Date of Submission


Date of Award


Institute Name (Publisher)

Indian Statistical Institute

Document Type

Doctoral Thesis

Degree Name

Doctor of Philosophy

Subject Name



SQC and OR Unit (Bangalore)


Ramamurthy, K. G. (SQCOR-Bangalore; ISI)

Abstract (Summary of the Work)

Reliability theory has nequired a special significance during the last few decades. It is mainly because of rapid advancement in technology which has given rise to complex and sophisticated systems. These systema suf- fer from derign flawe and wenknesses and their failures not only result in monetary lom but also pose a serious threat to human life and national se curity. Hence product reliability and safety is of paramount importance to us. Since system effectiveness can be optimised during design and develop- ment phases, it therefore calls for a detailed reliability engineering program during initial stanges. Often we find that due to lack of non-avnilabilty of reliability data, the design and reliability engineers are handicapped and no quantitative nasessment is poesible during system development phases. In such situations, it is of a great practical significance to know the relative importance of components of the system so that proper allocation of the resourcen can be made with a view to optimise system effectiveness.Different mensures have been proposed in the reliability theory to quan- tify the relative importance of componente in the system. These mensures can be classified as structural importance mensures or reliability importance messuren. Structural importance mensures require only the knowledge of the structure funetion of a system wherens reliability importance measuresrequire additional information about component reliabilities.A similar problem is encountered in other fields like game theory. In game theory, simple games are often used for modelling voting situations. The problem of quantifying the power of a player in simple games was first considered by Shapley and Shubik (47) in 1954 and they defined the Shapley-Shubik power index which was rediscovered in reliability theory by Barlow and Proechan (4) in 1975 as a measure of structural importance of components. Banzhaf (2] while analywing legal and constitutional problems defined another power index of a player in 1965 which was rediscovered by Birnbaum (8] in 1969 as a measure of structural importance of components.In reliability theory, a consecutive-k-out-of-n:F system has been studied since 1980. It consists of n linearly ordered and interconnected components. The system fails if and only if it has at least k consecutive failed compo- nents. This eystem finds applications in telecommunication, pipeline net- work, design of integrated circuits etc. Grifith and Govindarajulu (26] first considered the problem of calculating Birnbaums mensure of reliability importance of componente in a consecutive-k-out-of-n:F system. Papas tavridis (39] also studied this problem and incorrectly asserted that for the i.i.d. case, the most important components are located in the middle of the sequence of componente. We give a counterexample to show that his result is not correct. It can be shown that in a consecutive-2-out-of-6:F system component 2 ha more Birnbaum reliability importance than component 3. This provided the main background and motivation for studying the com-ponent importance in a consecutive-k-out-of-n:F system and consequently resulted in this dissertation. It is divided into five chapters.Chapter 1 covers the preliminarien needed for understanding the work done. Section 2 includes coherent systems, dual structures and related results. Section 3 provides a brief introduction to the simple games and highlights its eimilarities with semi-coherent structures.


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