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
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 .
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 , and Zuo  derived a recurrence equation
system reliability. to compute the reliability of a linear system in time:
Con/k/n:F consecutive-k-out-of-n:F (system)
with boundary condition for .
The approach of Gries & Levin  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
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 , and extensively studied
in –. Hwang & Wright  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 .
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:
email@example.com). , 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.  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 , 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  states that Case 2) .
the independent elements of are the elements in its last row,
for any .
Theorem 1 : For
for from Eq.
Proof: Clearly, and
for . from Eq.
Case 1) .
Case 2) . The theorem proposed by Du & Hwang  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-
, 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
6 IEEE TRANSACTIONS ON RELIABILITY, VOL. 53, NO. 1, MARCH 2004
TABLE I REFERENCES
EXPERIMENTAL RESULTS FOR n = 2000 AND p = 0:02
 J. M. Kontoleon, “Reliability determination of a r-successive-out-of-n:F
system,” IEEE Trans. Rel., vol. R-29, p. 437, Dec. 1980.
 F. K. Hwang, “Fast solutions for consecutive-k-out-of- n: F system,”
IEEE Trans. Rel., vol. R-31, pp. 447–448, Dec. 1982.
 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.
 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,
 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.
 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.
 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  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.
 M. J. Zuo, “Reliability of linear & circular consecutively-connected sys-
tems,” IEEE Trans. Rel., vol. R-42, pp. 484–487, Sept. 1993.
 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.
 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.
 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 , 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 . computing systems.