This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON RELIABILITY 1 Reliability and Reliability Importance of Weighted-r -Within-Consecutive-k -out- of-n : F System Kirtee K. Kamalja and Kalpesh P. Amrutkar Abstract—In this paper, we consider a weighted-r-within- consecutive-k-out-of-n : F system. The weighted system has in general n components, each one having a positive integer weight w i ,i =1, 2,...,n. The weighted-r-within-consecutive-k-out- of-n : F system fails if and only if the total weight of failed components among k consecutive components is at least r. We introduce a binomial-type weighted scan statistic and study the reliability, the Birnbaum, improvement potential importance and Bayesian reliability importance of the system taken into considera- tion. We develop an explicit closed-form formula for the evaluation of reliability and reliability importance measures of a weighted- r-within-consecutive-k-out-of-n : F system and demonstrate the results numerically. We present a study showing the effectiveness of the method in terms of CPU time. Index Terms—Binomial-type weighted scan statistic, Birn- baum reliability importance, improvement potential importance, weighted-r-within-consecutive-k-out-of-n : F system. NOMENCLATURE Acronyms and Abbreviations BT Binary trial. WMBT (MBT) Weighted Markov BT (Markov BT). B-importance Birnbaum importance. pgf (pmf) Probability generating function (probability mass function). i.i.d. Independently and identically distributed. C(k,n : F ) Consecutive-k-out-of-n : F system. C m (k,n : F ) m-consecutive-k-out-of-n : F system. C(r, k, n : F ) r-within-consecutive-k-out-of-n : F system. C w (k,n : F ) Weighted-consecutive-k-out-of-n : F system. C w m (k,n : F ) Weighted-m-consecutive-k-out-of-n : F system. C w (r, k, n : F ) Weighted-r-within-consecutive-k-out-of- n : F system. Manuscript received October 7, 2017; revised January 4, 2018; accepted April 5, 2018. The work of K. P. Amrutkar was supported by the Council of Scientific and Industrial Research, New Delhi, India, through award of Senior Research Fellowship (F. No. 09/728 (0033)/2014-EMR-I). Associate Editor: A. Romanovsky. (Corresponding author: Kirtee K. Kamalja.) The authors are with the Department of Statistics, School of Mathemati- cal Sciences, North Maharashtra University, Jalgaon 425001, India (e-mail:, kirteekamalja@gmail.com; amrutkarkp@gmail.com). Digital Object Identifier 10.1109/TR.2018.2826065 Assumptions The arrangement of components in a system is linear. The components and system states are binary. The component reliabilities are Markov-dependent (MD). Each component of a system is assigned a specific weight. Notations n Number of components in a system. X i State of the ith component of the system (X i also represents an outcome of BT). X i =1 if component i is functioning; X i =0 if it is failed. X (X 1 ,X 2 ,...,X n ), binary state random vec- tor (also represent a vector of n WMBT). w i Weight of the ith component/ith WMBT. w (w 1 ,w 2 ,...,w n ), vector of weights of n components in a system. w = ∑ n i =1 w i . Total weight of all components of the system. ϕ(X ) Binary structure function of a system: ϕ(X ) = 1(0) for working (failed) system. p 11 (p 00 ) Markov dependent reliability (unreliability) of a component given that its preceding com- ponent is working (failed). p 01 (p 10 ) Markov dependent reliability (unreliability) of a component given that its preceding com- ponent is failed (working). k Length of a window of consecutive compo- nents (k ≤ n). r The minimum total weight of failed compo- nents in a window of k consecutive compo- nents, which cause system failure. R(S) Reliability of the system S. I u B (S) B-importance of the uth component of the system S. I u IP (S) Improvement potential measure of the uth component of system S. I u Bay (S) Bayesian reliability importance of the uth component of the system S. M n,k,r The number of occurrences of overlapping scanning windows of length k containing at least r successes in n-BT. M w n,k,r Number of occurrences of overlapping scan- ning windows of length k containing failures with a total weight at least r in n-WMBTs. 0018-9529 © 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications standards/publications/rights/index.html for more information.