43 Dr. Reena Grover International Journal of Computer & Mathematical Sciences IJCMS ISSN 2347 – 8527 Volume 5, Issue 1 January 2016 Review of Reliability Theory, Analytical Techniques, and Basic Statistics Dr. Reena Grover Dept. of Mathematics SRM University, NCR Campus. Abstract: This report provides an introduction to the various probabilistic methods developed roughly for performing reliability or probabilistic uncertainty analysis on complex systems. The report centres on the relatively new and certainly less well known across the engineering community, analytical techniques. Discussion of the analytical methods has been broken into two reports. While a bit dated, methods described in the later portions of this report still dominate the literature and provide a necessary technical foundation for more current research. The flow of this report roughly follows the historical development of the various methods so each new technique builds on the discussion of strengths and weaknesses of previous techniques. To facilitate the understanding of the various methods discussed, a simple 2- dimensional problem is used throughout the report. The problem is used for discussion purposes only; conclusions regarding the applicability and efficiency of particular methods are based on secondary analyses. This document should be considered a ‘living document’ in the sense that as new methods or variations of existing methods are developed, the document and references will be updated to reflect the current state of the literature as much as possible. For those scientists and engineers already familiar with these methods, the discussion will at times become rather obvious. However, the goal of this effort is to provide a common basis for future discussions and as such, will hopefully be useful to those more intimate with probabilistic analysis and design techniques. There are clearly alternative methods of dealing with uncertainty (e.g. fuzzy set theory, possibility theory), but this discussion will be limited to those methods based on analytical methods. Keywords: Element reliability, system reliability, block diagram, system failure rate, system MTTF, redundancy. Introduction: Introduction to reliability contains rich blend of basic concepts and practical problems from the real world. “Reliability of a system is called its capacity for failure free operation for a definite period of time under the given operating conditions and for minimum time lost for repair and preventive maintenance”. Today reliability has become a key world in most of the industries. Products carry reliability specification such as M.T.T.F., failure rates etc. In most of the countries seminars and conferences on reliability are regular feature. The complexity of electronic system is increasing rapidly and the need of reliability in such complex electronic systems is very obvious. The principal of how to design for sustained performance (reliability) and the problem for how to design for the rapid diagnosis and removal of faults have not changed. However the growth of digital systems, software problems and complexity in general has spawned a large number of new techniques. These attempts to achieve, economically, the reliability formerly achieved by much simpler systems. 1.1 Discrete random variables: Distribution function i xj x PX x px , is the probability that X is less than or equal to the fixe value x . Associated with random variable X are two important characteristics of its distribution – the mean (or expected value) and the variance. The expected value of X is defined to be x EX xp x (1) 2 2 2 x x Var X E X EX xpx xp x (2)