CHAPTER - 1 INTRODUCTION AND BASIC CONCEPTS 1.1 INTRODUCTION The mathematical theory of reliability has grown out of the demands of the modern technology and particularly of the experiences in the World War II with complex military systems. In the early 1950’s, certain areas of reliability, especially life testing and electronic and missile reliability problems started to receive a great deal of attention both from mathematical statisticians and from the engineers in the military-industrial complex. Evidence of the intimate relationship between reliability and statistics is available in the significant number of papers written on statistical method in reliability. In December 1950, the Air Force formed an ad-hoc group on reliability of the electronic equipments to study the whole reliability situation and recommended measures that would increase the reliability of equipment and reduce maintenance. By late 1952, the department of defence had established the Advisory Group on Reliability of Electronic Equipment (AGREE). AGREE published its first report on reliability in June 1957. This report include minimum acceptability limits, requirements for reliability tests, effect of storage on reliability, etc. The overall scientific discipline that deals with general methods and procedures to which one needs to adhere during the planning, preparation, acceptance, transportation, use of manufactured articles to ensure their maximum
Transcript
CHAPTER - 1
INTRODUCTION AND BASIC CONCEPTS
1.1 INTRODUCTION
The mathematical theory of reliability has grown out of the demands of the
modern technology and particularly of the experiences in the World War II with
complex military systems. In the early 1950’s, certain areas of reliability,
especially life testing and electronic and missile reliability problems started to
receive a great deal of attention both from mathematical statisticians and from the
engineers in the military-industrial complex. Evidence of the intimate relationship
between reliability and statistics is available in the significant number of papers
written on statistical method in reliability. In December 1950, the Air Force
formed an ad-hoc group on reliability of the electronic equipments to study the
whole reliability situation and recommended measures that would increase the
reliability of equipment and reduce maintenance. By late 1952, the department of
defence had established the Advisory Group on Reliability of Electronic
Equipment (AGREE). AGREE published its first report on reliability in June
1957. This report include minimum acceptability limits, requirements for
reliability tests, effect of storage on reliability, etc.
The overall scientific discipline that deals with general methods and
procedures to which one needs to adhere during the planning, preparation,
acceptance, transportation, use of manufactured articles to ensure their maximum
2
effectiveness during use or examining a treatment to ensure that it is effective
enough to produce maximum life-time in a certain disease and develops general
methods of evaluating the quality of systems from known qualities of their
components parts or from strength and stress variable has received the name
reliability/survival theory. Obviously, reliability is an important consideration in
the planning, design and operation of systems.
The reliability theory is concerned with random occurrence of undesirable
events or failures during the life of a physical or biological system. Reliability is
an inherent attribute of a system just as is the system’s capacity or power rating.
The concept of reliability has been known for a number of years but has got
greater significance and importance during the past decade, particularly, due to the
impact of automation, development of complex missile and space programmes.
With increasing automation and the use of highly complex systems, the
importance of obtaining highly reliable systems has recently been recognized.
From a purely economic point of view, high reliability is desirable to reduce
overall costs. For example, the yearly cost of maintaining some military systems
in an operable state is as high as ten times the original cost of the system. The
failure of a component most often results in the breakdown of the system as a
whole. A space satellite may be rendered completely useless if a switch fails to
operate. Safety is an equally important consideration. A leaky brake cylinder
could result in personal injury and undue expenses. Also, caused by unreliability
are scheduled delays, inconvenience, customer dissatisfaction and perhaps also the
loss of national security. The need for reliability has been felt both by the
3
government and industry. For example, the Department of Defence and NASA
(USA) impose some degree of reliability requirements. MIL-STD-785
(Requirements for Reliability program for system and Equipments) and NASA
NPC 250-1 (Reliability program provisions for space system contractors), provide
in detail the requirements for a reliability programme to achieve reliable products.
Everyone has experienced the frustration of waiting in lines to obtain service.
It usually seems like an unnecessary waste of time. In our private lives, we have
the option of seeking service elsewhere or going without the service. Such
defections have direct economic consequences for the organization providing the
service. When a customer leaves a waiting line, he becomes an opportunity cost,
the opportunity to make a profit by providing the service is lost. An important
aspect of system design is to balance this cost against the expense of additional
capacity. The study of waiting lines, called the ‘queuing theory’, is one of the
oldest and most widely used operation research techniques. The first recognized
effort to analyze queues was made by a Danish engineer, A.K. Erlang, in his
attempts to eliminate bottlenecks created by telephone calls on switching circuits.
A queuing situation is basically characterized by a flow of customers arriving at
one or more service facilities. On arrival at the facility the customer may be
serviced immediately or if willing, may have to wait until the facility is made
available. The service time allocated to each customer may be fixed or random
depending on the type of service. Situations of this type exists in every-day life.
A typical example occurs in a barbershop. Here, the arriving individuals are the
customers and the barbers are the servers. Another example is represented by
4
letters arriving at a typist’s desk represent another example. The letter represents
the customers and the typist represents the server.
In general a queue is formed when either units requiring services –
commonly referred to as customers, wait for service or the service facilities, stand
idle and wait for customers. Some customers wait when the total number of
customers requiring service exceeds the number of service facilities, some service
facilities stand idle when the total number of service facilities exceeds the number
of customers requiring service. But in a few situations waiting lines cause
significant congestion and a corresponding increase in operating costs. For
example, ships wait to be unloaded at docks, project wait attention by the
engineering staff, aircraft wait to land at an airport and breakdowns await repair by
maintenance crews. These examples show that the term “Customer” may be
interpreted in a variety of ways. Also, a service may be performed by moving
either the server to the customer or the customer to the server. Such service
facilities are difficult to schedule "optimality" because of the presence of the
randomness element in the arrival and service patterns. A mathematical theory has
thus evaluated that provides means for analyzing such situations. This is queuing
(or waiting line) theory, which in based on describing the arrival and/or departure
(service) patterns by the appropriate probability distribution. Operating
characteristics of a queuing situation are then derived by using probability theory.
Examples of these characteristics are the expected waiting time until the service of
a customer is completed or the percentage of idle time per server. Availability of
such measures enables analysis to make inferences concerning the operation of the
5
system. The parameter of the system (such as the service rate) may then be
adjusted to ensure a more effective utilization from the viewpoints of both
customer and server.
Queuing theory analysis involves the study of a system's behaviour over
time. A system is said to be in transient state when its operating characteristics
(behaviour) vary with time. This usually occurs at the early stages of the system's
operation where its behaviour is still dependent on the initial condition. However,
since one is mostly interested in the "long-run" behaviour, most attention in
queuing theory analysis has been directed to steady state results. A steady state
condition is said to prevail when the behaviour of the system becomes independent
of time.
1.2 BASIC CONCEPTS IN RELIABILITY THEORY
Reliability engineering is a branch of science and every branch of science is
studied systematically i.e., first of all, its basic concepts are understood. The
following subsections include the definitions and mathematical expressions of
some important concepts which are necessary to understood before entering the
reliability theory.
System: A system is defined as an arbitrary device performing an activity. By
nature, systems are classified as under:
(i) Man Made or Engineering System - As a result of advancement of science,
today man has to his credit so many sophisticated systems which are fully
6
designed by his hands and brain. As for example, computer system, electric power
supply system, television system, etc. are some man made systems.
(ii) Natural or God Made System - Besides, the man made systems, the universe
have some other systems whose existence is independent of human hands and thus
called the natural or God made system. Human body system, solar energy system,
weather changing system etc. are some examples of God made systems. Generally,
when we perform the life-testing experiments with man made systems, we call it
'Reliability Analysis' while on the other hand, when we deal with God made
systems, we name it 'Survival Analysis'. Hence Reliability and Survival are
interchangeable terms. The concepts of reliability characteristics defined in many
ways by different authors. In the present study, the following definitions of
various reliability characteristics have been used.
1.2.1 Reliability
According to Bozovasky (1961), the reliability is a yardstick of the capability
of an equipment to operate without failure when put into service. A more rigorous
definition of reliability is as follows -
'Reliability' of a component (or a system) is the probability that the
component performs its intended function adequately for a specified period of time
without a major breakdown under the stated operating conditions or environment.
Mathematically if T is the time till the failure of the unit occurs, then the
probability that it will not fail in a given environment before time 't' is
R(t) = P (T>t) = Pt1 F (t) ...(1)
7
where F(t) is the cumulative distribution function (c.d.f.) of T, called unreliability
R(t) of the system so that F (t) + R (t) = 1. Thus, the reliability is a function of
time and depends on environmental conditions which may or may not vary with
time.
Noticeable features of reliability functions are
(i) 0 < R(t) < 1
(ii) t 0 tLim R(t) = 1 and Lim R(t) 0
(iii) R(t), in general, is a decreasing function of time.
1.2.2 Failure Rate or Hazard Rate Function
Experiences have shown that a very good way to present the failure data is to
compute and plot either the failure density function or the hazard-rate as a function
of time. The failure density is a measure of the overall speed at which failures are
occurring whereas the hazard-rate is a measure of instantaneous speed of failure.
As the time passes on, the unit get out worn and begin to deteriorate. There are
several causes of failures of components such as :
(a) Careless planning, substandard equipment and raw material used, lack poor
quality control, etc.
(b) Human errors
(c) Random or chance causes. Random failure occurs quite unpredictable
random intervals and cannot be eliminated by taking necessary steps the
planning production or inspection stage.
8
(d) Poor manufacturing techniques
Since the item is likely to fail at any time, it is quite customary to assume
that the life of the item is a random variable with a distribution function F(t),
where F(t) is the probability that the item fails before time T. A failure is the
partial or total loss or change in the properties of a device (or system) in such a
way that its functioning is seriously affected or completely stopped. Hazard
function of a system is a useful concept in life testing experiments. It provides the
instantaneous failure rate of a unit at time t, given that the unit has survived up to
time t. Hazard function is usually denoted by h(t). In economics, reciprocal of
this function is called "Mill's ratio" and in demography, its name is 'age specific
death rate'. This function is also known as force of mortality in actuarial and life
contingency problems. Mathematically expression is developed as under:
dt 0
dt 0
dt 0
P [a device of age t will fail in (t, t+dt)/it has survived upto t)]h(t) = Lim dt
P[t < T < t + dt/ T > t] = Lim dt
[T t+dt] P [T t] = Lim P[T t]. dt
=
dt 0
dt 0
F(t t) F(t)Lim[1 F(t)]dt
R(t) R(t+ t) = Lim R(t) dt
1 d = R (t) R(t) dt
f(t) = R(t)
...(2)
On integrating the expression in (2), one gets
t t
00
h(t) dt = log [1 F(t)] log R(t)
9
t
o R(t) = exp h(u) du
...(3)
It follows that the probability of failure free operation during the
interval (t1, t2) is expressed by
t21 2
t1R(t , t ) = exp h(t)d(t)
Let us denote t
0h(t) dt by H(t), represents the cumulative hazard function.
From (2) and (3), it follows that
t
0f (t) = h (t). exp h(t) dt
...(4)
From relation (4), it is clear that hazard function is helpful in deciding the form of
p.d.f. of failure time distribution. In general, failure phenomenon of a system can
be represented by
(i) a monotonically increasing hazard function or increasing failure rate
(IFR),
(ii) a monotonic decreasing hazard function or decreasing failure rate
(DFR),
(iii) a constant hazard function or constant failure rate (CFR),
(iv) a bath-tub shaped or U-shaped hazard function.
1.2.3 Mean Time To System Failure (MTSF)
Mean time to system failure or mean life of the system is the expected value
of the failure time distribution, i.e.
10
0
MTSF = E(T) = t. f (t).dt
But, d df(t) = F(t) = R(t)dt dt
Hence, 0
MTSF = t.dR(t)
00
= tR(t) R(t)dt
0
MTSF = R(t)dt ...(5)
1.3 SOME STATIC SYSTEM CONFIGURATIONS
A system is a combination of units or components forming an actively and to
find the system reliability, we must know the reliabilities of its units/components
and their network sufficiently well. Suppose that a system with life time T,
consists of n-different limits / components C1, C2, ..., Cn with life time Ti of the ith
unit/component. At time t the system reliability is
R (t) = P (T > t)
and the reliability of the ith unit/component is
(t)iR = P [Ti > t]
The system configurations generally considered are as follows:
11
1.3.1 Series Configuration
The simplest and the most common structure in the reliability analysis is the
series configuration. In this case, the functional operation of the system depends
on the proper operation of all system units/components. The examples of series
configuration are:
(i) The aircraft electronics system consists mainly of a sensor subsystem, a
guidance subsystem, computer subsystem and the fire control subsystem. This
system can only operate successfully if all these operate simultaneously.
(ii) Deepawali or Christmas glow bulbs, where if one bulb fails, it leads to entire
system failure.
The reliability of n units/components series system is given by
R (t) = P [ T1 > t, T2 > t, ... Tn > t]
n n
i ii=1 i=1
P [T > t] = R (t) ...(6)
If hi (t) is the instantaneous failure rate of the ith unit and h (t) is the instantaneous
failure rate. Then, we have,
n
ii=1
h (t) = h (t) ...(7)
Thus, in series configuration the units / components' reliabilities are multiplied to
obtain the system reliability and the unit's/component's hazard rates are added up
to the obtain the system hazard rate. An n-component series configuration is
shown in Fig. 1
12
FIG. 1
1.3.2 Parallel Configuration
In many systems, several signal paths perform the same operation. If the
system configuration is such that the function of atleast one path is sufficient to do
the job, the system can be represented by a parallel model. In this configuration,
all the units/components of the system are arranged in parallel and the failure of
the system occurs only when all the units/components of the system fail.
The reliability of the n-units/components parallel system is given by
n
ii=1
R(t) 1 (1 R (t)) ...(8)
An n-component parallel system configuration is shown in Fig. 2.
1.3.3 k-out of -n configuration
Another practical system is one where more than one of its parallel
components are required to meet the demands. For instance, two of the four
IN OUT Cn Ci C2 C1
Cn
Ci
C2
C1
OUT IN
FIG.2
13
generators in a generating station may be necessary to supply the required power
to the customers. The other two are added to increase the supply reliability.
Let us consider a such type of system which functions if at least k (l < k < n)
out of n units/components functions. For identical and statistically independent
units/ components the system reliability is given by
m i n i
c ci=k
R(t) = R (t) 1 R (t) ...(9)
where, Rc (t) is the reliability of each component at time t. In particular, the block
diagram for 2 out of 3 configuration having three identical components C1, C2, C3
can be represented as follows
C1
C1
C2 C3
. . OUT
C2
C3IN
FIG. 3
The series and parallel configurations are the particular cases of k-out of n
configuration when k = n and k = 1, respectively.
1.4 CENSORING IN LIFE TESTING EXPERIMENTS
Life testing experiments are usually much time consuming and
expensive. There are several situations where it is neither possible nor desirable to
14
use complete sampling. We, therefore, make the sample censored. Obviously, in
a censoring situation, only a portion of the sample of individual is studied. The
censoring is broadly classified in two ways.
1.4.1 Time Censoring or Type I Censoring
In this type of censoring plan, the amount of time required to obtain the
information from a complete sample is reduced. We may put the complete sample
to test and the test is terminated at a prefixed time. This type of censoring is called
Time Censoring or Type I Censoring. It is frequently arises in medical and
agricultural research. Mathematically, let n items are put on test and the test is
terminated at the t0. Let T be the r.v. denoting the lifetime of an item, such that
0 0F (t ) P[T t ]
Let m be the number of items failed before time t0. Then
(t )om ~ B n,F ; m = 0, 1, 2, ... n
Note that from type I censored sample we get the following information:
(i) Out of n items, m items failed before time t0 with lifetimes
(1) (2) (m)x x ...x
(ii) (n m) items did not fail time to. The likelihood of the sample
(1) (2) (m)(m, x x ...x ) is given by
(1) (2) (m) 0L x x ...x | F(t )
15
no
nn m
i 0i 1
[1 F(t )] ; m = 0 n f (x )[1 F(t )] ; m =1,2...n
(n m)!
where f(.) is the pdf of T.
1.4.2 Failure Time or Type II Censoring
If we put n individuals on test and terminate the experiment at the failure of
the first 'r' individuals, where r (< n) is pre-assigned number of failure. Then, the
terminated sample from such an experiment is called as Failure Censored Sample
or Type II Censored Sample.
In this type of censoring, the number of observations or failures 'r' is decided
before the data are collected. But the time of failure of r (fixed) individuals is a
random variable. It is mostly used in dealing with high cost sophisticated items
such as vacuum tube, X-ray machine, colour T.V. picture tubes, etc.
Suppose the failure times consist of the first 'r' smallest life times be
(1) (2) (r)x x ...x out of a r sample of n. Lifetimes x1, x2, ..., xn, which are i.i.d.
random variables having pdf f(t) and reliability function R(t). Then, it follows
from the general results on order statistic that the likelihood function of the sample
(1) (2) (r)x x ...x given by
n
n r(1) (2) (n) (1) (r)
i 1
n!L x x ...x f (x )[R(x )](n r)!
16
1.5 BAYESIAN APPROACH AT A GLANCE
Bayesian framework has several interesting features that make it more
attractive to applied statisticians than to its frequent counterpart. It combines the
prior information with the information contained in the data to formulate the
posterior distribution, which is the basis of Bayesian inference. It possesses a well
developed and straight forward procedure for facing the problem of optimal action
in a state of uncertainty. Application of Bayesian concepts and methods abound in
Econometrics, Sociology, Engineering, Reliability Estimation, O.R. and Quality
Control. It addresses the question of how the model underlying the data may be
revised in the light of new information and experience. Bayes analysis is an
essentially self-contained paradigm for statistics. It is an excellent alternative for
use of large sample asymptotic statistical procedures. Bayesian procedures are
almost always equivalent to classical large-sample procedure when the sample size
is very large. The foundation stone of this technique is Baye's theorem and
conditional probability. This theorem was presented by Reverend Thomas Bayes,
an English Minister who lived in the 18th century. Laplace modified the Bayes
basic theorem and the modified version is known as Bayes theorem.
1.5.1 Bayes Theorem
Consider an unobservable vector and observable vector y of length k and n
respectively with their density (, y). From standard probability theory, we have
p1 (, y) = p ( / y) p (y) ...(10)
p1 (, y) = p (y /) p () ...(11)
17
From (10) and (11), we get p ( / y) = p(y/θ) p(θ)p(y)
...(12)
Note that p(y) = p , y .d
p y / p( ). d = E [ p y/ ] ...(13)
where E indicates averaging with respect to distribution of [e.g. Box and Tiao
(1973); Gelman, Carlin, Stern and Rubin (1995) and Lee (1997)], it is clear that
p(y) is not a function of . As a result (12), can be rewritten as
p / y p (y/ ) p ( ) ...(14)
This is well known Bayes theorem. In the Baysesian terminology, p() is a
prior density of , which tells us what is known about without knowledge of
data. The density p(y/) is likelihood function of , which represents the
contribution of y (data) to knowledge about [e.g. Berger (1985), and Zellener
(1971)]. Finally, p(/y) is the posterior density, which tell us what is known about
given knowledge of data y.
A good feature of Bayesian analysis is that it takes explicit account of prior
information in the analysis and gives a satisfactory way of explicitly organizing
assumptions regarding prior knowledge or ignorance leading to posterior
distribution inferences. It creates the possibility of minimization of expected loss
consisting the parameters as random variables.
Suppose that n items are placed on test. It is assumed that their recorded life-
times form random sample say x1, x2, ..., xn which follow a distribution with
18
p.d.f. f(x, ). Here, we will assume to be a random variable. Let g() be the
p.d.f. of which is known or otherwise estimated by statistical techniques. Thus,
the failure time p.d.f. f (x, ) can be regarded as a conditional p.d.f. of x given .
Here g () is known as the prior p.d.f. Therefore, the joint p.d.f. of (x1, x2, ..., xn;
) is
n,1 2 n i 1 2 ni 1
H x x , ..., x ; f x , g ( ) = L x , x , ..., x ; g( )
...(15)
The marginal p.d.f. of (x1, x2, ..., xn) is given by
1 2 n 1 2 np(x , x , ..., x ) = H (x , x , ..., x ; ). d
...(16)
Therefore
1 2 n1 2 n
1 2 n
H (x , x , ..., x ; )g ( / x , x , ..., x ) =
p (x , x , ..., x )
1 2 n
1 2 n
L (x , x , ..., x ; ) g( ) =
L(x , x , ..., x ; ) g( ) d
...(17)
The variation in observed prior to the data (x1, x2, ..., xn) is represented by g()
and is known as prior p.d.f. The conditional distribution of given (x1, x2, ..., xn)
obtained posterior to the experiment is called posterior distribution and denoted by
g(|x1, x2,...,xn). Just as the prior distribution reflects beliefs about prior to the
experimentation, so g(|x1, x2,...,xn) reflects updated beliefs about the parameter
posterior to observing the sample x1, x2,...,xn. In other word, the uncertainty about
19
the parameter prior to the experiment is represented by the prior p.d.f. g () and
the same after the experiment is posterior p.d.f. denoted by (|x1, x2,...,xn).
This process is a straightforward application of Bayes theorem. After
obtaining the posterior distributions of the parameters involved in the parent
population, any statistical inference like estimation, testing of hypothesis about
these parameters may be drawn with the help of these distribution. In case when
prior distribution of is discrete, the integral sign in (16) and (17) are replaced by
the sign of summation over .
In Bayesian analysis, the estimator of i.e. * is one which minimizes
expected loss w.r.f. the posterior distribution, i.e., it depends on the loss function
chosen. If the loss function is taken as quadratic loss function defined as L (*,)
= (*)2, then the Bayes estimator that accomplishes the task of estimating is the
posterior mean, i.e.
*1 2 n 1 2 n = E |x , x , ..., x . |x , x , ..., x d
On using the posterior distribution, ( 100% Bayes confidence interval
(PPfor may be obtained from
P2
1 2 nP1
( | x , x , ..., x ) d = 1 ...(18)
The origin of Bayesian theory may be attributed to a very primary paper by
Rev. Thomas Bayes republished in 1958 due to its fundamental importance. The
details of Bayesian statistical theory can be found in Raifla and Schlaifer (1961),
20
Jeffreys (1961), Savage (1962), Lindeley (1965), Box and Tiao (1973), Berger
(1985), S.K. Sinha (1998) and Bernardo and Smith (1993).
1.6 CONCEPT OF PRIOR DISTRIBUTION
A prior distribution, which is supposed to represent, what is known about
unknown parameter before the data are available, plays an important role in
Bayesian analysis. The prior information concerning the parameter can be
summarized mathematically in the form of prior distribution g() on the parameter
space . A detailed discussion to obtain the solution to the problem concerning the
choice of a prior distributions of is given in Raiffa and Schlaifer (1961), but here
we shall confine ourselves just in defining them. Priors for the parameters in
distribution differ in respect of their widen properties.
The natural conjugate priors satisfy the closer property implying that the
prior and posterior distributions for the parameter belong to the same family. This
method of choosing the priors is much popular because it leads to mathematical
simplicity and tractability. This property of conjugate priors is also known as
'closer under sampling' property Weltherill (1961). Raiffa and Schlaifer (1961)
have considered a method of gathering prior densities on the parameter space. A
family of such densities has been called by them a 'natural conjugate family'. For
example in case of an exponential density, the gamma priors form such a family.
In case, when the decision maker does not have any prior knowledge about
the parameter, non-informative quasi density may be used. The role of non-
informative prior quasi densities in Bayesian analysis is available in Bhattacharya
21
(1967). Jeffreys (1961) proposed a general rule for obtaining the prior distribution
of . According to this rule, the unknown parameter , which is assumed to be a
random variable follows such a distribution which is proportional to the square
root of the fisher information on . Mathematically, we have
g ( ) I ( ) ...(19)
or g ( ) = constant I ( )
where 2 2
2 log f (x, ) log f (x, )I ( ) = E = E
...(20)
for the case when there is a single unknown parameter . For a situation when is
a vector valued parameter, the determinant of the information matrix i.e. |I()|
takes place where
2 log (f) I ( ) = E
i j
...(21)
A difficulty arises when the prior information about the parameter is vague or
worse still, there is no prior information whatever. This leads to the consideration
of what are known as improper or quasi prior distribution. For a proper prior we
have
g ( ) 0
and g ( ). d = 1,
when is a continuous random variable.
or g ( ) = 1,
when is a discrete random variable.
While for an improper prior
22
g ( ) 0
but g ( )d 1,
when is a continuous random variable.
or g ( ) 1,
when is a discrete random variable.
Jeffreys prior may or may not be proper. Various other rules have also
been suggested for the selection of a prior but no neat solution appears to the
problem until now.
In Bayesian analysis of reliability characteristics, the prior knowledge is
updated by using experimental data and variations in parameters are represented
by posterior distribution. The admissibility of the parameters of the prior
distribution can not be tested unless we make use of additional information on the
prior distribution. However, the variations in the parameters can be neutralized by
averaging as we do in the case of compound distribution of concerned variable.
Obviously, it seems more logical to infer about the parameters of the prior
distribution with the help of the compound distribution, which also involves these
parameters.
1.7 LOSS FUNCTION
Suppose be an unknown parameter of some distribution f (x|) and we
estimate by some statistics ˆ ˆ. Let L ( , ) represent the loss incurred when the
true value of the parameter is and we are estimating by the statistic ̂ . The
loss function denoted by ˆL ( , ) is defined to be real valued function satisfying:
(i) ˆL ( , ) 0 for all possible estimates ̂ and all
23
(ii) ˆL ( , )= 0 for ̂ = .
We now consider the following loss functions:
1.7.1 Quadratic Loss Function
A function defined as 2ˆ ˆ L ( , ) = k ( ) is called quadratic loss function.
Such a loss function is widely used in most estimation problems, if k is a function
of , the loss function is called the weighted quadratic loss function. If k = 1, we
have 2ˆ ˆ L ( , ) = ( ) , known as squared error loss function (SELF). Under the
SELF, the Bayes estimator is the posterior mean. The squared error loss function is
a symmetric function of ̂ and . The reason for the popularity of SELF is due to
makes the calculations relatively straightforward and simple.
1.7.2 Linex Loss Function
A symmetric loss function assumes that positive and negative errors are
equally serious. However, in some estimation problems such as assumption may
be inappropriate, Canfield (1970) points out that the use of symmetric loss
function may be inappropriate in the estimation of reliability function. Over
estimation of reliability function or average lifetime is usually much more serious
than under estimation of reliability function or mean time to failure (MTTF).
Also, an under-estimation of the failure rate results in more serious consequences
then the over estimation of the failure rate. This lead to the statistician to think
about asymmetrical loss function which has been proposed in statistical literature
Ferguson (1967), Zellner and Geisel (1968), Aitcheson and Dunsmore (1975) and
Berger (1980) have considered the linear asymmetric loss function. Varian (1975)
24
introduced the following convex loss function known as Linex (linear -
exponential) loss function:
aL ( ) = be c b; a, c 0, b > 0 ...(23)
where, ˆ = . It is clear that L (0) = 0 and the minimum occurs when
c = a.b, therefore,
aL ( ) = b e a 1 , d 0, b > 0
where a and b are the parameters of the loss function may be defined as shape and
scale respectively. This loss function has been considered by Zellner (1986), Rajo
(1987) and Base and Ebrahimi (1991) considered the L () as
aL( ) = be c b; a, c 0, b > O
where ˆ
=
And studied the Bayesian estimation under the Linex loss function for
exponential life time distribution. This loss function is suitable for the situations
where observations of is more costly than its underestimation. This loss function
L () have the following nice properties:
(i) For a = 1, the function is quite asymmetric about zero with overestimation
being more costly than underestimation.
(ii) For a < 0, L () rises exponentially when < 0 (underestimation) and almost
linearly when > 0 (overestimation).
(iii) For small value of |a|, L() is almost symmetric and not far from a squared
error loss function. Indeed, on expanding
25
2 2
e 1 + + z
or L() is a squared error loss function. Thus for small values of |a|, optimal
estimates are not far different from those obtained with a squared error loss
function.
1.7.3 Precautionary Loss
Norstrom (1996) introduced an alternative asymmetric precautionary loss
function and also present a general class of precautionary loss function with
quadratic loss function such as a special case. These loss functions approach
infinitely near the origin to prevent underestimation and thus giving a conservative
estimators, especially when, low failure rates are being estimated. These
estimators are very useful when underestimation may lead to serious
consequences. A very useful and simple asymmetric loss function is
ˆ( )ˆL ( , ) = ˆ
1.7.4 Entropy Loss
In many practical situations, it appears to be more realistic to express the loss
in terms of the ratio ˆ
. In this case, Calabria and Pulcini (1994) point out that a
useful asymmetric loss function is the entropy loss
peL ( ) p log ( ) 1 ...(27)
where ˆ
=
26
And who minimum occurs at ̂ when a positive error ( ̂ ) causes more
serious consequences than a negative error and vice-versa. For small |p| value, the
function is almost symmetric when both ˆ and are measured in a logarithmic
scale and approximately
22
e eˆL ( ) p log ( ) log ( ) ...(28)
Also, the loss function L () has been used in [Dey et al. (1987)] and [Dey and Liu
(1992)], in the original from having p = 1. Thus, L () can be written as
eL ( ) = b log ( ) 1 ; b > 0 ...(29)
where ˆ
=
1.8 SOME IMPORTANT FAILURE-TIME DISTRIBUTION MODELS
Life-testing is an important and useful section in the area of statistics for
engineering sciences. The analysis of failure time data over the years has given
birth to a number of parametric models. These models were found suitable for
representing a wide range of situation and particularly in problems related to the
modeling of various aging or failure process. Among univariate models, a few
particular distributions occupy a central role because of their demonstrated
usefulness. The choice of a failure-time model is largely a skill. In most
experiments, the measurements are assumed to be drawn from certain distribution.
The choice of the distributions depends on past experience with the process,
mathematical expediency and to some extent faith. However, in some cases, the
relationship between the failure-mechanism and the failure-time function may be
used in making a choice. A useful series of references for this purpose is the
27
Johnson and Kotz (1970), which extensively catalogs mathematical and statistical
properties of most of the distributions and provides additional references
concerning their areas of application. Some frequently used lifetime models are as
follows:
1.8.1 The Exponential Distribution:
In reliability theory, the exponential distribution plays an important role in
life testing experiments as the part played by the normal distribution in agricultural
experiments the effect of different treatments on the yield. Historically, the
exponential distribution was the first life time model for which statistical methods
were extensively developed. Early, Sobel (1953, 1954, 1955) and Epstein (1954,
1960) gave numerous results and popularized the exponential as a life time
distribution, especially in the area of industrial life testing. The desirability of the
exponential distribution is due to its simplicity and its inherent association with the
well developed theory of Poission Process.
The p.d.f. of one parameter exponential distribution is written as
x|1f (x, ) = e ; x, > 0
...(30)
where , the scale parameter, is the average or the mean life of the item.
In life testing, 1
, the first part of the density function in (30), is referred to
as constant hazard rate. The reliability function for time t of items whose life time
follow exponential distribution in (30) is
t / R(t) = P [X > t] = e ...(31)
for = 1, the density in (30) is called standard exponential distribution.
28
In many life testing problems, often it has been found useful to fit a two-
parameter exponential model to failure time data with p.d.f.
1 (x )f (x, , ) exp ; x > , > 0 ...(32)
where = 0 is called the guarantee time or threshold parameter. If = 0 , we get
the one parameter exponential density. Again, for this model the reliability
function for time t are
1 if t
R(t) exp[ (t ) / ] if t >
...(33)
1.8.2 The Gamma Distribution
The gamma distribution is a natural extension of exponential distribution.
The p.d.f. of two parameter gamma distribution is given by
k
k 1xf (x; , k) = e x ; x, , k > 0k
...(34)
where and k are the scale and the shape parameters respectively. For k =1, the
gamma distribution reduces to the one parameter exponential distribution with
parameter . For integer value of k, the gamma p.d.f. is also known as the
Erlangian p.d.f. Its reliability and hazard functions involve the incomplete gamma
function i.e.
f(t)R(t) = 1 I (k, t) and h(t) = , t > 0R(t)
...(35)
where I (k, t) is an incomplete gamma function defined as
t k 1
0
1I(k, t) = e dk
...(36)
29
It can be shown that h (t) is monotonic decreasing (increasing) for k < 0 (k>1)
constant for k = 1. The shape parameter k is also defined as the intensity of IFR or
DFR by Sharma and Rana (1990).
1.8.3 The Weibull Distribution
The Weibull distribution is perhaps the most extensively used life
distribution model in reliability theory. Weibull (1951) and Berretoni (1964)
advocated its applications in connection with life-times of many types of
manufactured items. It has been used as a model with diverse type of items such as
vacuum tubes (Kao, 1959), ball-bearings (Lieblein and Zelen, 1956) and electrical
insulation (Nelson, 1972). Weibull distribution is widely used in biomedical
applications like in studies on time to the occurrence of tumors in human
populations (Whittemore and Altschuler, 1976) or in laboratory animals (Pike
1966; Peto et al., 1972). The p.d.f. of the Weibull distribution is given by
kx kkf (x; ,k) = e x ; x, k, > 0
...(37)
Here, is referred to as a scale parameter and k is shape parameter. The random
variable X having p.d.f. in (37), is said to have two parameter (k and ) Weibull
distribution. For this distribution, the reliability function for time t, is given by
kt /R(t) = e ...(38)
and the instantaneous failure rate or hazard function becomes
k 1kh(t) = t
...(39)
30
Obviously, h(t) is monotonically decreasing (increasing) for k < 1 (k >1) leads to
the exponential distribution. This distribution was first introduced by a Swedish
Physicist Weibull (1939).
1.8.4 The Geometric Distribution
Generally, the life time distributions of the system's components are assumed
to be continuous. However, there exist systems whose components life time are
measured in terms of the number of completed stock cycles. Even for a
continuous operation, involving continuous measurement of life-time observations
made at periodic time point given rise to discrete situation and therefore a discrete
model may be more appropriate. Yakub and Khan (1981) Mishra (1982), Patel
and Gajjan (1990), Mishra and Singh (1992), Patel (2003), Dillip (2004), Krishna
and Jain (2004), and Anwar Hasan et al. (2007), etc. considered the geometric
distribution in analysis. Consider an experiment consisting of independent trials,
called Bernoulli trials such that there are only two outcomes E1 (occurrence of a
particular event) and E2 (non-occurrence of particular event). Thus, the sample
description space of this experiment is S = {E1, E2}. Define a r.v. Xi such that
th
1i th
2
0 : E occurrence on i trialX
1 : E occurrence on i trial
...(40)
Also, let P (Xi = 0) = and P (Xi = 1) = () i. Define another random
variate 'X' to denote the number of independent trials to the first non-occurrence.
The sample description space of X is S = {X: X = 1, 2, ...} and
xP(X x) (1 ). ; x = 0, 1, 2 ... ...(41)
31
The p.m.f. in (41) has been suggested as life-time distribution as it gives the
probability of x successful cycles of a system followed by one non-survival, where
is the probability of survival. For this distribution, the reliability function for
time t is
tR(t) = ...(42)
and the hazard rate becomes as
f(t, )h(t) = = (1R(t)
...(43)
Since the exponential distribution as a life time distribution, has some nice
properties. It represents constant hazard rate, has a memoryless property and
possesses much of the mathematical feasibility. Being a discrete analogue of the
exponential distribution, the geometric distribution also has a pride place among
life time distributions. It is the only discrete life time distribution having constant
hazard rate and following the momoryless property also.
1.9 COMPOUND DISTRIBUTION
These distributions are formed by 'mixtures of distributions'. The compound
distributions in the symbolic form can be written as 1 2ˆF F . Where F1 represents
the original distribution, the varying parameter and F2 is the compounding
(mixing) distribution. If the cumulative distribution function of a random variable
is F(x|1, 2,..., n), depending on n parameters 1, 2,..., n then a compound
distribution is constructed by ascribing to some or all of the 's, a probability
distribution. The new distribution has the cumulative distribution function E
32
[F(x|1, 2,..., n)], the expectation being taken with respect to the joint distribution
of 's.
Some of well known compound distributions are
(i) Exponential () ̂ gamma)
Means a compound exponential distribution formed by ascribing the gamma
distribution with parameter and to the expected value of of exponential
distribution. This compound distribution is a Pareto distribution and is defined by
x 1
0f (x, , ) = e e d
( +1) = ; , , x > 0
(x+ )
This is the p.m.f. of Pareto distribution with mean = 1
and
variance = 2
2 ; > 2
( 1) ( 2)
The co-efficient of variation is ( 2) and is independent of the scale
parameter .
(ii) Geo ̂ (u,v)
i.e. 1 x u 1 v 1
0
1f (x, u, v) = (1 ) (1 ) dB(u, v)
[x]
[1 x]v u ; x = 0, 1, 2 ...
(u v)
This is the p.m.f. of inverse Poly-Eggenberger distribution. Here
33
u[r] = u (u+1) ... (u+r
(iii) Paission () ̂ Gamma (, )
Means a compound Poission distribution formed by ascribing the gamma
distribution to the expected value of , of a Poission distribution. This compound
Poission distribution is a negative binomial distribution.
(iv) Binomial (n, p) p̂ Beta (, )
Means a compound binomial distribution formed by ascribing the beta
distribution of first kind to the expected value of of a binomial distribution. This
compound distribution is a Poly-Eggenberger distribution.
(v) Normal ˆ) Normal 2
2 μ ︵ θ,σ ︶
This compound normal distribution is also normal with parameters and
2 21 2( ) .
(vi) Hypergeometric (n, x, N) x̂ Binomial (N, p)
This compound distribution is a binomial with parameter n and p.
1.10 ROBUSTNESS
Robust statistics provides an alternative approach to classical statistical
methods. The motivation is to produce estimators that are not unduly affected by
the small departures from model assumptions. In statistics, classical methods rely
heavily on assumptions which are often not met in practice. In particular, it is
often assumed that the data residuals are normally distributed at last
approximately, or that the central limit theorem can be relied on to produce
normally distributed estimates. Unfortunately, when there are outliers in the data,
34
classical methods often have very poor performance. Robust statistics seeks to
provide methods that emulate classical methods but which are not unduly affected
by outliers or other small departures from model assumptions. The subject of
robustness is receiving considerable attention of late. The first theoretic approach
to the robust statistics was introduced by Huber (1981). His book on this topic
made this fundamental work to a wider audience. Other good books on robust
statistics are Hampel et al. (1986) and Rousseeuw and Leroy (1987). A modern
treatment is given by Maronna et al. (2006). Huber's book is quite theoretical,
whereas the book by Rousseeuw and Leroy is very practical. Hampel et al.
(1987) and Maronna et al. (2006) fall somewhere in the middle ground. Robust
parametric statistics tends to rely on replacing the normal distribution in classical
methods with the t-distribution with low degree of freedom (i.e. high Kurtosis;
degree of freedom between 4 and 6 have often been found to be useful in practice)
or with a mixture of two or more distributions. Examples of robust and non-robust
statistics are
(i) The 'median' is a robust measure of central tendency while the 'mean' is not.
(ii) The 'median absolute deviation' and inter-quartile range are robust measures
of statistical dispersion while the 'standard deviation' and 'range' are not Robust
statistics, in a loose, non-technical sense, is concern with the fact that assumptions
commonly made in statistics are almost approximations to reality. As a collection
of related theories, robust statistics is the statistics of the approximate parametric
methods. The moral is clear. One should check carefully to see that the underlying
assumptions are satisfied before using parametric method.
35
1.11 REVIEW OF LITERATURE
In recent years reliability has been formulated as the science of predicting,
estimating or optimizing the probability or survival, the mean life or more
generally the life distribution of components or systems. To study and solve
problems that arise in reliability theory, the knowledge of methods of probability
theory and mathematical statistics is necessary. At present, not only engineers and
scientists but also government leaders are concerned with increasing the reliability
of a system.
In order to obtain different parameters of interest like reliability (survival)
function, nature of hazard rate, mean time to system failure, availability, etc. called
reliability characteristics, the research area can be broadly classified into the
following two categories:
(1) In studies like Dhillon and Singh (1980), Govil (1983), and Balagurusamy
(1984), the authors developed stochastic models under the various assumption
which best fit to the engineering system used in day to day practical life. They
obtained the reliability characteristics and net expected profit during a finite
interval of time using the well known techniques such as regenerative point
technique, semi-markov process and supplementary variable technique.
(2) On the other hand, studies like Epstein and Sobel (1952), Barlow and
Proschan (1965, 1975), Mann et al. (1974), Kapur and Lamberson (1977), Elandt-
Johnson and Johnson (1980), Kalbfleich and Prentice (1980), Miller (1981), Cox
and Oakes (1984), Lawless (1982), Martz and Waller (1982), Sinha (1986),
include recording of lifetime data on individuals and then various inference
36
techniques are used to estimate various reliability characteristics of the system.
The reliability characteristics are analysed in respect of variation in the parameters
involved in lifetime distributions and repair time distributions.
The literature on reliability analysis includes, in a broad sense, the analysis
of a positive valued random variable representing time to failure of a physical or
biological system and its analysis is gaining importance for research workers in the
field of industry and engineering. Obviously, the nature of problems in such
analysis is extremely varied. In this category, experiments are conducted to record
life time data and these are used for analysing the life phenomenon of various
systems (human made or biological system) in terms of reliability or survival
function, increasing or decreasing hazard rate, mean time to survival etc. In other
words, the recorded life time data are used to draw inferences on the reliability
characteristics of the system or subunit to see its worth in accomplishing an
intending task and therefore, we can rightly call it as "Inferential Reliability
Analysis". A vast literature on this aspect is available in Lawless (1982), Sinha
(1986), Kapoor and Lamberson (1977), Mann et al., (1974), Martz and Walheer
(1982), Harris and Soms (1983), Nandi and Aich (1974), Basu and Ebrahimi
(1991), Sharma and Krishna (1994, 1995).
Life testing experiments are costly and time consuming phenomenon and
therefore it should be recognized that the parameters characterizing reliability
characteristics in a life time distributions are bound to follow some random
variations due to environmental changes. Therefore, it is a factor which should be
considered with the experimental data for analysing the reliability characteristics
of the system. Thomas Bayes (1763) introduced Bayesian inference in his famous
37
research paper entitled, "An essay towards solving a problem in the Doctrine of
Chance". Further, for basic theory and foundations one can also refer to Jeffreys
(1961) and Savage (1962). Lindley (1965) and Box and Tiao (1973) have
popularized and given this approach an unique important place in the field of
statistics. They developed a literature based on Bayes's approach. Today a vast
literature on Bayesian analysis of life testing problems in terms of some standard
text is available. A few of them are Savage (1962), Bhattacharya (1967), Martz
and Waller (1982), Sinha (1986) and Gelman et al. (1995) presented the Bayesian
analysis of the system reliability using many prior distributions. Some, priors with
their inherent statistical properties are also given in the study by Raiffa and
Schlaifer (1961). Studies like Sharma et al. (1993, 1994, 1995) are also effort in
the same direction. Apostolakis (1990) reviewed the literature on Bayesian theory
to assess the probabilistic safety of various engineering system. But in many
practical situation it may happen that the operational experiment with the complete
system is limited, non-existent or very expensive to realize. Moreover, we often
need to predict the reliability of complete system at the early stage of designing.
In this regard, Kaplan et al. (1989) studied about the prediction of reliability of
complete system assuming that the operational experience with the complete
system is limited, non-existent or very expensive to realize by using the
information available on boxes or subunits of the system.
The study analysed the behaviour of various probability curves, which in
turn may be used to express our degree of confidence about the complete system
reliability and the way in which Bayes' theorem updates prior probability curves to
account for various evidences. Like reliability, availability is also a measure of
38
effectiveness of a system for long-term performance. Gray and Lewis (1967) and
Masters and Lewis (1987) have obtained confidence interval for steady state
availability after using failure and repair information on respective distributions.
However, this approach was not considered satisfactory and a modified approach
was discussed in later. In this modified approach confidence interval for
availability was developed by getting simultaneous intervals for MTBF and
MTTR. However, when failure and repair information are recorded over a large
interval of time, it may be reasonable to assume random variations in the
parameters of failure time and repair time distributions. These parametric random
variations might result due to environmental impact on operating conditions of the
system or component. Recent contributions in these directions are by Sharma and
Krishna (1994, 1995a,b), Sharma and Bhutani (1994a, b) and Sharma et al.
(2004).
Queuing theory is concerned with the statistical description of the behaviour
of queues with findings. For example, the probability distribution of the number
in the queue from which the mean and variance of queue length can be found. In
queuing theory, the investigators must measure the existing system to make an
objective assessment of its characteristics and must determine how changes may
be made to the system and what effect of various kinds of changes in the system's
characteristics would be there. Probability mass function (p.m.f.) obtained in a
steady state situation is the basis of analyzing various queue systems in respect of
their characteristics. Traffic intensity () defined as the ratio of the arrival rate to
service rate is an important parameter of the p.m.f. Saaty (1961), Ackoff and
39
Sasieni (1968) and Taha (1976) studied various queue characteristics which have
been defined using the parameter . D.G. Kendal (1953) introduced a useful
notation for multiple - server queuing models which describes the three
characteristics namely, arrival distribution, departure distribution and number of
parallel service channels. Later, A. Lee (1966) added the fourth and fifth
characteristics to the notation; that is, the service discipline and the maximum
number in the system. In Taha (1976), the Kendall-Lee notation is augmented by
the sixth characteristics describing the calling source. The complete notation has
thus appears in the following symbolic form:
(a / b / c) : (d / e / f)
where
a arrival (or interarrival) distribution
b departure (or service time) distribution
c number of parallel service channels in the service
d service discipline
e maximum number allowed in the system (in service + waiting)
f calling source.
The following conventional codes are usually used to replace the symbols a,
b and d.
Symbols a and b replaced by
M Poisson (Markovian) arrival or departure distribution (or equivalently
exponential inter-arrival or service time distribution).
D Deterministic inter-arrival or service times.
40
Ek Erlangian or gamma inter-arrival or service time distribution with parameter
k.
GI General independent distribution of arrivals (or inter-arrival times).
G General distribution of departures (or service times).
Symbol d:
FCFS first come, first served
LCFS last come, first served
SIRO service in random order
GD general service discipline
The symbol c is replaced by any positive number representing the number of
parallel serves. The symbol e and f represent a finite or infinite number in the
system and calling a source, respectively.
To illustrate, the use of this notation, consider (M/M/c):(FCFS/N/). This
