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# CHAPTER - 1 INTRODUCTION AND BASIC CONCEPTS

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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

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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

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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

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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

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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

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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) = 1- P [T < t] = 1 - F (t) ...(1)

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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 t
Lim R(t) = 1 and LimR(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.

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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 0 0
h(t) dt = log [1 F(t)] log R(t) - - = - ��

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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
t2 1 2 t1
R(t , t ) = exp h(t)d(t) ⌈ ⌉ -�� �� �� �� �� ⌊ ⌋ Let us denote
t 0
h(t) dt �� by H(t), represents the cumulative hazard function. From (2) and (3), it follows that
t 0
f (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.

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10
0
MTSF = E(T) = t. f (t).dt
��
�� But,
d d f(t) = F(t) = R(t) dt dt -
Hence,
0
MTSF = t.dR(t)
��
- ��
0 0
= 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 i
th
unit/component. At time t the system reliability is R (t) = P (T > t) and the reliability of the i
th
unit/component is
(t) i
R
= P [Ti > t] The system configurations generally considered are as follows:

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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 i i=1 i=1
P [T > t] = R (t) = �� ��
...(6) If hi (t) is the instantaneous failure rate of the i
th
unit and h (t) is the instantaneous failure rate. Then, we have,
n i i=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

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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 i i=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

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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 c i=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 C3 IN
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

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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 0
F (t ) P[T t ] = �� Let m be the number of items failed before time t0. Then
( )(t )o
m ~ 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) 0
L x x ...x |F(t ) �� ��

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15
n o n n m i 0 i 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)!
- =
�� �� = -
��

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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 18
th
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)

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17 From (10) and (11), we get p (�� / y) = p(y/��) p(��) 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

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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 n i 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 n
p(x , x , ..., x ) = H (x , x , ..., x ; ). d
��
�� �� ��
...(16) Therefore
1 2 n 1 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

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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 Ω. 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

�� �� = �Ȧ� �� �� ��
On using the posterior distribution, (1-��) 100% Bayes confidence interval (P1, P2) for �� may be obtained from
P 2 1 2 n P 1
( |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),

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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 Ω. 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

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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

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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 ˆ�� . The loss function denoted by ˆL ( , )
�� �� is defined to be real valued function satisfying:
(i) ˆL ( , ) 0
�� �� ��
for all possible estimates ˆ�� and all �� �� ��

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23 (ii) ˆL ( , )= 0
�� ��
for ˆ�� = ��. We now consider the following loss functions: 1.7.1 Quadratic Loss Function A function defined as
2
ˆ 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
2
ˆ 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 ˆ�� 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)

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24 introduced the following convex loss function known as Linex (linear - exponential) loss function:
a
L ( ) = 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,
a
L ( ) = 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
a
L( ) = be c b; a, c 0, b > O

∆ - ∆ - �� where ˆ = �� ∆ -1 �� 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

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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
p e
L ( ) p log ( ) 1 ⌈ ⌉ �� �� �� - �� - ⌊ ⌋
...(27) where ˆ = �� �� ��

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26 And who minimum occurs at ˆ�� = �� when a positive error (ˆ�� - ��) 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
2 2 e e
ˆ L ( ) 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
[ ]
e
L ( ) = 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

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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|
1 f(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.

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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 1
x f (x; , k) = e x ; x, , k > 0 k
-�� -
�� �� �� �� ...(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 > 0 R(t) - �� ...(35) where I (k, ��t) is an incomplete gamma function defined as
t k 1 0
1 I(k, t) = e d k
�� - -��
�� �� �� �� ��
...(36)

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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 k
k f (x; ,k) = e x ; x, k, > 0
- �� -1
�� �� ��
...(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 1
k h(t) = t
-
��
...(39)

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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 1 i th 2
0 : E occurrence on i trial X 1 : E occurrence on i trial ⎧ �� = ⎨ ��⎩
...(40) Also, let P (Xi = 0) = �� and P (Xi = 1) = (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
x
P(X x) (1 ). ; x = 0, 1, 2 ... = = -�� �� ...(41)

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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
t
R(t) = �� ...(42) and the hazard rate becomes as f(t, ) h(t) = = (1 R(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 2
ˆF 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

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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 (��) ˆ �� 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 0
f(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 (1-��) ˆ �� ��1 (u,v) i.e.
1 x u 1 v 1 0
1 f (x, u, v) = (1 ) (1 ) d B(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

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33 u
[r]
= u (u+1) ... (u+r-1) (iii) Paission (��) ˆ �� 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) ˆp 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 1
(��,��
2 2
�� ︵��,�� ︶ This compound normal distribution is also normal with parameters �� and
2 2 1 2
( ) �� + �� . (vi) Hypergeometric (n, x, N) ˆx 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,

 Page 34
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.

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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

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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

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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

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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

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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.

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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 denotes Poisson arrival (exponential inter-arrival), Poisson departure (exponential service time), c parallel servers, "first come, first served" discipline, maximum allowable number N in the system, and infinite calling source. In all such analysis, �� is assumed to be constant. However, over a long period of time the assumption about constant �� seems to be restrictive. With the advancement in science and technology over a period, the parameters involved in queue characteristics can not be considered as constant. Here, it should be recognized that the investigator has considerable a-priori knowledge about the variations in these parameters. On the repeated analysis of various queue systems, we can have a strong base for collecting prior information showing variations in ��.

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41 Following the concept, Muddapur (1972) and Armero (1985) presented a Bayesian analysis of some queue characteristics. The primary aim of these studies has been updated the prior knowledge about the parameters using experimental data. The studies in Ackoff and Sasieni (1968) and Taha (1976) have not considered estimation and testing of hypothesis aspects of queue characteristics. Sharma and Kumar (1999) studied the statistical inferences on various important technological performances measures for a (M/M/1) (�� /FIFO) queue system model. Of late, Maurya (2004), has confined his considerable attention to analyze a more generalized queue (M/G/ ��) (�� /GD) regarding statistical inferences on its useful characteristics. The arrival and servicing patterns in the system are greatly influenced by a number of factors which can not be controlled or assessed in advance and therefore, the sample information may be used to draw valid inferences about queue characteristics. Here, it should be recognized that the investigator has a priori knowledge about the variations in these parameters. Now, the investigator needs to combine this a-priori knowledge with the operational data. On the queue's system, obviously the objective can be met with the Bayesian analysis of various queue characteristics. Studies like Apostoolakis (1990), Kaplan et al. (1989) include the conceptual framework and methodology for such analysis. Some more studies (Martz and Waller, 1982; Sharma and Bhutani,1992, 1994; Krishna and Sharma, 1995; Sharma et al., 2004) include the classical and Bayesian analysis of steady state, point-wise and interval availability of the system.

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42 Reviewing all the above studies, the investigator has been able to highlight some vital problems of very practical nature in Engineering Reliability and Queuing Theory. 1.12 THESIS AT A GLANCE The present thesis includes six chapters. The contents, developments and findings in different chapters are discussed below. The Chapter I contains a brief account of introduction and development of reliability and queuing theory. The important statistical techniques and concepts like Bayesian inference, concept of prior distributions, life time models or - life time distributions and compound distributions have been discussed in a concise form. This thesis also include work on queue systems. A brief summary of the queue system is also included in this chapter. In the end, we have important matter on review of literature and thesis at a glance providing a brief outline of the results in the present research work. For time consuming life testing experiments, it seems unrealistic to treat the parameters involve in the life time distribution as constant throughout. Thus, the parameters in the life time distribution are treated as a random variable. Following this concept, the chapter-2 of the thesis deals with the development of statistical methodology useful in the analysis of the robust character of various static system configurations with Geometric life time of components. The chapter-3 of the present thesis deals with the reliability analysis of different types of system configurations such as series, parallel, m - out of n and non-series parallel complex system with Raileigh life time of component. In this chapter an easier alternative method for the construction of the structure function

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43 of different types of systems configurations are suggested. Further, evaluation as well as estimation of the different reliability characteristics have been carried out. The chapter-4 deals with the analysis of the robust character of queue system in a power supply problems when traffic intensity is treated as a random variable with beta distribution of second kind. Here, it is assumed that the random variable X follows binomial distribution with parameter 1 ⎛ ⎞ �� �� �� + �� ⎝ ⎠ and the prior belief about �� is assumed to be beta distribution of the second kind. Then the compound distribution of X, comes out to be Polya-Eggenberger distribution. The analysis depends upon the information available on units of system. Now for developing the updated compound distribution/predictive basic distribution, the posterior distribution of �� is developed for given data. The predictive basic distribution is also comes out to be Polya-Eggenberger distribution. The study reveals that the value of the estimates with the predictive distribution is uniformly higher as compared to the estimates obtained by using compound distribution. It is also noted that estimates tend to be more precise and consistent in the case of predictive basic distribution as that compared with the simple compound distribution. Chapter -5 in its first section presents the Bayesian analysis of various queue characteristics in a power supply system model. The analysis depends upon the operational data on the queue system. The arrival and service time distributions for system are taken to be exponential. Prior belief about arrival and service rate of the system have been employed in the analysis. The posterior distribution of traffic intensity is essentially a powerful tool to analyze the system characteristics in the Bayesian framework. Also for this purpose, the Squared Error Loss Function

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44 (SELF) and Linex Loss Function (LLF) have been used in the analysis. The second section of this chapter deals with the development of methodology to study the effect of random variations in the parameter of arrival and service time distributions on various queue characters in power supply system. The sixth chapter of the present thesis considered the (M/M/1) (��/FCSF) queue system model. Since over a long period of time, the assumption about constant arrival and service rate seems to be restrictive. To overcome this situation, the parameter involved in arrival and service time distributions are treated as a random variable. To study the robust character of various queue characteristics of a (M/M/1) (��/FCSF) queue system, this chapter deals with the developments of methodology for updating the basic arrival and service time distributions in respect of their prior variations. These updated distributions have been used to study the robust character of various queue characteristics of the (M/M/1) : (��/FCSF) queue system.
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