International Journal of Operations Research International Journal of Operations Research Vol. 4, No. 2, 110-117 (2007)
Reliability Analysis of Consecutive-k, r-Out-Of-n: DFM System using GERT Manju Agarwal* and Pooja Mohan
Department of Operational Research, University of Delhi, Delhi-110007, India Received September 2006; Revised January 2007; Accepted March 2007
Abstract¾Koutras (1997) analyzed reliability of a consecutive-k, r-out-of-n: DFM system consisting of n components linearly arranged which fails if and only if at least k consecutive components are failed-open or at least r consecutive components are failed-short. In this paper Graphical Evaluation and Review Technique (GERT) has been applied to model and analyze the reliability of the above system. One of the strengths of the GERT network is the graphical representation, which is intuitive and easy to understand. The components are assumed to be i.i.d. Furthermore, numerical computations are conducted using Software Mathematica to determine the actual computation times, which are almost negligible. Keywords¾Consecutive-k, r-out-of-n: DFM system, Reliability, GERT
ACRONYMS CDFM(k,r,n): Consecutive-k, r-out-of-n: DFM system GERT: Graphical Evaluation and Review Technique NOTATIONS n: number of components p: survival probability of a component q1: probability of a component in failed-open mode q2: probability of a component in failed-short mode Rk ,r ( n ) : Reliability of the system Further, the three modes of operation (working, failed-open, failed-short) of a component are supposed to be mutually exclusive and exhaustive, i.e. p + q1 + q2 = 1 1. INTRODUCTION The study of dual failure mode (DFM) or three state devices has received continuing research interest since mid-1950s (Dhillon and Rayapati (1986), Jenney and Sherwin (1986), Malon (1989), Page and Perry (1987, 1988, 1989), and Satoh et al. (1993)). Several redundant structures as well as methods of calculating system reliability have been researched in order to improve their reliability. The major areas of substantial advance are the reliability evaluation and optimal design of various redundant DFM structures. These systems have wide applicability in nuclear industry where the common terminology used is “failure to safety” and “failure to danger”; in fluid flow control networks where a defective valve could be either “stuck open” or “stuck closed”; in *
electronic/electrical engineering studies, where the modes of failure are usually labeled as “failed-open” and “failed-short” (Koutras (1997)). Many research results have been reported on reliability evaluation of consecutive-k-out-of-n systems; for example, see Chiang and Niu (1981), Kuo et al. (1994) and Chao et al. (1995). A survey of consecutive-k systems and its various generalizations can be found in Chang et al. (2000), Kuo and Zuo (2003), and Pham (2003). The consecutive-k, r-out-of-n: DFM system is an extension of well known consecutive-k-out-of-n: F system subject to dual failure mode environment. Koutras (1997) studied the reliability of CDFM(k, r, n) in which the system fails if and only if at least k consecutive components are failed-open or at least r consecutive components are failed-short with independent but not necessarily identical components providing recurrence relation. Further, upper and lower bounds are also derived, for a quick assessment of the order of magnitude of the system’s reliability. In this paper, CDFM(k, r, n) (Koutras, 1997) has been analyzed through GERT. The components are assumed to be i.i.d. It can be observed that GERT not only provides the visual picture of the system but also helps to determine the generating function for the reliability of the system in a much easier way. GERT is easier to use than minimal cut set method. In GERT one has to evaluate a W function, the generating function of the waiting time for the occurrence of the system failure, whereas in minimal cut set method one has to...
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