Throughout this work, conditional failure rates for discrete positive integer-valued random variables and some of their applications are considered in some detail. Conditional failure rates are of fundamental importance in the study of lifetime distributions and many of their properties. All the notions introduced and the results derived here can be used in reliability theory, operations research, inventory theory, biometry, etc. Chapter 1 begins with the concept of conditional failure rate of a discrete random variable. Then, it is shown how to obtain explicit expressions for probability densities and survival distributions in terms of this notion. Next, extensions of the univariate results are discussed for bivariate discrete random vectors. Finally, some multivariate concepts and results are outlined. One of the fundamental applications of conditional failure rates is the mathematical representation of ageing. In Chapter 2, several univariate notions of ageing are given for discrete random variables. Such notions constitute the starting point for the classification and study of lifetime distributions that have significant importance in reliability theory, biometry, and several other areas. In Chapter 3, three important ordering relations, and a chain of implications among them, are discussed; the likelihood ratio ordering, the failure rate ordering, and the stochastic ordering. These orderings are useful in applied probability, stochastic processes, statistics, etc. In particular, they are an essential tool in the study and analysis of systems with dependent components, specially when the components are associated. No attempt is made, however, to consider specific applications of these orderings here. Finally, Chapter 4 contains an application of conditional failure rates in the analysis of repairable systems. A random mechanism of repair of failed units, called imperfect repair, is introduced and some simplified models are considered in some extent. These models can be used in the analysis and design of maintenance policies.

The finite mixed Weibull distribution is an appropriate distribution in modeling the lifetime of the units having more than one possible failure cause. Due to the lack of a systematic statistical procedure of fitting the distribution to a data set, it has not been widely used in lifetime data analyses. Many areas on this subject have been studied in this research. The following are the findings and contributions. Through a change of variable, 5 parameters in a two Weibull mixture can be reduced to 3. A parameter'vector (p₁, η, β) defines a family of two-Weibull mixtures which have common characteristics. Numerous probability plots are investigated on Weibull probability paper (WPP). For a given p₁ the η-β plane is partitioned into seven regions which are labeled by A through F and S. The Region S represents the two Weibull mixtures whose cdf curves are very close to a straight line. The Regions A through F represent six typical shapes of the cdf curves on WPP, respectively. The two-Weibull mixtures in one region have similar characteristics. Three important features of the two-Weibull mixture with well separated subpopulations are proved. Two existing methods for the graphical estimation of the parameters are discussed, and one is recommended over the other. The EM algorithm is successfully applied to solve the MLE for mixed Weibull distributions when m, the number of subpopulations in a mixture is known. The algorithms for complete, censored, grouped and suspended samples with non-postmortem and postmortem failures are developed accordingly. The developed algorithms are powerful, efficient and they are insensitive to the initial guesses. Extensive Monte Carlo simulations are performed. The distributions of the MLE of the parameters and of the reliability of a two Weibull mixture are studied. The MLEs of the parameters are sensitive to the degree of separation of the two subpopulation pdfs, but the MLE of the reliability is not. The generalized likelihood ratio (GLR) test is used to determine m. Under H₀: m=1 and H₁: m=m₁>1, ζ, the GLR is independent of the parameters in the distribution of H₀. The distributions of ζ or -21n(ζ) with n=50, 100 and 150 are obtained through Monte Carlo simulations. Compared with the chi-square distribution, they fall in between x²(4) and x²(6), and they are very close to x²(5). A FORTRAN computer program is developed to conduct simulation of the GLR test for 1 ≤ m₀ < m₁ ≤ 5.

We consider the impact that various shocks and stresses have on the reliability of networks with radial topology, such as an electrical power grid. We incorporate the effects of aging, geographical risk, and local dependence between components into a model of overall system reliability. We also simulate how the system fares under extreme weather events, such as hurricanes. Our model gives a flexible and general understanding of how outside forces affect network reliability and can be adapted to a range of specific uses. We run a simulation using this model which yields realistic results.

Wireless Sensor Network consists of tens or thousands of sensor nodes scattered in a physical space and one or more Base stations or Sinks. Even thought developed for military applications now they find a wide variety of civilian applications also. Some of the applications are Target tracking, Animal monitoring, Vehicle monitoring. The need (or lack thereof) for reliability in a sensor network is firmly dependent upon the specific application the sensor network is used for. Some applications like re-tasking or reprogramming sensor nodes [upgrading software or algorithms, adding codes, scripts etc] over -the-air requires assured delivery of high-priority events to sinks. We believe that as the number of sensor network applications grows, there will be a need to build more powerful general-purpose hardware and software environments capable of reprogramming or retasking sensors to. / Wireless sensor network is a special form of wireless networks dedicated to surveillance and monitoring applications Reliability in wireless sensor network is application specific. The specific form of reliability might change from application to application. Our idea is to generate reliability based transport protocol that is customizable to meet the needs of emerging reliable data applications in sensor networks and is also adaptive when the nodes are mobile. In our approach, clusters are formed for minimizing energy dissipation. The nodes maintain a neighbor list to forward data and any changes in the local topology can trigger updates to a node’s neighbor list. If a node notices that its neighbor list has changed, it can spontaneously re-advertise all of its data thus providing reliable transport in mobility conditions also. Our approach has five phases-setup, relaying, relay initiated error recovery, selective status reporting and node supervising. Our simulation results prove that the proposed approach can outperform existing related techniques and is highly responsive to the various error and mobility conditions experienced in sensor networks.

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