## On Some Important Reliability Aspects of General Coherent Systems

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##### Date

2023-11

##### Researcher

Sahoo, Tanmay

##### Supervisor

Hazra, Nil Kamal

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##### Publisher

Indian Institute of Technology Jodhpur

##### Abstract

In practice, we use di?erent kinds of systems that are structurally equivalent to
various well-established systems available in the literature, namely, ordinary coherent systems,
ordinary r-out-of-n systems, sequential r-out-of-n systems, fixed weighted coherent
systems, fixed weighted r-out-of-n systems, random weighted coherent systems, random
weighted r-out-of-n systems, etc. In this thesis, we study di?erent reliability aspects of
these systems under various scenarios.
In most real-life scenarios, the components of a system work in the same environment
and share the same load. As a consequence, there may exist two di?erent types of
dependencies between the components of a system, namely, interdependency and failure
dependency. The interdependency structure between the components of a system is usually
modeled by a copula. The family of Archimedean copulas is commonly used to serve this
purpose as it describes a wide spectrum of dependence structures. On the other hand, the
failure dependancy for a system means that the failure of one component has an e?ect on
the lifetimes of the remaining components of the system. This failure dependency is often
modeled by assuming distributional changes in the lifetimes of the remaining components,
upon each failure, of the system. The sequential order statistics (SOS) and the developed
sequential order statistics (DSOS) are two models that are commonly used to describe the
failure dependency of a system. There is a one-to-one relationship between order statistics
and the lifetimes of systems. For example, the lifetime of an ordinary r-out-of-n system
is the same as the (n ? r + 1)-th order statistic of the lifetimes of the components of the
system. Similar relationships exist for SOS and DSOS. Thus, the study of order statistics is
the same as the study of the lifetimes of systems. In this thesis, we study various ordering
and ageing properties of ordinary r-out-of-n systems formed by dependent and identically
distributed components, where the dependence structure is described by an Archimedean
copula. Further, we study the ordering properties of the developed sequential order statistics
(DSOS) with the dependence structure described by the Archimedean copula. Similar to
the DSOS model, we introduce the notion of developed generalized order statistics (DGOS)
which is an extended generalized order statistics (GOS) model formed by dependent random
variables. This model contains all existing models of ordered random variables. We study
various univariate and multivariate ordering properties of the DGOS model governed by the
Archimedean copula. Further, we consider the SOS model with non-identical components
and study several univariate and multivariate stochastic comparison results. The basic structures of many real-life systems match with random weighted
coherent systems. The performance of a random weighted coherent system is usually measured
by its total capacity. However, the major drawback of this measure is that it does
not take into account the structure of a system. To overcome this drawback, we introduce
a new structure-based performance measure, namely, the survival capacity. Based on this
measure, we define three survival mechanisms (namely, Types-I, II and III) for random
weighted coherent systems. We develop a methodology to evaluate the reliability of a random
weighted coherent system, and provide a signature-based reliability representation for
this system. Further, we study di?erent reliability importance measures for the components
of a random weighted coherent system. We study the optimal allocation strategy of active
redundancies and the optimal assembly method of random weights in a random weighted
coherent system. By developing the results for random weighted coherent systems, we generalize
many well-established results available for ordinary coherent systems and weighted
coherent systems in the literature.

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##### Citation

Sahoo, Tanmay. (2023).On Some Important Reliability Aspects of General Coherent Systems (Doctor's thesis). Indian Institute of Technology Jodhpur, Jodhpur.