Document Sample

IEEE TRANSACTIONS ON RELIABILITY, VOL. 53, NO. 1, MARCH 2004 3 An O(k2 log(n)) Algorithm for Computing the Reliability of Consecutive-k-out-of-n: F Systems Min-Sheng Lin Abstract—This study presents an O(k2 log(n)) algorithm for time, and is therefore more efficient & much computing the reliability of a linear as well as a circular consecu- simpler than that of Hwang & Wright [11]. tive-k-out-of-n: F system. The proposed algorithm is more efficient and much simpler than the O(k3 log(n k)) algorithm of Hwang & Wright. II. RELIABILITY OF A LINEAR SYSTEM Index Terms—Algorithm, consecutive-k-out-of-n:F system, Du & Hwang [7], and Zuo [9] derived a recurrence equation system reliability. to compute the reliability of a linear system in time: ACRONYMS (1) Con/k/n:F consecutive-k-out-of-n:F (system) with boundary condition for . NOTATION The approach of Gries & Levin [12] can be used to accel- n number of components in the system erate the computation of (1), which yields an efficient algorithm, k minimum number of consecutive failed components based on a matrix representation, to compute the order- Fi- to cause system failing bonacci numbers using the recurrence equation, p probability that a component works . Let be a matrix, probability that a component fails ; reliability of a linear Con/k/n:F system reliability of a circular Con/k/n:F system a matrix ; I. INTRODUCTION This matrix has the interesting property that, A CONSECUTIVE -out-of- (Con/k/n:F) system is an ordered sequence of components which fail if and only if at least consecutive components in the system fail. Two cases . . . . . . exist: linear or circular, according to whether the components are arranged on a straight line or on a circle. [from Eq. (1)] This system can represent telecommunications, oil pipelines, computer ring networks, and other kinds of systems. Such sys- Applying the above equation times yields, tems were introduced by Kontoleon [1], and extensively studied in [2]–[11]. Hwang & Wright [11] derived an time algorithm for computing the reliability of a Con/k/n:F . . . . . . system with i.i.d. components. In the model of such a system, the components are assumed to be statistically independent, and each component has the same probability of working, and probability of failing. This study takes the same approach to obtain the reliability of Con/k/n:F systems for both . . (2) . linear & circular systems. The proposed algorithm requires Manuscript received April 26, 2002; revised October 11, 2002. Responsible Thus, computing requires only that is com- Editor: K. Chien. puted. Clearly, requires matrix multiplications. The author is with the Department of Electrical Engineering, National However, a significantly faster method is available to calculate Taipei University of Technology, Taipei 10626, Taiwan, R.O.C. (e-mail: mslin@ee.ntut.edu.tw). , which only requires matrix multiplications. Digital Object Identifier 10.1109/TR.2004.823845 The method is as follows. 0018-9529/04$20.00 © 2004 IEEE 4 IEEE TRANSACTIONS ON RELIABILITY, VOL. 53, NO. 1, MARCH 2004 Fast-Matrix-Power Algorithm III. RELIABILITY OF A CIRCULAR SYSTEM Input: Several recursive algorithms have been proposed to compute Output: the reliability of circular systems in terms of the reliability of begin linear systems. Derman et al. [3] showed that: is a identity matrix for to do if the th bit of the binary representation of is 1 then ; end for (4) return This section proposes a new recurrence equation which in- end Algorithm volves only the reliability of circular systems. Let for , yielding the following theorem. can easily be computed in , because the Theorem 2: product of two matrices in the above algorithm can be per- formed in time by conventional matrix multiplications. In (5) fact, according to Gries & Levin [12], rather than computing elements in the product matrix of two matrices, only in- dependent elements need to be appropriately computed, so that with the boundary condition for . the other elements in the resulting product matrix can be derived Proof: from these independent elements. Without loss of generality, assume that is the resulting product matrix, (or Case 1) . Let for . ), in the above algorithm. Theorem 1 [12] states that Case 2) . the independent elements of are the elements in its last row, for any . Theorem 1 [12]: For for from Eq. for (3) Proof: Clearly, and for . from Eq. Case 1) . because from Eq. because Case 2) . The theorem proposed by Du & Hwang [7] is similar to (5), but it does not hold for the case of . Notably, (5) has the same form as (1). Consequently, because can be calculated by the same approach as for linear systems, as described in Section II. Using the boundary conditions for cir- cular systems, , instead of those in (2); this approach yields, by the definition of because . . . . . . Now, can be obtained in time by calculating the elements of the last row of using conventional matrix mul- Therefore, the same approach can be followed to obtain the tiplication and then filling in the rest of using (3). Accord- reliability of a linear as well as a circular Con/k/n:F system in ingly, can be computed in time. time. LIN: AN ALGORITHM FOR COMPUTING THE RELIABILITY OF CONSECUTIVE-k-OUT-OF-n: F SYSTEMS 5 from Eq. from Eq. 6 IEEE TRANSACTIONS ON RELIABILITY, VOL. 53, NO. 1, MARCH 2004 TABLE I REFERENCES EXPERIMENTAL RESULTS FOR n = 2000 AND p = 0:02 [1] J. M. Kontoleon, “Reliability determination of a r-successive-out-of-n:F system,” IEEE Trans. Rel., vol. R-29, p. 437, Dec. 1980. [2] F. K. Hwang, “Fast solutions for consecutive-k-out-of- n: F system,” IEEE Trans. Rel., vol. R-31, pp. 447–448, Dec. 1982. [3] C. Derman, G. J. Liberman, and S. Ross, “On the consecu- tive-k-out-of-n: F system,” IEEE Trans. Rel., vol. R-31, pp. 57–63, Apr. 1982. [4] R. C. Bollinger, “An algorithm for direct computation in consecu- tive-k-out-of-n:F systems,” IEEE Trans. Rel., vol. R-35, pp. 611–612, Dec. 1986. [5] A. N. Philippou, “Distributions and fibonacci polynomials of order k, longest runs, and reliability of consecutive-k-out-of-n: F systems,” Fi- bonacci Numbers and Their Applications, pp. 203–227, 1986. [6] I. Antonopoulou and S. Papastavridis, “Fast recursive algorithm to eval- uate the reliability of a circular consecutive-k-out-of-n: F system,” IEEE Trans. Rel., vol. R-36, pp. 83–84, Apr. 1987. [7] D. Z. Du and F. K. Hwang, “A direct algorithm for computing reliability of a consecutive-k-cycle,” IEEE Trans. Rel., vol. R-37, pp. 70–72, Jan. TABLE II 1988. EXPERIMENTAL RESULTS FOR k = 50 AND p = 0:2 [8] W. Kuo, W. Zhang, and M. Zuo, “A consecutive-k-out-of-n:G system: The mirror image of a consecutive-k-out-of-n:F system,” IEEE Trans. Rel., vol. R-39, pp. 224–253, June 1990. [9] M. J. Zuo, “Reliability of linear & circular consecutively-connected sys- tems,” IEEE Trans. Rel., vol. R-42, pp. 484–487, Sept. 1993. [10] M. T. Chao, J. C. Fu, and M. V. Koutrces, “Survey of reliability studies of consecutive k-out-of-n:F & related systems,” IEEE Trans. Rel., vol. R-44, pp. 120–127, Jan. 1995. [11] F. K. Hwang and P. E. Wright, “An O(k log(n=k)) algorithm for the consecutive-k-out-of-n:F system,” IEEE Trans. Rel., vol. R-44, pp. 128–131, Mar. 1995. [12] D. Gries and G. Levin, “Computing fibonacci numbers (and similarly defined functions) in log time,” Information Processing Lett., vol. 11, no. 2, pp. 68–69, 1980. IV. EXPERIMENTAL RESULTS Tables I and II compare the running time (in seconds) be- tween the algorithm of Hwang & Wright [11], and that pro- posed in this paper. The tables also show the values of & Min-Sheng Lin received his M.S. in 1991 and Ph.D. in 1994 in Computer . Both algorithms were implemented using Perl program- Science & Information Engineering from National Chiao Tung University ming language and run on an Intel Pentium-III/1000 PC under a (HsinChu, Taiwan). He is currently an Associate Professor at National Taipei Windows/XP operating system. The proposed algorithm clearly University of Technology (Taipei, Taiwan). His research interests include the analysis of network reliability and performance evaluation of distributed outperforms that of Hwang & Wright [11]. computing systems.

DOCUMENT INFO

Shared By:

Tags:
Index Terms—Asymptotic variance, confidence interval, cumulative
exposure model, exponential distribution, Fisher information, maximum likelihood, optimum test plan, Censored data, competing risks, exponential distribution, maximum likelihood, missing cause.

Stats:

views: | 33 |

posted: | 9/23/2012 |

language: | English |

pages: | 4 |

Description:
High quality scientific paper

OTHER DOCS BY MichaelABarron1234

How are you planning on using Docstoc?
BUSINESS
PERSONAL

By registering with docstoc.com you agree to our
privacy policy and
terms of service, and to receive content and offer notifications.

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.