RELIABILITY ANALYSIS OF M-ORDER SYSTEMS WITH NONIDENTICAL AND s-DEPENDENT UNITS

Abstract

In recent years, reliability consideration in system design has received much attention and such factors as growing complexity of systems, increasing degree of automation, and ever-increasing competetion for quality products have all combined to make higher reliability an economic necessity. In desxgning for higher reliability, in most cases, a design engineer has to resort to redundancy in addition to other techniques such as system simplification, creative design, etc. There are various forms of accomplishing redundancy which necessitates reliability comparison between alternative schemes and selection of one which best satisfies the objectives, This needs 'appropriate' mathematical models and computer programmes for efficient and meaningful numerical evaluation. A redundant system (or subsystem) having m units connected in parallel Is referred to as an m-order system. In practice, system models that are frequently encountered are of k-out-of-m type and such systems function successfully if and only if at least k, K k < m, of the m units are good and operating. Series and parallel systems are the special cases of the above model. First, systems with identical and statistically independent (s-independent) units are reviewed and a graphical technique for determining some reliability parameters of both maintained and nonmaintained systems - 11 - is presented. Recursive formulae have been developed to facilitate the graphical solution v/hich provides quick and accurate results. However, in most of the applications, particularly electrical and electronic systems, the failure probabilities are unequal and s-dependent the facts v/hich have not been treated hitherto in detail. The primary objective of the present work is to consider the above factors in detail and propose more general mathematical models for analysing the reliability characteristics of m-order systems. This includes sj/sterns with stress-dependent units as well as equipment hierarchy. Eor m-order systems with nonidentical units, two approaches, namely, logic diagram approach and event realization approach are proposed for evaluating the exact reliability and mean time to failure. An explicit formula in terms of m and k has been obtained and a summation coefficient triangle has been developed to help evaluate the coefficients of the proposed formula. Systems with onen and short circriit failures are also considered. The analysis is further extended to maintained systems with unequal failure and repair rates. Frequency and duration concepts of Hall, Ringlee and Wood are employed to determine mean uptime, mean downtime, and frequency of system failures. - iii - In analysing systems with stress-dependent units, the following cases are considered in detail: (1) Constant hazard rate with nonlinear stress-hazard model. (2) Constant hazard rate with linear stress-hazard model. (3) General hazard function with linear stress-hazard model. (4) General hazard function with nonlinear stress-hazard model. Renewal theory approach and Laplace transform methods are employed for Case 1. The familiar Sandler's dependency model when failure rate of units increases linearly with load has been disproved. Another simple method.known as Effective Hazard Rate method has been suggested for case 2. A conditional reliability approach Is proposed for analysing systems con taining units with time dependent hazard rates when they operate in the linear region of stress versus hazard rate characteristic (case 3). For case 4, the concept of mean hazard rate is employed. Two stress models, namely, power function and exponential models are considered. Derating charts v/hich will be useful to design engineers have been developed. The effect of stress-dependency of units on reliability characteristics of systems has been illustrated through examples. The models .presented here are explicit functions of mand k and therefore are more general than those of Shooman, and Yang and Lin.

For stress-dependent m-order systems with repair, models for evaluating availability, failure frequency, mean uptime, mean time to first system failure,etc., are developed. Both parallel and fractional repair facilities are considered. Optimum maintenance policies are also discussed. An important class of systems known as hierarchical (seaii-redundant) systems is discussed in detail. Tie set logic diagrams and theory of conditional probabilities are found suitable for analysing such systems. Reliability and cost models for m-order, n-level hierarchical systems are formulated. Reliability-cost characteristics and optimal regions are studied through examples. Wherever possible, suitable approximations v/hich will be useful in obtaining quick and reasonably accurate solutions for comparison of alternative design schemes are suggested. In short, the nresent work provides appropriate mathematical models which have not been available till now for exact analysis of m-order systems with nonidentical and s-dependent units. The derating charts and approximations presented in this thesis would serve as useful design tools for reliability engineers and system designers.