Stochastic Ordering Results for Consecutive k-out-of-n : F Systems by chrondiaz


									IEEE TRANSACTIONS ON RELIABILITY, VOL. 53, NO. 1, MARCH 2004                                                                                               7

             Stochastic Ordering Results for Consecutive
                        k-out-of-n : F Systems
                                                   Philip J. Boland and Francisco J. Samaniego

    Abstract—A linear (circular) consecutive -out-of- : F system                     systems. In the intervening years, a huge literature of research has
is a system of n linearly (circularly) ordered components which fails                been generated on consecutive -out-of- systems, addressing
if and only if at least k consecutive components fail. We use recur-                 problems of reliability computation (e.g., Derman et al. [10],
sive relationships on the reliability of such systems with indepen-
dent identically distributed components to show that for any fixed                   Du & Hwang [12]), dependency between components (Boland
  , the lifetime of a (linear or circular) consecutive -out-of- : F                  et al. [3], Gera [13]), component importance (e.g., Zuo [18],
system is stochastically decreasing in . This result also holds for                  Chang et al. [7]), optimal arrangement of components (Du et al.
linear systems when the components are independent and not neces-                    [11], Tong [17]), lifetime distributions (Aki and Hirano [1]), and
sarily identically distributed, but not in general for circular systems.             various generalizations (Koutras and Papastavridis [14]). For re-
  Index Terms—Linear and circular consecutive -out-of-                       sys-    views of the topic, one would do well to consult Chang, Fu, &
tems, stochastic order, system reliability.                                          Koutras [9], and Chang, Cui, & Hwang [8].
                                                                                        The concept of stochastic order is a useful tool in comparing
                                                                                     system lifetimes. There are many types of stochastic relation-
                                                                                     ships which are commonly used in comparisons, although here
                       vector of state components of system                          we shall restrict our considerations to what is sometimes re-
                                                                                     ferred to as the usual stochastic order. Shaked & Shanthikumar
                       state of component (i.e.,                        if com-      ([16]) provides a comprehensive treatment of stochastic order
                       ponent is failed (working))                                   relationships. If       &      are random variables with cumulative
                       path (cut) sets of system                                     distribution functions          & , respectively; then we say that
                                                                                         is greater than       in the (usual) stochastic order (           )
                                                                                     if                   for all , where                 . Simply put, for
                       reliability of the linear or circularconsecutive              any ,      is more likely to exceed (survive) than           is.
                         -out-of-       system                                          A linear (circular) consecutive -out-of-                system is a
                                       when        are i.i.d.,        and            system of linearly (circularly) ordered components which fails
                       system is linear                                              if and only if at least consecutive components fail. For given
                                       when        are i.i.d.,        and               and , one should carefully note the difference between the
                       system is circular                                            linear and circular consecutive -out-of-               systems. In the
                       lifetime of linear consecutive -out-of-                       circular case, the components & 1 are in some sense adjacent
                       system                                                        (or consecutive), while in the linear case they are not – in fact they
                       lifetime of circular consecutive -out-of-                     are as distant as they could be. Consider for example consecutive
                          system                                                     2-out-of-4: systems. If the pair of components                    fails
                       linear consecutive -out-of-             system                for any         , 2, 3, then both the linear and circular consecutive
                       circular consecutive -out-of-             system              2-out-of-4: systems fail. However, when only the components
                       longest sequence of failures in                 in-           {4,1} fail, then the circular consecutive 2-out-of-4: system fails
                       volving              component                                but the corresponding linear one does not. In general for a given
                                                                                        & , the circular consecutive -out-of-             system has more
                                                                                     failure possibilities than the corresponding linear consecutive
                                                                                       -out-of-        system, and consequently lower reliability.
                             I. INTRODUCTION                                            One would expect that for a fixed , the larger the number
                                                                                     of components in a consecutive -out-of-                   system, the
C    HIANG & NIU [6] introduced the concept of the consecu-
     tive -out-of-      system in 1981, indicating its relevance
to modeling, for example, oil pipelines, and telecommunication
                                                                                     less reliable the system is. In this note, we use the concepts
                                                                                     of path/cut sets & recursive relationships on the reliability of
                                                                                       -out-of-         systems to show that for any fixed , when the
                                                                                     components are independent and identically distributed, the life-
   Manuscript received October 9, 2001; revised July 19, 2002. Responsible Ed-       time of such a system stochastically decreases with (Boland &
itor: D. Dietrich. This work was supported in part by grant DAAD 19-99-1-1082
from the U.S. Army Research Office.                                                  Samaniego [4] used the concept of the signature of a system to
   P. J. Boland is with the Department of Statistics, National University of Ire-    prove this stochastic ordering relationship for linear consecutive
land, Dublin 4, Ireland (e-mail:                            2-out-of-          systems). This result also holds for linear sys-
   F. J. Samaniego is with the Department of Statistics, University of California,
Davis, CA 95616 USA (e-mail:                               tems when the components are independent and not necessarily
   Digital Object Identifier 10.1109/TR.2004.824830                                  identically distributed, but not in general for circular systems.
                                                                 0018-9529/04$20.00 © 2004 IEEE
8                                                                               IEEE TRANSACTIONS ON RELIABILITY, VOL. 53, NO. 1, MARCH 2004

             II. RELIABILITIES FOR CONSECUTIVE                          This can also be seen by noting that
                    -OUT-OF-      SYSTEMS                               (because a cut set for a linear consecutive -out-of-
   Consider a consecutive -out-of-               system where the       system may also be viewed as a cut set for a linear consecutive
components are independent with respective reliabilities                  -out-of-           system) and then using (1). This then estab-
(            ). We generally assume that                  (for when     lishes that if one adds any additional independent component
        we have a series system and when          we have a parallel    (at either end) to a linear consecutive -out-of-         system,
system, both of whose properties are well known). Computing             then the resulting linear consecutive -out-of-            system
the reliability of such a system (linear or circular) has received      has a stochastically smaller lifetime.
considerable attention, and many of the methods for doing so               Recursively, continuing to pivot on the            component
depend on recursive relationships for                    . A set of     and sequentially others, one sees that
components is a path (cut) set for a system if functioning (failure)
of the components in the set implies that the system functions
(fails). The reliability                of any coherent system of
order may be expressed as follows by making use of the path
( ) and/or cut ( ) sets of the system:


  We note that by pivoting on the component in any coherent
system (see for example Barlow & Proschan [2]), we can also

We will employ this basic technique to derive various recursive
expressions for the reliability of consecutive -out-of-          sys-
                                                                        In particular for a linear consecutive 2-out-of-         system, one
   We let          (respectively            ) represent the path sets
for a linear (circular) consecutive -out-of-             system, and
similarly          (respectively            ) the corresponding col-                                                                       (4)
lections of cut sets. Because any path set for a (linear or circular)
consecutive -out-of-             system is also a path set for a con-   and for the situation when all components have the same relia-
secutive         -out-of-          system (                       and   bility ,
                       ), it follows that

                                                        and                                                                                (5)

                                                                        The latter two equations provide vehicles for computing the reli-
In particular one has that both                           and
                                                                        ability polynomial of a linear consecutive -out-of-            system
                                                                        in terms of systems of lower order, thereby offering the possi-
   Let us now consider the case where is fixed but varies.
                                                                        bility of establishing properties of these systems by mathemat-
For a linear consecutive -out-of-     system, it is clear that
                                                                        ical induction.
for any        ,
                                                                           Next we consider the circular consecutive -out-of-
                                                                        system with reliability                        . In this situation, as
                                                                        previously remarked, we regard components & 1 as being ad-
                                                                        jacent, so there is no first or last component. Because a circular
(We use the convention that for any      ,           ).                 consecutive -out-of-             system places more requirements
Pivoting on the last component in a linear consecutive                  on its components (than the corresponding linear consecutive
 -out-of-          system, it follows that                                -out-of-         system) to function (for example
                                                                                ), it follows that in general
                                                                                             . A natural question is under what circum-
                                                                        stances does                                                   ? This
                                                                        clearly holds when            or , because in either of these cases
                                                                                           . On the other hand if               , then the set
                                                                                                                          (and hence
                                                                 (2)    is a proper subset of           ). Therefore if                 for all
BOLAND AND SAMANIEGO: STOCHASTIC ORDERING RESULTS FOR CONSECUTIVE -OUT-OF-             SYSTEMS                                            9

              , then                                           . It      For the circular consecutive -out-of-          system with non-
is also clear that for any           ,                                homogeneous components, the situation is more subtle. Con-
                                                                      sider placing an additional             component with reliability
                                                                             between the      & 1st components in a circular consecu-
                                                                      tive -out-of-         system to form a new circular consecutive
                                                                        -out-of-           system. Intuitively one should expect that if
                                                                      the new component is sufficiently reliable (relative to the relia-
                                                                      bilities of the other components), then it should strengthen the
   The following theorem gives an explicit and useful expression
                                                                      resulting system. An extreme situation would be where the new
for the reliability of circular consecutive -out-of-
                                                                      component is completely reliable (                 ), in which case
systems in terms of linear consecutive -out-of-         systems
                                                                      the new system is essentially reduced to the (stronger) linear
of smaller order , when all components are equally reliable.
                                                                      consecutive -out-of-            system of the original compo-
   Theorem 1: For a circular consecutive -out-of-
                                                                      nents. On the other hand, if the new component is not (rela-
system where all components have reliability and               ,
                                                                      tively speaking) sufficiently reliable, it should result in a weaker
one has that
                                                                      system. The following shows that there is a threshold value
                                                                      for        determining when                           is greater than
                                                              (7)                                    in the usual stochastic order.
                                                                         Theorem          3: Let                                        and
     Proof: Let us use the random variable           to denote the                                  be circular consecutive -out-of-
longest sequence of failures (in the circular sense) in a consecu-    systems in which the first components of each have the same
tive -out-of-          system involving the            component      probabilities              of functioning. Then there exists a
(but at the same time noting that we could equally concentrate        threshold value
on any other component in a similar way). If the circular con-
secutive -out-of-           system is functioning, then we must
have that                   (note that                corresponds
to the event where the               component is functioning).       such that         is superior to           , that is
Clearly                           and
for           . Using the law of total probability, we obtain
                                                                      if and only if


                                                                           Proof: We assume that                  and               for all
                                                                                     . Now
  In the case where              , one has that
                                . In particular it is easy to show
                                                                      Therefore solving for      (the reliability of the new compo-
                                                              (9)     nent), the newly formed system
                                                                      will be less reliable than                   if and only if
More generally we have that
   Theorem 2: Let                    be the lifetime of a consec-
utive linear (circular) -out-of-        system with independent
(independent and identically distributed) component lifetimes.                                                                        (11)
Then                        and                        .
      Proof: The result in the linear case is immediate from (2),       Example 1: Let us consider in more detail the circular con-
while the circular case can be seen by using (2) together with        secutive 2-out-of-4 system, and determine the expression for
(7), as follows:                                                      the threshold parameter                    in this case. This pa-
                                                                      rameter has the property that
                                                                                      . Now if        , then for       to function, one
                                                                      must have                , and hence
                                                                                     . By pivoting on component 4, and then compo-
                                                                      nent 3, one obtains
10                                                                                    IEEE TRANSACTIONS ON RELIABILITY, VOL. 53, NO. 1, MARCH 2004

                             TABLE I                                                                         REFERENCES
                 CONSECUTIVE 2-OUT-OF-n : F SYSTEMS                            [1] S. Aki and K. Hirano, “Lifetime distributions of consecutive-k -out-
                                                                                   of-n : F systems,” Nonlinear Anal.-Theor., vol. 30, no. 1, pp. 555–562,
                                                                               [2] R. E. Barlow and F. Proschan, Statistical Theory of Reliability and Life
                                                                                   Testing: Probability Models Silver Spring, MD, USA, 1981.
                                                                               [3] P. J. Boland, F. Proschan, and Y. L. Tong, Linear Dependence in Consec-
                                                                                   utive k -out-of-n Systems, 1990, vol. 4, Prob. Eng. Inf. Sc., pp. 391–397.
                                                                               [4] P. J. Boland and F. Samaniego, “The signature of a coherent system
                                                                                   and its applications in reliability,” in Mathematical Reliability: An Ex-
                                                                                   pository Perspective, R. Soyer, T. Mazzuchi, and N. Singpurwalla, Eds:
                                                                                   Kluwer, 2003, vol. 67, International Series in Operations Research and
                                                                                   Management Science, pp. 1–29.
and                                                                            [5] P. J. Boland, M. Shaked, and J. G. Shanthikumar, Stochastic Ordering
                                                                                   of Order Statistics. San Diego, CA: Academic Press, 1998, vol. 16,
                                                                                   Handbook of Statistics, ch. 5, pp. 89–103.
                                                                               [6] D. Chiang and S. C. Niu, “Reliability of consecutive k-out-of-n,” IEEE
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                                                                               [7] C. J. Chang, L. R. Cui, and F. K. Hwang, “New comparisons in birnbaum
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                                                                               [8]       , Reliabilities of Consecutive k -out-of-n Systems. Dorrecht:
   The expression for the threshold parameter therefore takes the                  Kluwer Academic Publishers, 2000.
form                                                                           [9] M. T. Chao, J. C. Fu, and M. V. Koutras, “Survey of reliability studies
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                                                                              [10] C. Derman, G. J. Lieberman, and S. Ross, “On the consecutive
                                                                                   k -out-of-n : F system,” IEEE Trans. Reliability, vol. R-31, pp. 57–63,
                                                                              [11] D. Z. Du, F. K. Hwang, and Y. Jung, “Optimal consecutive-k -out-of-
   For example                                          , and hence we             (2k + 1) : G cycle,” J. Global Optim., vol. 19, no. 1, pp. 51–60, 2001.
can stochastically improve on               where                             [12] D. Z. Du and F. K. Hwang, “A direct algorithm for computing relia-
                     by adding a 5th component between compo-                      bilities of consecutive k -out-of-n cycles,” IEEE Trans. Reliability, vol.
                                                                                   R-37, no. 2, pp. 70–72, 1988.
nents 4 & 1, if and only if its reliability satisfies                    .    [13] A. E. Gera, “A consecutive k -out-of-n : G system with dependent el-
One may also show for example that                                                 ements – A matrix formulation and solution,” Reliab. Eng. Syst. Safe.,
        , demonstrating that                         is clearly not per-           vol. 68, no. 1, pp. 61–67, 2000.
                                                                              [14] M. Koutras and S. G. Papastavridis, “Consecutive k -out-of-n : F sys-
mutation invariant in                     .                                        tems and their generalizations,” in New Trends in System Reliability
   In the situation where           for all                 , let us write         Evaluation, K. B. Misra, Ed. Amsterdam: Elsevier, 1993, pp. 228–248.
                                . Using (8) & (9), one may readily            [15] A. Satyanarayana and M. K. Chang, “Network reliability and the fac-
                                                                                   toring theorem,” Networks, vol. 13, pp. 107–120, 1983.
establish that                              &                            .    [16] M. Shaked and J. G. Shanthikumar, Stochastic Orders and their Appli-
Table I gives these threshold points for various values of . As an                 cations. San Diego, CA: Academic Press, 1994.
example, note that                                                            [17] Y. L. Tong, “A rearrangement inequality for the longest run, with an
                                                                                   application to network reliability,” Journal of Applied Probability, vol.
if and only if            . The results in Table I suggest that, in the            22, pp. 386–393, 1985.
homogeneous case,                     for all              and arbitrary      [18] M. J. Zuo, “Reliability and component importance of a consecutive-
        . That this is actually the case is stated in the following                k -out-n system,” Microelectronics Reliab., vol. 33, no. 2, pp. 243–258,
   Corollary 1: For homogeneous circular consecutive
  -out-of-       systems, the threshold parameter                  , given
in (11) with            for                 , satisfies the inequality       Philip J. Boland received his Ph.D. (1972) from the University of Rochester.
               for all           .                                           He has worked in the Departments of Mathematics and Statistics at NUI-Dublin
      Proof: From Theorem 2 we know that in the homogeneous                  since 1972, and has served as Professor and Head of the Department of Statistics
                                                                             since 1986. He became the first President of the Irish Statistical Association
case                         , and hence from Theorem 3 it follows           in 1997/98, and was elected an Honorary Member of the Society of Actuaries
immediately that                   for any            .                      in Ireland in 1997. His main research interests are reliability theory, actuarial
                                                                             statistics, and statistical education.

   The authors are very grateful for the positive suggestions
they received from the referees and associate editor, which have             Francisco J. Samaniego received his Ph.D. in mathematics from UCLA in
added to the readability and value of the manuscript. They would             1971, and has been on the faculty at the University of California, Davis, since
                                                                             1972. His research interests include decision theory, reliability and sampling
also like to acknowledge the generosity of facilities provided by            theory. He is a Fellow of the American Statistical Association, the Institute of
Trinity College Dublin during the preparation of this work.                  Mathematical Statistics and the Royal Statistical Society.

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