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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- NOTATION 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: Philip.J.Boland@ucd.ie). 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: fjsamaniego@ucdavis.edu). 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: (1) We note that by pivoting on the component in any coherent system (see for example Barlow & Proschan [2]), we can also write (3) 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 tems. obtains 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 (6) 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 (10) Proof: We assume that and for all . Now In the case where , one has that . In particular it is easy to show that (8) and 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 THRESHOLD VALUES FOR IMPROVING CIRCULAR 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, 1997. [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 Trans. Reliabil., vol. R-30, pp. 87–89, 1981. [7] C. J. Chang, L. R. Cui, and F. K. Hwang, “New comparisons in birnbaum importance for the consecutive-k-out-of-n systems,” Probab. Eng. In- form. Sc., vol. 13, no. 2, pp. 187–192, 1999. [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 of consecutive k -out-of-n : F and related systems,” IEEE Trans. Relia- bility, vol. R-44, no. 1, pp. 120–127, 1995. [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, 1982. [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, 1993. Corollary. 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. ACKNOWLEDGMENT 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.