Competing Causes of Failure and Reliability Tests for Weibull Lifetimes Under Type I Progressive Censoring

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					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                 .
                                                                                 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:                                                               , 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
                                                                                     Several recursive algorithms have been proposed to compute
                                                                                  the reliability of circular systems in terms of the reliability of
                                                                                  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
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
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)      .


                                                                                                                                          from Eq.

    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,
                                                                                  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.

                                                                                         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.
                                                                  [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.

  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.