Vol.:(0123456789) 1 3 Life Cycle Reliability and Safety Engineering https://doi.org/10.1007/s41872-018-0039-7 ORIGINAL RESEARCH Reliability estimation of s-out-of-k system for non‑identical stress–strength components Azeem Ali 1  · Shama Khaliq 2  · Zeeshan Ali 2  · Sanku Dey 3 Received: 14 December 2017 / Accepted: 13 February 2018 © Society for Reliability and Safety (SRESA) 2018 Abstract This paper takes into account the estimation of system reliability of a multi-component stress–strength model of a s-out- of-k system, where both strength and stress are non-identical components and follow Weibull and Burr-III distributions, respectively. The reliability of such a system is obtained by the methods of maximum likelihood and Bayesian approach. We consider Bayes estimation under squared error loss function using conjugate priors for the parameters involved in the models. We propose Markov Chain Monte Carlo (MCMC) techniques to generate samples from the posterior distributions and in turn computing the Bayes estimator of the system reliability. Simulation study shows that Bayes estimator works better as compared to maximum likelihood estimator. Keywords Weibull distribution · Laplace transformation · Maximum likelihood method · Bayesian estimation 1 Introduction The term, stress–strength, in the context of reliability, was introduced by Church and Harris (1970). Since then, a lot of work has been done in this context by several authors to discuss single-component stress–strength model in numerous lifetime distributions. The stress–strength model, in its simplest form, defnes the reliability of a component as the probability that the strength of the unit (X) is greater than the stress (Y) imposed on it, that is, R = P(X > Y ). The estimation of stress–strength reliability is very common in the statistical literature. The reader is referred to Kotz et al. (2003) for other applications and motivations for the study of the stress–strength reliabil- ity. Bhattacharya and Johnson (1974) were frst to discuss the reliability of a multi-component stress–strength model. They studied the situation where a system, consisting of k com- ponents, functions when at least s(1 ≤ s ≤ k) of the compo- nents survive with a common random stress. This situation corresponds to the s-out-of-k:G system. Draper and Guttman (1978) considered stress–strength models with diferent dis- tributions to estimate system reliability in which the compo- nent strengths were assumed to be i.i.d. random variables. Ebrahimi (1982) considered a series system consisting of p components where in the system was subjected to q diferent s-independent stresses. Pandey and Borhan (1992) studied a system having independent, but non-identical distribution for strength subjected to the common stress; they assumed that component strength is divided into two parts and both follow Weibull distributions but with diferent parametric values. Paul and Uddin (1997) considered that the strength has distributed non-identically subjected to common stress for estimating reliability of s-out-of-k system; they divided the component strength into two parts each having exponential distribution with diferent values of the parameter. In recent past, using classical method of estimation to estimate the reliability in multi-component stress–strength for the log-logistic, inverse Rayleigh and Burr-Type XII distributions were considered by Rao and Kantam (2010) and Rao et al. (2013, 2015), respec- tively; however, recently, Kizilaslan and Nadar (2015), Dey * Sanku Dey sankud66@gmail.com Azeem Ali syedazeemali.qau@hotmail.com Shama Khaliq shamakhaliq1@gmail.com Zeeshan Ali 901zeeshas@gmail.com 1 National College of Business Administration and Economics, Lahore, Pakistan 2 Department of Statistics, Government College University Lahore, Lahore, Pakistan 3 Department of Statistics, St. Anthony’s College, Shillong, Meghalaya, India