Microelectron.Reliab., Vol. 34, No. I, pp. 179-181, 1994. 0026-2714/9456.00 + .00 Printed in Great Britain. © 1993Pergamon Press Ltd TECHNICAL NOTE TOLERANCE LIMITS FOR THE RELIABILITY OF A k-OUT-OF-m SYSTEM K. K. SHARMAand HAREKRISHNA Department of Statistics, Meerut University, Meerut, India (Receivedfor publication 19 June 1992) Al~traet--The reliability characteristics of various system configurations have been studied in detail, with assumptions about the lifetime distribution involved. The present study deals with the development of non-parametric tolerance limits for the reliability of a k-out-of-m system. The limits are useful in economically optimising the number of redundant components in the system. 1. INTRODUCTION The reliability level of a ~ystem is determined during the design process, and subsequent production, as- sembly and delivery of the system will not improve upon this inherent reliability level. The design process also dictates the system configuration; the configur- ation chosen influences the reliability level as well as the cost of achieving this level. Thus, a preliminary reliability analysis as well as many other design factors should be considered during the design phase when changes are most easily and economically made. In the analysis of the reliability characteristics of a system, one needs an assumption about a parametric model for the lifetime distribution of the system or of its components. Many authors [1] have studied this aspect in detail. However, in many real situations, one cannot think of an example of a parametric model. In such cases one can only develop non-parametric tech- niques to analyse the reliability characteristics in order to achieve the maximum reliability possible with reasonable confidence. In a recent study [2] tolerance limits were developed for system reliability in the cases of series, parallel, series-parallel and parallel-series system configurations. The present study develops non-parametric toler- ance limits for system reliability in the case of a k-out-of-m system. With cost constraints in view, these tolerance limits may be analysed to obtain the optimum number of redundant components to be installed in the system so that the pre-assigned re- liability level is achieved with the desired confidence. 2. k-OUT-OF-m SYSTEM CONFIGURATION --A DEFINITION A k-out-of-m system consists of m components such that the system operates as long as at least k ofits components operate. Thus m -k components in such a system are re- dundant and give longevity to the system, i.e. re- liability can be increased by introducing a greater number of redundant components. The system reliability Rs(t) of a k-out-of-m system having independent and identical components can be expressed as a polynomial of the component re- liability R(t) as follows: Rs(t)=~(7) (1) Particular cases (a) For k = m, a k-out-of-m system reduces to a series system which fails as soon as any one of its components fails. (b) Similarly for k = 1, the system reduces to a parallel system which functions as long as any one of its components operates. 3. STATISTICAL BACKGROUND For the development of the desired tolerance limits we make use of the following well-known definition and results. (i) Uniform distribution U(0, 1): a random variable U is said to follow a uniform distribution U(0, 1), written as U ~ U(0, 1), if its probability density function (p.d.f.) is 1, 0<u<l f(u) = 0, otherwise. (ii) If U ~ U(0, 1), then (1 - U)~ U(0, I). (iii) Probability integral transformation: let X be a continuous random variable with the cumulative distribution function F(. ), then U = F(X) is a uniform variate U(0, 1). 179