In this paper, we consider the reliability of k-out-of-n: Fsystems and its conditional form with exchangeable components in the stress-strength setup. We assume a random stress common to all the components in the system level. Applications of these results to illustrate the reliability for the system consisting of three components with the multivariate FGM and multivariate Marshall-Olkin distributions are also given.Publisher's Versio

Gürler, Selma

Özler, Burçin Şeyda

TWMS Journal of Applied and Engineering Mathematics

English

Isik University Academic Open Access

TWMS J. App. Eng. Math. V.2, N.1, 2012, pp. 109-115ON THE RELIABILITY OF k-OUT-OF-n: F SYSTEMS WITHEXCHANGEABLE COMPONENTS IN THESTRESS-STRENGTH SETUPB. S. OZLER 1, S. GURLER 2 §Abstract. In this paper, we consider the reliability of k-out-of-n: F systems and itsconditional form with exchangeable components in the stress-strength setup. We assumea random stress common to all the components in the system level. Applications of theseresults to illustrate the reliability for the system consisting of three components with themultivariate FGM and multivariate Marshall-Olkin distributions are also given.Keywords: Reliability, Dependence, k-out-of-n:F system, Stress-strength model.AMS Subject Classification: 62N05, 60K101. IntroductionAn important method for improving the reliability of a system is to build redundancyinto it. A common structure of redundancy is the k-out-of-n system. A k-out-of-n:Fsystem consisting of n components works if and only if (n − k + 1) of the n componentswork and fails if k or more components fail. Thus, the system breaks down at the time ofthe kth component failure. It finds wide applications in various fields such as industrialand military systems. Some examples of such systems can be found in Kuo and Zuo(2003).In the reliability theory, the stress-strength model is generally interested in the reliabilityof a component with strength X, which is under the random stress Y . When it is assumedthat X and Y are independent random variables, then the component fails if the stressexceeds the strength of the component, i.e. X < Y . There are numerous papers onthe reliability of a component which has been generally concerned with the probabilityP (X > Y ). Examples of such results and references can be found in Bhattacharyya andJohnson (1974, 1975), Greco and Ventura (2009), Hanagal (1999a), Johnson (1988), Kotz1 Ondokuz Mayıs University, Faculty of Arts and Sciences, Department of Statistics, Kurupelit,Samsun, Turkeye-mail: burcinseydaozler@windowslive.com2 Dokuz Eylül University, Department of Statistics, Faculty of Science, Tinaztepe Campus, 35160Buca, Izmir, Turkeye-mail: selma.erdogan@deu.edu.tr§ Manuscript received 19 February 2012.TWMS Journal of Applied and Engineering Mathematics Vol.2 No.1 c⃝ Işık University, Departmentof Mathematics 2012; all rights reserved.109110 TWMS J. APP. ENG. MATH. V.2, N.1, 2012et al. (2003) and additional results can be found in Devanji and Rao (2001), Ebrahimi(1982), Eryılmaz (2008, 2010), Hanagal (1999b) for the reliability of systems in this setup.In this paper, we consider the reliability of k-out-of-n: F systems with exchangeablecomponents in the stress-strength setup. We assume that the stress and strengths areindependent random variables and a random stress common to all the components. InSection 2, we present the reliability for the k-out-of-n: F system with exchangeable com-ponents in the stress-strength setup. We also give two examples for the 2-out-of-3: Fsystem using the multivariate FGM and multivariate Marshall-Olkin models under theexponential distributed stress Y . In Section 3, we evaluate the conditional reliability ofthe system for the case of dependent components.2. Reliability of the k-out-of-n: F SystemThe random vector X1, X2, . . . , Xn is exchangeable for each n, if the joint distributionP (X1 ≤ x1, . . . , Xn ≤ xn) = P (Xπ(1) ≤ x1, . . . , Xπ(n) ≤ xn),for any finite permutation π(1), . . . , π(n) of the indices {1, 2, . . . , n}. Then, the jointsurvival function is represented asFX(x1, x2, . . . xn) = P (X1 > x1, X2 > x2, . . . , Xn > xn).Let us consider a k-out-of-n:F system with n exchangeable strengths represented bya random vector X1, X2, . . . , Xn. If we assume that X1:n ≤ X2:n ≤ · · · ≤ Xn:n are theordered strengths of the components, the failure of the k-out-of-n:F system depends onthe failure of the kth component. Hence, the strength of the system is represented by thecorresponding kth order statistic Xk:n. The survival function of the system is given byF k:n(t) = P (Xk:n > t)=n∑i=n−k+1(−1)i−n+k−1(i− 1n− k)(ni)FX(t, . . . , t︸ ︷︷ ︸i), (2.1)for which FX(t, . . . , t︸ ︷︷ ︸i) = P (min(X1, . . . , Xi) > t), 1 ≤ i ≤ n, (see David and Nagaraja,2003). Navarro et al. (2006) have shown that F k:n(t) with exchangeable componentlifetimes is the generalized mixture of series and parallel systems. For some results onk-out-of-n systems taken into account the dependence among the lifetimes of components,see e.g. Bairamov and Parsi (2010), Eryılmaz (2011), Navarro and Balakrishnan (2010),Navarro et al. (2005), Navarro and Rychlik (2007).Let us consider a k-out-of-n:F system with n exchangeable strengths and the strengthof each component is subjected to a common random stress Y . Let also the random stressbe independent of the random vector of the strengths.Lemma 2.1. If X1, X2, . . . , Xn are the exchangeable strengths, the reliability of the k-out-of-n:F system under the random stress isB. S. ÖZLER, S. GÜRLER: ON THE RELIABILITY OF K-OUT-OF-N : F SYSTEMS WITH ... 111RYk:n =∫y n∑i=n−k+1(−1)i−n+k−1(i− 1n− k)(ni)FX(y, . . . , y︸ ︷︷ ︸i) dGY (y),where GY (y) is the distribution function of the stress for y > 0.Proof. From (2.1), we haveP (Xk:n > Y ) =∞∫0n∑i=n−k+1(−1)i−n+k−1(i− 1n− k) ∑1≤j1<···<ji≤nP (Xj1 > Y, . . . ,Xji > Y |Y = y )dGY (y).Then the proof follows noting that RYk:n = P (Xk:n > Y ). Remark 2.2. A special cases for k = n and k = 1 are equivalent to a parallel and seriessystems, respectively. In these cases, Lemma 2.1 reduces to the reliability of the paralleland series systems under the random stress.Example 2.3. Let X1, X2, X3 have an exchangeable FGM distribution with the joint sur-vival functionFX(x1, x2, x3) =3∏i=1FX(xi)1 + α ∑1≤i1<i2≤3(1− FX(xi1)) (1− FX(xi2))−3∏i=1(1− FX(xi)) ,where xi > 0 and α is the common dependence parameter (see, Kotz et al., 2000). Ifwe assume that X1, X2, X3 have exponential marginals with common parameter λ1 andthe stress Y is also exponential with parameter λ2, then we have for the reliability of a2-out-of-3:F system from Lemma (2.1)RY2:3 =∞∫0(3FX(y, y)− 2FX(y, y, y))dGY (y)=3λ2(1 + α)2λ1 + λ2− 2λ2(1 + 5α)3λ1 + λ2+9λ2α4λ1 + λ2− 2λ2α6λ1 + λ2,where 0 < α < 1/2, λ1, λ2 > 0.Figure 1 shows that when the λ1 (λ2) increases then the reliability of the system decreases(increases) as expected.Example 2.4. Let X1, X2, X3 have Marshall and Olkin multivariate exponential distribu-tion with the joint survival functionFX(x1, x2, x3) = exp(−3∑i=1λixi − λ0max(x1, x2, x3)),where xi > 0, λi > 0 and λ0 ≥ 0 (see, Kotz et al., 2000).The reliability of the 2-out-of-3:F system consisting of three components each have ex-ponential ( λ1) marginals with the exponential stress Y with the parameter λ2 is givenby:RY2:3 =(3λ22λ1 + λ2 + λ0− 2λ23λ1 + λ2 + λ0).112 TWMS J. APP. ENG. MATH. V.2, N.1, 2012Figure 2 shows that when the λ1 increases then the reliability of the system decreases.RYk:n is a decreasing function of λ0 for Marshall and Olkin multivariate exponential distri-bution.Figure 1. Reliability curves of the 2-out-of-3 system for FGM multivariate dis-tribution with exponential marginals (α = 0.2)Figure 2. Reliability curves of the 2-out-of-3 system for Marshall-Olkin’s mul-tivariate exponential distribution (λ0 = 0.5).3. Conditional Reliability Under the StressFor some systems consisting of n components which are subjected to a random stress,the operator may consider some maintenance or replacement procedure when all the com-ponents are working. The conditional random variable (Xk:n − Y | X1:n > Y ) representsthe remaining strength under the condition that all components are working under thestress.B. S. ÖZLER, S. GÜRLER: ON THE RELIABILITY OF K-OUT-OF-N : F SYSTEMS WITH ... 113Lemma 3.5. If X1, X2, . . . , Xn are exchangeable strengths, the conditional reliability in ak-out-of-n:F system when all the components are working under the stress Y isS1,k:n(x|y) =∞∫0n∑i=n−k+1(ni)pi,n(y, y + x)dGY (y)∞∫0FX(y, . . . , y︸ ︷︷ ︸n)dGY (y),wherepi,n(y, y + x) =FX(y + x, . . . , y + x︸ ︷︷ ︸i, y, . . . , y︸ ︷︷ ︸n−i)−n−i∑j=1(−1)j+1(n− jj)FX(y + x, . . . , y + x︸ ︷︷ ︸i+j, y, . . . y︸ ︷︷ ︸n−i−j).Proof. Consider the random variable Xk:n − Y |X1:n > Y and its survival function withexchangeable strengths X1, X2, . . . , Xn given byP (Xk:n − Y > x|X1:n > Y ) =1∞∫0P (X1 > y, . . .Xn > y)dGY (y)∞∫0n∑i=n−k+1(ni)P (X1 > y + x,× . . . , Xi > y + x,Xi+1 ∈ (y, y + x], . . . , Xn ∈ (y, y + x]) dGY (y)The probability in the nominator can be represented aspi,n(y, y + x) = P (A ∩Bc) = P (A)− P (A ∩B)where A ≡ {X1 > y + x, . . . ,Xi > y + x,Xi+1 > y, . . . ,Xn > y}, B ≡n−i∪j=1{Xj > y + x}.Thus the proof is completed. References[1] Bairamov, I. and Parsi, S., (2011), Order statistics from mixed exchangeable random variables, Journalof Computational and Applied Mathematics, 235 (16), 4629-4638.[2] Bhattacharyya, G.K., Johnson, R.A., (1974), Estimation of reliability in a component stress-strengthmodel, J. Amer. Statist. Assn., 69, 966-970.[3] Bhattacharyya, G.K., Johnson, R.A., (1975), Stress-strength models for system reliability, Proceedingsof the symposium on Reliability and Fault Tree Analysis SIAM, 509-532.[4] David, H.A., Nagaraja, H.N., (2003), Order Statistics, Wiley Series in Probability and Statistics, NewJersey.[5] Dewanji, A., Rao, T.S., (2001), On system reliability under stress-strength modeling, Commun.Statist-Theory Meth., 30(6), 1185-1196.[6] Ebrahimi, N., (1982), Estimation of reliability for a series stress-strength system, IEEE Transactionson Reliability, 31, 202-205.[7] Eryılmaz, S., (2008), Consecutive k-out-of-n: G System in Stress-Strength Setup, Communications inStatistics-Simulation and Computation, 37, 579-589.[8] Eryılmaz, S., (2010), On system reliability in stress-strength setup, Statistics and Probability Letters,80(9-10), 834-839.[9] Eryılmaz, S., (2011), Circular consecutive k-out-of-n systems with exchangeable dependent compo-nents, Journal of Statistical Planning and Inference, 141, 725-733.114 TWMS J. APP. ENG. MATH. V.2, N.1, 2012[10] Greco, L. & Ventura, L., (2011), Robust inference for the stress-strength reliability, Statistical Papers,52, 773-788.[11] Hanagal, D. D., (1999a), Estimation of reliability of a component subjected to bivariate exponentialstress, Statistical Papers, 40, 211-220.[12] Hanagal, D. D., (1999b), Estimation of system reliability, Statistical Papers, 40, 99-106.[13] Johnson, R. A., (1988), Stress-strength Models for Reliability, Handbook of Statistics, Vol 7, QualityControl and Reliability.[14] Kotz, S., Balakrishnan, N., Johnson, N. L., (2000), Continuous Multivariate Distributions: Modelsand Applications, Vol.1, 2nd edn, John Wiley and Sons, Inc. New York.[15] Kotz, S., Lumelskii, Y., Pensky, M., (2003), The Stress-Strength Model and its Generalizations :Theory and Applications, World Scientific Singapore.[16] Kuo, W., Zuo, M. J., (2003), Optimal Reliability Modeling, Principles and Applications, John Wileyand Sons, Inc. New York.[17] Navarro, J., Balakrishnan, N., (2010), Study of some measures ofdependence between order statisticsand systems, Journal of Multivariate Analysis, 101, 52-67.[18] Navarro, J., Ruiz, J. M., Sandoval, C. J., (2005), A note on comparisons among coherent systemswith dependent components using signatures, Statist. Probab. Lett., 72, 179-185.[19] Navarro, J., Ruiz, J. M., Sandoval, C. J., (2006), Systems with exchangeable components and Gumbelexponential distribution. In Advances in Distribution Theory, Order Statistics And Inference. Statis-tics for Industry and Technology, Balakrishnan, N., Castillo, E., Sarabia, M., Eds., Birkhauser Boston,291-306.[20] Navarro, J., Rychlik, T., (2007), Reliability and expectation bounds for coherent systems with ex-changeable components, Journal of Multivariate Analysis, 98, 102-113.Burçin Şeyda Özler got her BS degree in 2009 and her MS degree in 2011both in Statistics from Dokuz Eylül University, İzmir, Turkey. She is still aPhD student and currently works as a Research Assistant at the Departmentof Statistics in Ondokuz Mayıs University, Samsun, Turkey. Her researchinterests focus in reliability of a system, risk analysis, fuzzy time series,artificial neural networks systems, genetic algorithm.B. S. ÖZLER, S. GÜRLER: ON THE RELIABILITY OF K-OUT-OF-N : F SYSTEMS WITH ... 115Selma Gürler received her BS, MS and PhD degrees in Statistics fromthe Dokuz Eylül University, İzmir, Turkey (1998, 2002, 2006). She hasworked as a Research Assistant in the same department between 1999 and2004. She has worked as an Assistant Professor between 2009 and 2011. In2008, she was a visiting researcher at Institute of Statistics in RWTH AachenUniversity in Germany. She currently works as an Associate Professor at theDepartment of Statistics in Dokuz Eylül University, Turkey. Her researchinterests are in reliability theory, order statistics, applied probability andstatistics.

On the reliability of k-out-of-n: f systems with exchangeable components in the stress-strength setup

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On the reliability of k-out-of-n: f systems with exchangeable components in the stress-strength setup

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