Date of Submission


Date of Award


Institute Name (Publisher)

Indian Statistical Institute

Document Type

Doctoral Thesis

Degree Name

Doctor of Philosophy

Subject Name



Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)


Seth, Aravind

Abstract (Summary of the Work)

Present day technology has been characterized by development of complex systems or equipments containing a large number of subaystems and com- ponents. Reliability, as a buman attribute, has been praised for a very long time. For technical systems, however, the reliability concept has not been applied for more than about 50 years. Reliability is the concern of all scien- tists and engineers engaged in developing a system, from design, through the manufacturing, to its ultimate use. Reliability technology has a potentially wide range of application areas like safety or risk analysis, environmental protection, quality, optimization, maintenance, engineering design, etc.For a highly complex system, formal optimisation of system reliability may not be possible. In such cases, in attempting to achieve high reliability. a basic problem facing the system analyst and reliability engineers is that of evaluating the relative importance of the various components comprising the system. Measuring the relative importance of components may permit the analyst to determine which components merit the most additional research and development efforts to improve overall system effectiveness. A number of different importance measures have been defined to quantify the relative importance of components of a system and provide component ranking in order of importance. These measures can be classified as atructural impor- tance measures and rekability importance measures. Structural importance measures require only the knowledge of the atructure function of the system. Whereas reliability importance measures require the additional informationabout component reliabilities. Structural measures of importance are more suitable during system design and development phases.It is an undesirable fact that the reliability of a series system is low and, an the other hand, the parallel system has high reliability but tends to be very expensive. In last two decades a new system, a consecutive-k-out-of-n:F system, has caught the attention of many engineers and researchers because of its high reliability and low cost.The present work deals with problems of reliability analysis and comp0- nent importance measures of a consecutive-k-out-of-n:F system. In reliability theory, a consecutive-k-out-of-n:F system has been studied since 1980. It was first introduced by Kontoleon [37]. It consists of n linearly ordered and in- terconnected components. The system fails if and only if it has at least k consecutive failed components. If components be arranged on a circle we then have a circular consecutive-k-out-of-n:F system. Such systems find ap- plications in telecommunication systems and pipeline networks [21], design of integrated circuits (11], vacuum systems in accelerators (35), compater networks (32], spacecraft relay stations [20], etc. A consecutive-k-out-of-n:F system is always more reliable than a conventional k-out-of-n:F system, in which the system fails if and only if at least k components fail. Since the family of minimal cut sets of the former is a subset of the family of minimal cut sets of the latter. However for k =1 both the systems reduce to a series system and for k =n reduce to a parallel system.In this dissertation, the entire work has been divided into five chapters.Chapter 1 covers the preliminaries needed for understanding the work done. A brief history of the development of reliability is traced and concepts and notations used in the subsequent chapters are defined and described. It also describes different measures of component importance. Chapters 2, 3, 4 and 5 mainly present the work of the author.Chapter 2, starts with a review of the literature on a consecutive-k-out- of-n:F Rystem. We then study the path seta of a consecutive-k-out-of-n:F system and its applications.In Section 2, we present a new and direct formula for determining the number of path sets with known size in a consecutive-k-out-of-n:F system.Section 3, considers reliability function of the system with L.i.d. comp- nents and examines other results in the literature.In Section 4, we study the stractural matriz and its applications. Seth and Ramamurthy (61] introduced the concept of structural matrix and presented a unified approach for determining different structural importance measuren. We give a combinatorial expression for the elements of the structural matrix and apply this to a consecutive-k-out-of-n:F system.


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