Afrka Statstka Vol. 101, 2015, pages 795 805. DOI: http://dx.do.org/10.16929/as/2015.795.70 Afrka Statstka ISSN 2316-090X Relablty of lnear and crcular consecutve-kout-of-n systems wth shock model Besma Bennour and Soher Belalou, Department of Mathematcs and Informatcs, Larb Ben M hd unversty, Oum El Bouagh, Algera Department of Mathematcs, Constantne 1 Unversty, Algera Receved January 31, 2015; Accepted October 14, 2015 Copyrght c 2015, Afrka Statstka. All rghts reserved Abstract. A consecutve k-out-of-n system conssts of an ordered sequence of n components, such that the system functons f and only f at least k k n consecutve components functon. The system s called lnear L or crcular C dependng on whether the components are arranged on a straght lne or form a crcle. In the frst part, we use a shock model to obtan the relablty functon of consecutve-k-out-of-n systems wth dependent and nondentcal components. In the second part, we treat some numercal examples to show the derve results and deduce the falure rate of each component and the system. Résumé. Un système k-consécutfs-parm-n est un système consttué de n composants, tel que ce système fonctonne s et seulement s au mons k k n composants consécutfs fonctonnent. Le système est dt lnéare L ou crculare C suvant la dsposton des composants en lgne ou en cercle. Dans la premère secton, en utlsant le modèle de chocs, on établt la fablté du système en queston ayant des composants dépendants et non dentques. Dans la deuxème secton, on trate des exemples numérques qu llustrent les résultats obtenus tout en dédusant le taux de panne de chaque composant et du système. Key words: Lnear and crcular consecutve-k-out-of-n system; Felablty functon; Falure rate; Shock model. AMS 2010 Mathematcs Subject Classfcaton :62N05; 68M15; 68M20; 90B25; 60K10. 1. Introducton In some envronments, the falure of the system depends not only on the tme, but also upon the number of random shocks. So many applcatons n relablty analyss can be Correspondng author Soher Belalou : s belalou@yahoo.fr Besma Bennour: besma bennour@yahoo.fr
B. Bennour, and S. Belalou, Afrka Statstka, Vol. 101, 2015, pages 795 805. Relablty of lnear and crcular consecutve-k-out-of-n systems wth shock model. 796 descrbed by shock models. Shocks may refer for example to damage caused to bologcal organs by llness or envronmental causes of damage actng on a techncal system, see e.g Hameed and Proschan 1973. Also, as an example, a press machne produces external frames n automobles or refrgerators. The machne can fal due to unexpected causes such as accdental changes n the temperature or electrcal voltage, defectve raw materals, or errors by human operators. In the lterature, the concept of models shock on systems was treated by many researchers, we can quote some of them. Marshall and Olkn 1967 consdered forms of shock to deduce the bvarate exponental dstrbuton. Where, ther man objectve s stated that two components are subjected to shocks from three dfferent ndependent sources. One shock destroys one component and the thrd shock destroys both components. Shock model s also used by Grabsk and Sarhan 1995 and Sarhan 1996 to obtan the relablty measures estmatons for seres and parallel systems wth two nonndependent and nondentcal components. They assumed that the tmes at whch the shocks occur are exponentally dstrbuted. Sarhan, A.M. and Abouammoh 2000 used the shock model to derve the relablty functon of k-out-of-n systems wth nonndependent and nondentcal components. They assumed that a system s subjected to n + m ndependent types of shocks. Lu et al. 2008 proposed a model to evaluate the relablty functon of seres and parallel systems wth degradaton and random shocks. In ther model, the system s assumed to be faled when nternal degradaton or cumulatve damage from random shocks exceed random lfe thresholds. There are other several types of the shock models whch have been consdered for the falure of a system as: extreme shock model, cumulatve shock model, run shock model, or δ-shock model. For example, δ-shock model s based on the length of the tme between successve shocks. In ths model, the system fals when the tme between two consecutve shocks falls below a fxed threshold δ. Ths shock model has been studed by L and Zhao 2007 and Erylmaz 2012. The man objectve n ths paper s to use a shock model to obtan the relablty functon of consecutve-k-out-of-n systems, whch have a wde range of applcatons; e.g: telecommuncatons, gas and ol ppelnes, transport network...etc, wth dependent and nondentcal components n the two topologes, lnear and crcular confguratons. Then to deduce the falure rate of the system and also the relablty and the falure rate of each component n the system. In the end, we treat some examples to llustrate the provded results. 2. Notatons and assumptons Notatons n: number of components n a system. k: the mnmum number of consecutve components requred to be good for the system to be good. n + mm 1: number of ndependent sources of shocks. s : the source, c : the component. S = {s 1,..., s n,..., s n+m }: the set of all sources. C: the set of sources whch destroy component.
B. Bennour, and S. Belalou, Afrka Statstka, Vol. 101, 2015, pages 795 805. Relablty of lnear and crcular consecutve-k-out-of-n systems wth shock model. 797 { n k + 1 In the lnear case J = n In the crcular case I = {c,..., c +k 1 }, = 1,..., J, the th mnmal path. S = +k 1 l= C l, = 1,..., J. { S j S f j = 1 = j l=1 S l f j = 2,..., J where: 1 1 < < l < < j J. U : the random tme of shock from s. Q t = PU t: the dstrbuton functon of U. Q t = PU > t. T = mn j C U j : the lfetme of the component, = 1, 2,..., n. A : the event {T > t}. B = +k 1 j= A j. p: denotes L lnear or C crcular. T p : the lfetme of consecutve-k-out-of-n system. R p t = PT p > t: the relablty functon of consecutve-k -out-of-n system. λ p t = R p t R pt : the falure rate of consecutve-k-out-of-n system. Assumptons 1. The T, = 1,..., n, are nonndependent and nondentcal dstrbuted. 2. The system s subjected to n + m ndependent sources of shocks. 3. s destroys c, = 1,..., n. At least one shock from the remanng m sources destroys all components, and shocks from the other sources m 1 destroy a group of components. The system s subjected to a set of varous shocks, where a shock from source occurs at a random tme U. The random varables U, = 1,..., n + m, are ndependent. 3. Relablty of Consecutve-k-out-of-n System In ths secton, we consder both lnear and crcular consecutve-k -out-of- n systems wth dependent and nondentcal components. Defnton 1. A consecutve k-out-of-n system conssts of n lnearly or crcularly arranged components, ths system works f and only f at least k consecutve components work. In other words, there s at least a mnmal path whch s workng. Defnton 2. A mnmal path vector s a set of mnmum number of components n workng state whch ensures the system s functonng. In the lnear case, we have: I1 = {c 1, c 2,..., c k 1, c k } I2 = {c 2, c 3,..., c k, c k+1 }. In k + 1 = {c n k+1, c n k+2,..., c n 1, c n }
B. Bennour, and S. Belalou, Afrka Statstka, Vol. 101, 2015, pages 795 805. Relablty of lnear and crcular consecutve-k-out-of-n systems wth shock model. 798 But n the crcular case, we have: Where: c n+ = c, = 1,..., k 1. I1 = {c 1, c 2,..., c k 1, c k } I2 = {c 2, c 3,..., c k, c k+1 }. In k + 1 = {c n k+1, c n k+2,..., c n 1, c n } In k + 2 = {c n k+2, c n k+3,..., c n, c n+1 } In k + 3 = {c n k+3, c n k+4,..., c n+1, c n+2 }. In = {c n, c n+1,..., c n+k 2, c n+k 1 } Theorem 1. Let R p t be the relablty functon of consecutve-k-out-of-n systems, and T p denotes ts lfetme. We have: J [ } R p t = { 1 j 1 Q l t 1 j=1 If p = L, then J = n k + 1. If p = C, then J = n. 1 1<...< j J l S j Proof. The lnear or crcular consecutve k-out-of-n system works f at least k consecutve components work. Then, we have: R p t = PT p > t = P J =1 {T > t, T +1 > t,..., T +k 1 > t} = P J =1 { +k 1 j= A j } = P J =1 B we apply the addton theorem for J events: J R p t = P B P B 1 B 2 where, + =1 1 1< 2< 3 J 1 1< 2 J P B 1 B 2 B 3 + + 1 J 1 P B 1 B 2 B J P B = P T > t, T +1 > t,..., T +k 1 > t = P mn U l > t, mn U l > t,..., mn U l > t l C l C +1 l C +k 1 = P U l > t l C, U l > t l C +1,..., U l > t l C +k 1 = P {U l > t, l C C +1... C +k 1 } = P { mn U l > t} l C C +k 1 = P l +k 1 j= C j Ul > t = l S Q l t = l S 1 2 Q l t 3
B. Bennour, and S. Belalou, Afrka Statstka, Vol. 101, 2015, pages 795 805. Relablty of lnear and crcular consecutve-k-out-of-n systems wth shock model. 799 Smlarly, and P B 1 B 2 = P B 1 B 2 B 3 = l S 2 Q l t 4 l S 3 Q l t 5 P B 1 B 2 B J = Q l t 6 Substtutng the equatons 3, 4, 5, and 6 nto equaton 2 we obtan: l S R p t = + = J =1 l S 1 1 1< 2< 3 J Q l t l S 3 1 1< 2 J J { 1 j 1 j=1 l S 2 Q l t Q l t + + 1 J 1 l S Q l t 1 1<...< j J [ l S j } Q l t. 7 4. Numercal examples 4.1. Lnear consecutve-k-out-of-n system In partcular t s assumed that: A1 k > 1, n, and s = 2n + 1. A2 s destroys c, = 1,..., n. A3 the sources between n and 2n + 1 destroy k consecutve components: s n+1 destroys {c 1,..., c k } = I1.... s 2n k+1 destroys {c n k+1,..., c n } = In k + 1. s 2n k+2 destroys {c n k+2,..., c n, c 1 }.... s 2n destroys {c n, c 1,..., c k 1 }. A4 s 2n+1 destroys all components. A5 the random varables U, = 1, 2,..., 2n + 1 are Webull dstrbuted wth parameters α, β respectvely: Q t = exp α t β Example 1: Lnear consecutve-2-out-of-3 system We consder a lnear consecutve-2-out-of-3 system and the assumptons A1-A5 are satsfed, we have: S = {s 1,..., s 7 }
B. Bennour, and S. Belalou, Afrka Statstka, Vol. 101, 2015, pages 795 805. Relablty of lnear and crcular consecutve-k-out-of-n systems wth shock model. 800 C 1 = {s 1, s 4, s 6, s 7 }, C 2 = {s 2, s 4, s 5, s 7 }, C 3 = {s 3, s 5, s 6, s 7 } I1 = {c 1, c 2 }, I2 = {c 2, c 3 }. S 1 = 2 j=1 C j = S \ {s 3 } = S 1 1, S 2 = 3 j=2 C j = S \ {s 1 } = S 1 2 S 2 = 2 l=1 S l = S 1 S 2 = S 1 S 2 = S, where 1 1 < 2 2. Fg. 1. the lnear consecutve-2-out-of-3 system and 7 sources Usng equaton 1, t follows that: R L t = 2 { 1 j 1 j=1 = l S 1 1 Q l t + 1 l 7 1 1<...< j 2 l S 1 2 = exp [ l S j Q l t l S Q l t l {1,3} and λ L t; the falure rate of the lnear system s gven by: λ L t = 1 l 7 } Q l t exp, l {1,3} α l β l t βl 1 α lβ l t βl 1 exp 1 l {1,3} exp α lt β l 1 1 In general, f n = k + 1, then: R L t = exp l {1,n} exp 8 λ L t = l {1,n} α l β l t βl 1 α lβ l t βl 1 exp 1 l {1,n} exp α lt β. 9 l 1 1
B. Bennour, and S. Belalou, Afrka Statstka, Vol. 101, 2015, pages 795 805. Relablty of lnear and crcular consecutve-k-out-of-n systems wth shock model. 801 Example 2: Lnear consecutve-3-out-of-5 system For ths system and the assumptons A1-A5, we have: S = {s 1,..., s 11 }. C 1 = {s 1, s 6, s 9, s 10, s 11 }, C 2 = {s 2, s 6, s 7, s 10, s 11 }, C 3 = {s 3, s 6, s 7, s 8, s 11 }, C 4 = {s 4, s 7, s 8, s 9, s 11 }, C 5 = {s 5, s 8, s 9, s 10, s 11 }. I1 = {c 1, c 2, c 3 }, I2 = {c 2, c 3, c 4 }, I3 = {c 3, c 4, c 5 }. S 1 = S \ {s 4, s 5 } = S 1 1, S 2 = S \ {s 1, s 5 } = S 1 2, S 3 = S \ {s 1, s 2 } = S 1 3. S 2 = 2 l=1 S l = S 1 S 2, where 1 1 < 2 3, then: S 2 = S 1 S 2 = S \ {s 5 } S 1 S 3 = S S 2 S 3 = S \ {s 1 } S 3 = 3 l=1 S l, where 1 1 < 2 < 3 3, then: S 3 = S. Usng equaton 1, we obtan: R L t = exp 1 l 11 l 5 + exp 2 l 11 l {1,4} exp [ exp α 2 t β2, and the the falure rate of the system s as followng: λ L t = exp + [ l {1,4} 2 l 11 1 l 11 l 5 In general, f n = k + 2, then: 1 l 11 l 5 α l β l t β l 1 l {1,4} α l β l t βl 1 exp 1 exp exp 2 l 11 α l β l t β l 1 exp α 2 t β2 + α 2 β 2 t β2 1 exp α 2 t β2 1 R L t = exp l n + exp 2 l 2n+1 l {1,n 1} exp [ exp α 2 t β2, 10
B. Bennour, and S. Belalou, Afrka Statstka, Vol. 101, 2015, pages 795 805. Relablty of lnear and crcular consecutve-k-out-of-n systems wth shock model. 802 and: λ L t = exp + l {1,n 1} [ 2 l 2n+1 l n l n α l β l t β l 1 l {1,n 1} α l β l t βl 1 exp 1 exp 2 l 2n+1 exp α l β l t β l 1 exp α 2 t β2 + α 2 β 2 t β2 1 exp α 2 t β2 1.11 Example 3: Lnear consecutve-5-out-of-8 system We consder a lnear consecutve-5-out-of-8 system and the assumptons A1-A5 are satsfed, we have: S = {s 1,..., s 17 }. C 1 = {s 1, s 9, s 13, s 14, s 15, s 16, s 17 }, C 2 = {s 2, s 9, s 10, s 14, s 15, s 16, s 17 }, C 3 = {s 3, s 9, s 10, s 11, s 15, s 16, s 17 }, C 4 = {s 4, s 9, s 10, s 11, s 12, s 16, s 17 }, C 5 = {s 5, s 9, s 10, s 11, s 12, s 13, s 17 }, C 6 = {s 6, s 10, s 11, s 12, s 13, s 14, s 17 }, C 7 = {s 7, s 11, s 12, s 13, s 14, s 15, s 17 }, C 8 = {s 8, s 12, s 13, s 14, s 15, s 16, s 17 }. I1 = {c 1,..., c 5 }, I2 = {c 2,..., c 6 }, I3 = {c 3,..., c 7 }, I4 = {c 4,..., c 8 }. S 1 = S \ {s 6, s 7, s 8 } = S 1 1, S 2 = S \ {s 1, s 7, s 8 } = S 1 2, S 3 = S \ {s 1, s 2, s 8 } = S 1 3, S 4 = S \ {s 1, s 2, s 3 } = S 1 4, S 2 = 2 l=1 S l = S 1 S 2, where 1 1 < 2 4, then: S 1 S 2 = S \ {s 7, s 8 } S 1 S 3 = S \ {s 8 } S 2 S = 1 S 4 = S S 2 S 3 = S \ {s 1, s 8 } S 2 S 4 = S \ {s 1 } S 3 S 4 = S \ {s 1, s 2 }. S 1 S 2 S 3 = S \ {s 8 } S 3 = 3 l=1 S l = S 1 S 2 S 3 = S 4 = 4 l=1 S l = S. Usng equaton 1, we have: R L t = exp 1 l 17 l 8,l 7 + exp 3 l 17 S 1 S 2 S 4 = S S 1 S 3 S 4 = S S 2 S 3 S 4 = S \ {s 1 } l {1,6} exp exp
B. Bennour, and S. Belalou, Afrka Statstka, Vol. 101, 2015, pages 795 805. Relablty of lnear and crcular consecutve-k-out-of-n systems wth shock model. 803 and so: λ L t = exp l {1,6} 1 l 17 l 7 exp + 1 l 17 l 8,l 7 α l β l t β l 1 exp + 3 l 17 3 l 17 α l β l t β l 1 exp 1. l {1,6} α l β l t β l 1 α l β l t β l 1 exp 1 exp In general, f n = k + 3, then: Rt = exp l n,l n 1 + exp 3 l 2n+1 l {1,n 2} exp exp 12 λ L t = exp l {1,n 2} l n 1 exp + l n,l n 1 α l β l t β l 1 exp + 3 l 2n+1 3 l 2n+1 l {1,n 2} α l β l t β l 1 α l β l t β l 1 exp 1 exp α l β l t β l 1 exp 1. 13 Remark 1. For all precedent cases, let R t et λ t, = 1,..., n, denote respectvely the relablty functon and the falure rate of the component, whch are gven by the followng expressons: R t = exp, λ t = α l β l t βl 1. l S l S 4.2. Crcular consecutve-k-out-of-n system Example 4: Crcular consecutve-2-out-of-3 system We consder a crcular consecutve-2-out-of-3 system and 7 ndependent sources act on the system see fg.2. S = {s 1, s 2, s 3, s 4, s 5, s 6, s 7 } C 1 = {s 1, s 4, s 6, s 7 }, C 2 = {s 2, s 4, s 5, s 7 }, C 3 = {s 3, s 5, s 6, s 7 },
B. Bennour, and S. Belalou, Afrka Statstka, Vol. 101, 2015, pages 795 805. Relablty of lnear and crcular consecutve-k-out-of-n systems wth shock model. 804 I1 = {c 1, c 2 }, I2 = {c 2, c 3 }, I3 = {c 3, c 1 }. S 1 = S \ {s 3 } = S 1 1, S 2 = S \ {s 1 } = S 1 2, S 3 = S \ {s 2 } = S 1 S 1 S 2 = S S 2 = 2 l=1 S l = S 1 S 2 = S 1 S 3 = S S 2 S 3 = S S 3 = S. 3, Fg. 2. The crcular consecutve-2-out-of-3 system and 7 sources U Webα, β, = 1;... ; 7, and usng equaton 1, t follows that: R C t = exp 1 l 7 1 l 3 then λ C t; the falure rate of the crcular system, s gven by: λ C t = 1 l 7 exp 1 2, 1 l 3 α l β l t βl 1 α lβ l t βl 1 exp 1 1 l 3 exp α lt β. l 1 2 Example 5: Crcular consecutve-3 -out-of-5 system We consder a crcular consecutve-3-out-of-5 system and 11 ndependent sources act on the system. S = {s 1,..., s 11 }. C 1 = {s 1, s 6, s 9, s 10, s 11 }, C 2 = {s 2, s 6, s 7, s 10, s 11 }, C 3 = {s 3, s 6, s 7, s 8, s 11 }, C 4 = {s 4, s 7, s 8, s 9, s 11 }, C 5 = {s 5, s 8, s 9, s 10, s 11 }. I1 = {c 1, c 2, c 3 }, I2 = {c 2, c 3, c 4 }, I3 = {c 3, c 4, c 5 }, I4 = {c 4, c 5, c 1 }, I5 = {c 5, c 1, c 2 }. S 1 = S \ {s 4, s 5 }, S 2 = S \ {s 1, s 5 }, S 3 = S \ {s 1, s 2 }, S 4 = S \ {s 2, s 3 }, S 5 = S \ {s 3, s 4 }.
B. Bennour, and S. Belalou, Afrka Statstka, Vol. 101, 2015, pages 795 805. Relablty of lnear and crcular consecutve-k-out-of-n systems wth shock model. 805 S 2 = 2 l=1 S l = S 1 S 2, where 1 1 < 2 5, then: S 1 S 2 = S \ {s 5 } S 1 S 5 = S \ {s 4 } S 2 S = 2 S 3 = S \ {s 1 } S 3 S 4 = S \ {s 2 } S 4 S 5 = S \ {s 3 } S for the other cases. S 3 = 3 l=1 S l = S, where 1 1 < 2 < 3 5. S 4 = 4 l=1 S l = S, where 1 1 <... < 4 5. S 5 = S. Usng equaton 1, we obtan: R C t = Q l t + 1 l 5 l {,+1} = 1 l 5 exp l {,+1} 1 l 11 Q l t [1 1 l 5 + exp 1 l 11 Q l t 1, [1 where Q 6 t = Q 1 t 1 l 5 exp 1. References Erylmaz, S., 2012. Generalzed δ-shock model va runs. Statstcal and Probablty Letters 82, 326-331. Grabsk, F. and Sarhan, A.M., 1995. Bayesan estmaton of two non ndependant components exponental seres system. Zagadnena Eksploatacj Maszn Zeszyt. 175-187. Hameed. A.M.S. and Proschan, F., 1973. Nonstatonary shock models, Stock Proc Appl, 1. 333-404. L, Z.H. and Zhao, P., 2007. Relablty analyss on the δ -shock model of complex systems. LEEE Trans. On Relablty. 56, 340-348. Lu, Y., Huang H.Z. and Pham, H., 2008. Relablty evaluaton of systems wth degradaton and random shocks. Proceedngs of the Relablty and Mantanablty Symposum. Las Vegas, USA: IEEE, 328-333. Marshall, A. and I.Olkn, I., 1967. A multvarate exponental dstrbuton. J Amer. statst. Assoc. 30-44. Sarhan, A.M. and Abouammoh, A.M., 2000. Relablty of k-out-of-n nonreparable systems wth nonndependent components subjected to common shocks. Mcroelectroncs Relablty. 41, 617-621. Sarhan, A.M., 1996. Bayesan estmaton n relablty theory. PH.D. dssertaton, Insttute of Mathematcs, Gdansk Unversty, Gdansk, Poland.