On Computing Signatures of k-out-of-n Systems Consisting of Modules

Transcription

1 On Computing Signatures of k-out-of-n Systems Consisting of Modules Gaofeng Da Lvyu Xia Taizhong Hu Department of Statistics Finance, School of Management University of Science Technology of China Hefei, Anhui People s Republic of China dagf@ustc.edu.cn thu@ustc.edu.cn January, 2012 Revised September, 2012 Supported by the NNSF of China Nos , ,

2 Abstract In this paper, we study how to compute the signature of a k-out-of-n coherent system consisting of n modules. Formulas for computing the signature the minimal signature of this kind of systems based on those of their modules are derived. Examples are presented to demonstrate the applications of our formulas. The main results obtained in this paper generalize some related ones in recent literature. Mathematics Subject Classifications 2000: 60K10; 90B25 Keywords: Signature; Minimal signature; Module; k-out-of-n system; Reliability

3 1 Introduction The concept of system signature, introduced by Samaniego 1985 for coherent systems with independent identically distributed i.i.d. components, has become a very useful tool in the analysis of system reliability. A monograph for the signature its applications in reliability was written by Samaniego The signature of a coherent system whose n components have i.i.d. lifetimes X 1,..., X n with a common absolutely continuous distribution function, is defined as an n-dimensional probability vector with ith element s i = PT = X i:n, where T denotes the lifetime of the system X 1:n,..., X n:n are the order statistics of the rom sample X 1,..., X n. By means of the signature, the system s reliability function can be represented as PT > t = n s i PX i:n > t, t. 1.1 i=1 In terms of 1.1, the signature was used to evaluate the system reliability or to compare performance of different system structures so on; see, for example, Kochar et al. 1999, Navarro et al. 2005, 2007, 2010, Navarro Eryilmaz 2007, Navarro Rychlik 2007, related references therein. The concept of signature 1.1 have been generalized to non-i.i.d. cases such as systems with exchangeable components Navarro et al., 2008, independent heterogeneous components Navarro et al., 2011 arbitrarily dependent components Marichal Mathonet, 2011; Marichal et al., Analogously to the signature-based representation 1.1, Navarro et al proved that the reliability function of a coherent system with exchangeable components can be alternatively written as PT > t = n α i PX 1:i > t, t. i=1 The vector of coefficients α = α 1,..., α n, which depends only on the system structure, is called minimal signature. Although some of the elements of α may be negative, it is noteworthy that the vector satisfies the constraint n i=1 α i = 1. For details of the minimal signature its applications in theory of reliability, one may refer to Navarro Rychlik 2010, Eryilmaz 2010, Navarro et al. 2011, related references therein. It is known that the computation of signatures is not simple especially when a system has a large number of components. Triantafyllou Koutras 2008 provided formulas that facilitate the evaluation of the signature of a reliability structure by a generating function approach, while Eryilmaz 2010 developed some interesting expressions for the computation of minimal maximal signatures of a coherent system consisting of exchangeable components. Usually, the 1

4 signature of a system with n i.i.d. components can be computed by making use of the ratio s i = a i /n!, where a i is the number of orderings of failure times of all components for which ith failure causes the system failure, i = 1,..., n. It seems that in this expression we need to identify every ordering of failure times of all components in order to determine the signature vector of the system. Another useful expression for s i introduced by Bol 2001 is given by s i = a n i+1 n a n i n, i = 1,..., n, 1.2 where a i n = r in, i = 1,..., n, n i r i n is the number of path sets of size i of the system, a i n represents the proportion of component subsets of size i which are path sets for the system. From 1.2, it follows that r i n = n n s j, i = 1,..., n. 1.3 i j=n i+1 The expression in 1.2 is based on counting the number of path sets of systems. Recently, Da et al considered the computation of the signature of a coherent system consisting of two subsystems modules based on signatures of its modules, derived two basic formulas for computing the signatures of series parallel structures with two modules respectively, which facilitate the computation of the signatures of some systems with a large number of components. Eryilmaz 2012 reconsidered the above problem exploited expression 1.2 to derive formulas with alternative forms obtained a similar formula for computing the minimal signature of the overall system as well. Gertsbakh et al calculated the signature of a system with an arbitrary organizing subsystem identical modules, they presented signatures of the series parallel of two arbitrary subsystems, independently of Da et al Eryilmaz As an extension of series parallel systems, k-out-of-n systems are well-known models have been studied quite extensively in reliability theory, life testing, statistical inference many other areas. A k-out-of-n system works if only if at least k of the n components work hence its lifetime is actually n k + 1th order statistic of lifetimes of the n components. Clearly, k = n k = 1 correspond to the series parallel systems, respectively. One may refer to Kuo Zuo 2003 for a comprehensive discussion on k-out-of-n systems. In the present paper, we consider the computation of the signature of a k-out-of-n coherent system consisting of n modules. Let C, φ denote a coherent system with the index set of 2

5 components C the structure function φ. Consider a modular decomposition of the system with the set of disjoint modules {C 1, χ 1,..., C n, χ n } the organizing structure ψ, where C i χ i are respectively the index set of components the structure function of module i, i = 1,..., n. One may refer to Barlow Proschan 1981, Chapter 1 for definitions of the notions above. Throughout this paper, we assume that ψ is the structure function of a k-out-of-n system. We will derive a formula for computing the signature of the overall system C, φ based on given signatures of its modules. As in Erylimaz 2012, we first determine the number of path sets of the overall system in terms of the numbers of path sets of its all modules, then combine this result with to obtain the desired formula. Furthermore, a similar formula for computing the minimal signature of overall system is also derived. The main results are given in Section 2. In Section 3, several examples are given to demonstrate how to use our formulas to compute the signature the minimal signature of a system. In what follows, the system C, φ is called overall system the module C i, χ i is called module i for short, i = 1,..., n. 2 Main Results In this section we derive formulas for computing the signature the minimal signature of the overall system C, φ based on given signatures minimal signatures of its modules, respectively. Suppose that module i of the overall system has m i components, i.e. C i = m i, i = 1,..., n, C = n i=1 m i. In order to obtain simple formulas for the signature of the overall system, we should observe the following facts. Let d min,i k min,i be the minimum numbers of failed components that may cause failure of module i of working components that guarantee the functioning of module i, respectively; that is, m i k min,i + 1 is the maximum number of sizes of minimal cut sets of module i k min,i is the minimum of sizes of minimal path sets of module i. Then, for the signature p i = p i,1,..., p i,mi of module i, we have p i,j = 0 for j < d min,i or j > m i k min,i + 1. Denote by d 1 min d2 min... dn min k1 min k2 min... kn min the ordered sequences of {d min,1,..., d min,n } {k min,1,..., k min,n }, respectively. Based on the definition of a k-out-of-n system, it follows that the minimum number of failed 3

6 components that may cause failure of the overall system is n k+1 d min = min d min,πi = π P n i=1 n k+1 i=1 d i min, 2.1 where π = π 1,..., π n P n, the permutation group of Λ = {1, 2,..., n}. Similarly, the minimum number of working components that assures the operation of the overall system is k min = min π P n k k min,πi = i=1 k i=1 k i min. 2.2 So the signature of the overall system C, φ has the following form s = 0,..., 0, s dmin,..., s C kmin +1, 0,..., To compute the signature of the overall system, as in Eryilmaz 2012, the key problem is to determine the number of path sets of size i of the overall system in terms of the number of path sets of the n modules. For this purpose, we first derive the formulas for determining the number of path sets cut sets of size i of the overall system with the series parallel organizing structure in the following lemma. For each j, denote by r i C j the number of path sets of size i of module j, while r i C j = mj i r i C j represents the number of sets of size i whose complements in C j are cut sets of module j. Lemma 2.1. i The number of path sets of size i of a system consisting of modules j J in the series structure is where I J i = A J i = τ = τ j, j J : τ A J i r τj C j ; 2.4 j J τ j = i, τ j k min,j, j J. ii For a system consisting of modules j J in the parallel structure, the number of sets of size i with their complements being cut sets of the system is ξ J i = j J τ B J i j J r τj C j ; 2.5 where B J i = τ = τ j, j J : τ j = i, m j τ j d min,j, j J. j J 4

7 Proof. First observe that the overall system works if only if all modules j J work. Thus, for a path set of size i of the overall system, each of modules works its i working components include τ j working ones from module j such that τ j k min,j, j J, j J τ j = i. The proof of the parallel case is similar. Now we present the formula for the number of path sets of size i of the overall system in terms of the number of path sets of the n modules. Some notations are as follows. For each J Λ, J c = Λ\J represents the complement of J in Λ. Define m J = j J m j kmin J = j J k min,j for each J Λ. Theorem 2.2. Let r i C denote the number of path sets of size i of the overall k-out-of-n system with node i being module i for each i Λ. Then r i C = M i l=k J Λ, J =l min{i,m J } u=max{k J min,i m J c } I J u ξ J ci u 2.6 for i k j=1 kj min, where I J ξ J are defined by , respectively, m M i = sup m : k j min i, m Λ. j=1 Proof. According to the definition of the k-out-of-n system, the overall system works if only if at least k of its n modules work. Then, for a path set of size i i k j=1 kj min ; otherwise there are not k working modules of the overall system, it ensures that the i working components lead to at least k at most M i modules work. Thus, we have M i r i C = Q l, 2.7 where Q l denotes the cardinality of set Q l, Q l is the set of path sets of size i of the overall l=k system which lead to exactly l modules work. Observe that Q l = J Λ, J =l Q J, 2.8 where Q J is the subset of Q l, all components in Q J working will lead to modules j J function while the rest of modules don t function. We can further divide Q J according to the number of working components in modules j J which leads to modules j J function. So Q J = min{i,m J } where Q J u is the subset of Q J satisfying that u=max{k J min,i m J c } Q J u, 2.9 5

8 1 u of i working components are from modules j J these modules work; 2 i u of i working components are from the rest of modules, j J c, these modules don t work. Thus, the number of path sets of size i of the overall system satisfying the above conditions is just the product of the number of path sets of size u of the system consisting of modules j J in the series structure the number of sets of size i u with their complements being cut sets of the system consisting of modules j J c in the parallel structure. From Lemma 2.1, it follows that Combining , we conclude 2.6. Q J u = I J u ξ J ci u Combining Theorem 2.2 with 1.2, , we can obtain the formula for computing the signature of the overall system in terms of the signatures of its modules. p i = p i,1,..., p i,mi be the signature of module i for i Λ. Corollary 2.3. The ith element of the signature vector s = s 1,..., s C of the overall system is given by s i = r C i+1c r C ic C C i 1 i for i = d min,..., C k min + 1, s i = 0 otherwise, where r i C is defined in 2.6 with for i C j j Λ. r i C j = mj i m j k=m j i+1 Next we establish a formula for computing the minimal signature of the overall system based on the minimal signatures of its modules. In order to obtain the aforementioned result, the following lemma, which states the relationship between the minimal signature the number of path sets of size i of a coherent system, will be proved useful in the sequel Eryilmaz, 2012; Samaniego, Lemma 2.4. Let α 1,..., α n be the minimal signature of a coherent system of order n p j,k r i n be the number of path sets of size i of this system. Then α i = n k min l=n i l 1 i+l n r n l n n i 6 Let

9 for i k min α i = 0 for i < k min ; r i n = i j=k min n j α j i j for i k min, where k min is the minimum number of sizes of path sets of the coherent system. By Theorem 2.2 Lemma 2.4, the formula for the computation of the minimal signature of the overall system can be obtained immediately. Corollary 2.5. Let α j = α j,1,..., α j,mj be the minimal signature of module j for each j Λ. Then the ith element of the minimal signature vector α = α 1,..., α C of the overall system is given by α i = C k min l= C i l 1 i+l C r C i C l C for i k min α i = 0 for i < k min, where k min r i C are given by , respectively, with for i C j j Λ. 3 Examples r i C j = i l=k min,j mj l α j,l i l Four examples are presented in this section to demonstrate the applications of our formulas proved in the previous section. Example 3.1. Consider the overall system C, φ consisting of three modules with respective structure functions given by χ 1 x 1, x 2, x 3 = max{min{x 1, x 2 }, min{x 2, x 3 }}, χ 2 x 4, x 5, x 6, x 7 = max{min{x 4, x 5 }, min{x 5, x 6 }, min{x 6, x 7 }} χ 3 x 8, x 9, x 10, x 11, x 12 = max{min{x 8, x 9, x 10 }, min{x 9, x 10, x 11 }, min{x 10, x 11, x 12 }}. Here, χ 1, χ 2 χ 3 are linear consecutive 2-out-of-3:G, 2-out-of-4:G 3-out-of-5:G systems, respectively. A linear [circular] consecutive k-out-of-n:g system consists of n linearly [circularly] 7

10 ordered components such that the system functions if only if at least k consecutive components function see Chiang Niu, The signatures minimal signatures of these modules are respectively given by 1 p 1 = 3, 2 3, 0, α 1 = 0, 2, 1, p 2 = 0, 12, 12, 0, α 2 = 0, 3, 2, 0 It is easy to see that p 3 = 1 5, 1 2, 3 10, 0, 0, α 3 = 0, 0, 3, 2, 0. k min,1 = 2, d min,1 = 1, k min,2 = 2, d min,2 = 2, k min,3 = 3, d min,3 = 1. Eryilmaz 2012 computed the signatures the minimal signatures for the overall systems of these three modules with series parallel organizing structures respectively. Here we compute the signature the minimal signature of the overall system consisting of the three modules with 2-out-of-3 organizing structure. From the given signatures or the minimal signatures of the modules, it follows that the vectors of the number of path sets the number of the sets with their complements being cut sets of the modules are given by rc 1 = 0, 2, 1, rc 1 = 3, 1, 0, rc 2 = 0, 3, 4, 1, rc 2 = 4, 3, 0, 0 rc 3 = 0, 0, 3, 4, 1, rc 3 = 5, 10, 7, 1, 0, respectively. By the formulas proved in the previous section, the vector of the numbers of path sets, the signature the minimal signature vectors of the overall system can be computed as s = r = 0, 0, 0, 6, 56, 207, 372, 361, 200, 65, 12, 1, 1 0, 66, 5 66, , , , , , 6, 0, 0, α = 0, 0, 0, 6, 8, 17, 30, 66, 40, 8, 0, 0. 8

11 Example 3.2. Consider the overall system C, φ consisting of 3 modules with structure functions given by χ 1 x 1 = x 1, χ 2 x 2, x 3 = max{x 2, x 3 } χ 3 x 4, x 5 = min{x 4, x 5 }, respectively. Then the signatures minimal signatures of the three modules are respectively given by p 1 = 1, α 1 = 1; p 2 = 0, 1, α 2 = 2, 1; p 3 = 1, 0, α 3 = 0, 1. By conducting a computer program based on the formulas proved in the previous section R codes can be downloaded from dagf/signature.r, we can easily obtain the vector of the number of path sets r, the signature vector s, the minimal signature vector α of the overall system with i-out-of-3 organizing structure φ, i = 1, 2. For 1-out-of-3 structure φ, we have r = 3, 10, 10, 5, 1 s = 0, 0, 0, 2 5, 3, α = 3, 2, 2, 3, 1. 5 For 2-out-of-3 structure φ, we have r = 0, 2, 8, 5, 1 1 s = 0, 5, 3 5, 1 5, 0, α = 0, 2, 2, 5, 2. Navarro Rubio 2010 computed the signatures of all the coherent systems with five components, summarized these results into a table. Here we compute the signatures the minimal signatures of the overall system consisting of the three modules with 1-out-of-3 2- out-of-3 organizing structures, which coincide with the 5-order coherent system #179 #90 in Table 2 in Navarro Rubio 2010, respectively. Example 3.3. Consider the overall system consisting of four modules with respective structure functions χ 1 x 1,..., x 8 = maxminmaxx 1, x 5, maxx 2, x 6, minmaxx 2, x 6, maxx 3, x 7, minmaxx 3, x 7, maxx 4, x 8, χ 2 x 9,..., x 13 = maxminx 9, x 10, x 11, minx 10, x 11, x 12, minx 11, x 12, x 13, minx 12, x 13, x 9, minx 13, x 9, x 10, χ 3 x 14,..., x 18 = maxminx 14, x 15, minx 16, x 17, minx 14, x 16, x 18, minx 15, x 17, x 18 χ 4 x 19,..., x 22 = maxminx 19, x 20, minx 20, x 21, minx 21, x 22. Here, χ 1 is a linear consecutive 2-out-of-4:G system with componentwise redundancy, χ 2 is a circular consecutive 3-out-of-5:G system, χ 3 is a bridge system, χ 4 is a linear consecutive 9

12 2-out-of-4:G system. The signatures minimal signatures of the four modules are respectively given by p 1 = 0, 0, 0, p 3 = 0, 3 70, 6 35, 5 14, 3 1 5, 3 5, 1 5, 0 7, 0, p 2 =, p 4 = 0, 0, 1 2, 1 2, 0, 0, 1 2, 1 2, 0 α 1 = 0, 12, 28, 27, 12, 2, 0, 0, α 2 = 0, 0, 5, 5, 1, α 3 = 0, 2, 2, 5, 2, α 4 = 0, 3, 2, 0. By conducting a computer program based on the formulas from previous section R codes can be downloaded from dagf/signature.r, we can easily obtain the vector of the number of path sets, the signature vector the minimal signature vector of the overall system consisting of the four modules with 2-out-of-4 organizing structure as follows r = 0, 0, 0, 66, 1135, 8884, 41655, , , , , s = , , , , 74583, 26334, 7315, 1540, 231, 22, , 0, 0, 0, 0, 24871, , , , , , , , , , , , , 6, 0, 0, α = 0, 0, 0, 66, 53, 313, 15, 1911, 2208, 1616, 1938, 9434, 24581, 28481, 19665, 8590, 2334, 360, 24, 0, 0, 0. Example 3.4. It is known that the signature of the bridge system is p = 0, 1/5, 3/5, 1/5, 0. Now we consider the overall system consisting of three bridge modules with 2-out-of-3 organizing structure. By using the same code in the preceding examples R codes can be downloaded from dagf/signature.r, we can easily compute the vector of the number of path sets, the signature vector the minimal signature vector of the overall system as follows r = 0, 0, 0, 12, 156, 836, 2400, 4035, 4169, 2847, 1353, 455, 105, 15, 1 4 s = 0, 0, 0, 455, , , , , , , , 4 0, 0, α = 0, 0, 0, 12, 24, 64, 84, 171, 116, 264, 108, 442, 348, 120,

13 References [1] Barlow, R.E. Proschan, F Statistical Theorey of Reliability Life Testing: Probability Models. Silver Spring, Maryl: To Begin With. [2] Bol, P.J Signatures of indirect majority systems. Journal of Applied Probability, 38, [3] Chiang, D.T. Niu, S.C Reliability of consecutive k-out-of-n:f systems. IEEE Transactions on Reliability, R30, [4] Da, G., Zheng, B. Hu, T On computing signatures of coherent systems. Journal of Multivariate Analysis, 103, [5] Eryilmaz, S Mixture representations for the reliability of consecutive-k systems. Mathematical Computer Modelling, 51, [6] Eryilmaz, S On signatures of series parallel systems consisting of modules with arbitrary structures. Technical Report, Department of Mathematics, Izmir University of Economics, Turkey. [7] Gertsbakh, I., Shpungin, Y. Spizzichino, F Signatures of coherent systems built with separate modules. Journal of Applied Probability, 48, [8] Kochar, S., Mukerjee, H. Samaniego, F.J The signature of a coherent system its application to comparison among systems. Naval Research Logistics, 46, [9] Kuo, W. Zuo, M.J Optimal Reliability Modeling, Principles Applications. New York: Wiley. [10] Marichal, J-L. Mathonet, P Extensions of system signatures to dependent lifetimes: Explicit expressions interpretations. Journal of Multivariate Analysis, 102, [11] Marichal, J-L., Mathonet, P. Waldhauser, T On signature-based expressions of system reliability. Journal of Multivariate Analysis, 102, [12] Navarro, J. Eryilmaz, S Mean residual lifetimes of consecutive k-out-of-n systems. Journal of Applied Probability, 44,

14 [13] Navarro, J. Rubio, R Computation of signatures of coherent systems with five components. Communications in Statistics Simulation Computation, 39, [14] Navarro, J., Ruiz, J.M. Soval, C.J A note on comparisons among coherent systems with dependent components using signatures. Statistics Probability Letters, 72, [15] Navarro, J., Ruiz, J.M. Soval, C.J Properties of coherent systems with dependent components. Communications in Statistics Theory Methods, 36, [16] Navarro, J. Rychlik, T Reliability expectation bounds for coherent systems with exchangeable components. Journal of Multivariate Analysis, 98, [17] Navarro, J. Rychlik, T Comparisons bounds for expected lifetimes of reliability systems. European Journal of Operational Research, 207, [18] Navarro, J., Samaniego, F.J. Balakrishnan, N The joint signature of coherent systems with shared components. Journal of Applied Probability, 47, [19] Navarro, J., Samaniego, F.J. Balakrishnan, N Signature-based representations for the reliability of systems with heterogeneous components. Journal of Applied Probability, 48, [20] Navarro, J., Samaniego, F.J., Balakrishnan, N. Bhattacharya, D On the application extension of system signatures in engineering reliability. Naval Research Logistics, 55, [21] Samaniego, F.J On closure of the IFR class under formation of coherent systems. IEEE Transactions on Reliability, R-34, [22] Samaniego, F.J System Signatures Their Applications in Engineering Reliability. New York: Springer. [23] Triantafyllou, I.S. Koutras, M.V On the signature of coherent systems applications. Probability in the Engineering Informational Sciences, 22,