function for a
stated period of time under certain specified conditions. Four aspects of reliability are apparent
from this definition first, reliability is a probability of success-related concept; the numerical
value of this probability is always between 0 and 1. Second, the functional performance of the
product had to be measured under certain stipulated conditions. Product design is expected to
ensure development of a product that meets or exceeds the specified requirements under
specified operating conditions. For example, if the breaking strength of a nylon cord is expected
to be 1000 kg, then in the predefined operational conditions, the cord must be able to bear
weights of 1000 kg or more. Third, reliability implies successful operation over a certain period
of time (t). Although no product is expected to last forever, the time dimension ensures
satisfactory performance over at least a minimal stated period (say, 100 hours). In the context of
these three aspects, the reliability of the nylon cord might be described as having a probability of
successful performance of 0.92 in bearing loads of 1000 kg for 1 year under dry conditions.
It is observed that most manufacturing products go through three distinct phases (see Figure 38) from product inception to wear-out.
Figure 3-8 Bathtub Curve
The life-cycle curve of Figure 3-8 shows the variation in the failure rate as a
function of time in different phases. Conventionally the failure rate (  ) is plotted as a function
of time. This curve is often referred to as the bathtub curve; it consists of the debugging (infantmortality) phase, the chance-failure phase (useful life phase), and the wear-out phase.
The debugging phase, also known as the infant-mortality phase, exhibits a drop
in the failure rate as initial problems identified during prototype testing are removed.
The chance-failure phase, between times t1 and t2, is then encountered; failures occur
randomly and independently. This phase, in which the failure rate is constant, typically
represents the useful life of the product delivered to end customer. In the wear-out phase, an
increase in the failure rate is expected due to wear and tear of the product. Here, after the end of
their useful life, parts age and wear out.
For
the
random
chance-failure
phase,
which
represents
the
useful
life
of
the
product or component, the failure rate is assumed to be constant. As a result, the exponential
distribution is selected to describe the time-to-failure of the product for this phase.
An exponential distribution as a memory less property and its probability density function is
given by
f  t    e  t , t  0
The mean-time-to-failure (MTTF) for the exponential distribution can be expressed as
MTTF=1/
The reliability, at time t, say R(t), is the probability of the product lasting up to time t. It can be
expressed as,
R t   1  F t 
t
=1- e t dt  e  t
0
Here, F(t) represents the cumulative distribution function at any time t. Reliability decreases
exponentially with time (Figure 3-9) and the failure-rate function, say r(t) , is given by the ratio
of the time-to-failure probability density function to the reliability function. We have
r t  
f t 
R t 
Figure 3-9 Reliability v/s Time
Thus, assuming an exponential distribution, r (t) implying a constant failure rate, as
shown below.
r t  
 e  t
e  t

Let us consider a resistor component, which follows an exponential time-to-failure distribution
with a failure rate of 8% per 1000 hr. We are interested to calculate the reliability of the resister
at 5000 hr, and also we intend to calculate the mean-time-to-failure. Here the constant failure
rate  is obtained as
  008 /1000 hr
=0.00008/hr
Thus, the reliability for 5000 hr of survival is
R  t   e  t
=e 
- 0.000085000
=e-0.4  0.6703
Thus there is about 67% chance of survival (success) of the resister under stipulated conditions
and stipulated time (5000 hr). The mean (average) time-to-failure (assuming it cannot be repair)
of the resister will be
MTTF=1/  1 / 0.00008  12,500 h
System Reliability
Let us consider a system with three components (say three resister) in series as shown in Figure
3-10.
Figure 3-10 Series System
Without loss of generality, if the system components can be assumed to have a time-to-failure
distribution as exponential with each component has a constant failure rate, we can easily
compute the reliability of n-system in series. Suppose the system has n components and in series,
each with exponentially distributed time-to-failure with failure rates 1 , 2 , , n . The system
reliability is calculated as the product of the component reliabilities:
RS (t )  e  1t  e  2t 
 e  nt
  n
 
=exp -   i  t 
  i 1  
This
implies
that
the
time-to-failure
distributed with an equivalent failure rate of

n
i 1
of
the
system
is
exponentially
i . The mean time to failure for the system is
given by
MTTF 
1
n

i 1
i
Systems with Components in Parallel
System reliability can be improved by placing redundant components in parallel. The system
operates as long as at least one of the components operates. A three component parallel system is
represented as given in Figure 3-11.
Figure 3-11 A Parallel Component System
If the time-to-failure of each component follows exponential distributions, each with a constant
failure rate, i , i = 1, ... , n, the system reliability, assuming independence of component
operation (failure of one does not impact failure of any other component), is given by
n
RS (t )  1- 1  Ri (t )
i=1
n

=1- 1  e i t
i=1

In a special case, where all components have the same failure rate,  , the system
reliability for parallel component is given by

RS (t )  1  1  ei t

n
For such specific situation, the mean-time-to-failure for the system with n identical components
in parallel, and also assuming that each failed component is immediately replaced by an identical
component, can be expressed as
MTTF 
1
1 1
1
1    ...  
 
2 3
n
Systems with Components in Series and in Parallel
Real life systems often consist of components that are mixed and consist of both series and
parallel configuration. For such system, reliability calculation is primarily based on the
previously discussed concepts, and assumption of components operating independently. Parallel
systems are first collated to get a composite reliability, and then the overall components are
considered as series to calculate system reliability. Systems can also consist of standby
component, which operates as and when base component fails. Reader may refer to book by
Amitava Mitra (2008) or Besterfield et al (2004 ) for further details on. K-out-of-N system
(parallel system is 1 out of N system) is another possible system configuration.