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

References: 1. Chang, G.J., Cui, L.R., and Hwang, F.K. (2000). Reliabilities of Consecutive-k-Systems, Kluwer, Boston. 2. Chao, M.T., Fu, J.C., and Koutras, M.V. (1995). A survey of the reliability studies of consecutive-k, r-out-of-n: F systems and its related systems. IEEE Transaction Reliability, 44:120-127. 3. Cheng, C.H. (1994). Fuzzy consecutive-k-out-of-n:F system reliability. Microelectronics and Reliability, 34: 909-1922. 4. Chiang, D.T. and Niu, S.C. (1981). Reliability of a consecutive-k-out-of-n: F system. IEEE Transaction on Reliability, 30: 87-89. 5. Dhillon, B.S. and Rayapati, S.N. (1986). A method to evaluate reliability of three-state device networks. Microelectronics and Reliability, 26: 535-554. 6. Feller, W. (1985). An Introduction to Probability Theory and Its Applications, 4th edition, John Wiley & Sons, New York. 7. Jenney, B.W. and Sherwin, D.J. (1986). Open and short-circuit reliability of systems of identical items. IEEE Transaction on Reliability, 35: 532-538. 8. Koutras, M.V. (1997). Consecutive-k, r-out-of-n: DFM systems. Microelectronics and Reliability, 37: 597-603. 9. Kuo, W., Fu, J.C., and Lou, W.Y. (1994). Opinions on consecutive-k-out-of-n: F system. IEEE Transaction on Reliability, 43: 659-662. 10. Kuo, W. and Zuo, M. (2003). Optimal Reliability Modelling: Principles and Applications, John Wiley and Sons, New Jersey. 11. Malon, D.M. (1989). On a common error in open and short-circuit reliability computation. IEEE Transaction on Reliability, 38: 275-276. 12. Page, L.B. and Perry, J.E. (1987). Reliability of networks of three-state devices. Microelectronics and Reliability, 22: 175-178. 13. Page, L.B. and Perry, J.E. (1988). Optimal “series-parallel” networks of three-state devices. IEEE Transaction on Reliability, 37: 388-394. 14. Page, L.B. and Perry, J.E. (1989). A note on three-state