International Journal of Performability Engineering, Vol. 1, No. 1, July 2005, pp RAMS Consultants Printed in India

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1 Iteratioal Joural of Performability Egieerig, Vol. 1, No. 1, July 2005, pp RAMS Cosultats Prited i Idia LIYANG XIE 1, JINYU ZHOU 1, YONGYAN WANG 2, XUEMIN WANG 1 1 Northeaster Uiversity, Sheyag, , Chia, 2 Qigdao Uiversity of Sciece ad Techology, Qigdao, , Chia (Received o Ja. 15, 2005) 1. Itroductio Abstract : For s-depedet systems, reliability models capable of reflectig the effect of failure depedece, i.e. the so called commo cause failure are i demad, sice the system reliability depeds ot oly o the reliabilities of the idividual compoets but also o the iteractios amog the compoet failures. Based o the attribute of system failure, the meaig of order statistics, as well as the mechaism of failure depedece, this paper developed s-depedet system reliability models by icorporatig order statistics of compoet stregth ito load-stregth iterferece model. The models simply characterize the fact that es system fails as log as the applied load exceeds the first (the miimum) or der statistic of compoet stregth, parallel system (with i.i.d. compoets) fails whe ad oly whe load exceeds the th (the maximum) order statistic of compoet stregth, ad k-out-of-(f) system fails if load exceeds the k th order statistic of compoet stregth. Sice o assumptio about s-idepedet failure was applied, the models are i a positio to reflect the effect of failure depedece which may brig about commo cause failures. To evaluate the reliability of geeral system comprisig of o-i.i.d. compoets, a exteded load - stregth order statistics iterferece model was preseted, too. Key Words: order statistics, s-depedet system, commo cause failure, system reliability Reliability, as a probability for a system to fuctio successfully agaist expected eviromet, has log bee treated as oe of the most importat performace attribute. Traditioal system reliability models assumed the s-idepedet probabilities of basic evet failures. Violators of this assumed s-idepedece are called systems iteractios, depedecies, commo mode failures, or commo cause failures. It has bee widely recogized that compoet failures i a system are ormally s-depedet of each other, i.e., there exists the failure mechaism called commo cause failure (CCF) or commo mode failure (CMF) by which several compoets will fail to the same cause simultaeously. Correspodig author Liyag_Xie@mail.eu.edu.c 23

2 24 L. Xie, J. Zhou, Y. Wag ad X. Wag Compoet reliability is a fuctio of load ad stregth. For a group of compoets subjected to the same or correlated load eviromet, the degree of s-depedece amog compoet failures is determied by both the characteristic of load ad that of stregth, too. If all the compoets are exactly the same (e.g., all the compoets have the same determiistic stregth) ad are subjected to the same load, these compoets will always fail simultaeously whe load exceeds their stregth. I a geeral coditio that the idividual compoets are statistically differet from each other (e.g., the stregth of the compoets follows the Weibull distributio), the radomess i load is the root cause for failure depedece. Oly whe load is a determiistic costat, the compoet failures are s-idepedet of each other [1]. It is the ucertaity i load that itroduces the failure depedece amog compoet failures. Coversely, dispersio i compoet stregth is helpful to mitigate failure depedece. Geerally, failure depedece ( failure depedece metioed i this paper is the failure-couplig mechaism that might iduce commo cause failure) is uavoidable for the majority of systems. The existece of such failure depedece is a challege to the covetioal system reliability aalysis approach. It also makes system reliability estimatio much more complicated. Covetioal approach to modelig system reliability was geerally based o the assumptio of s-idepedece amog compoet failures. With such a assumptio, the covetioal procedure to evaluate system reliability is from compoet to system. That is, load-stregth iterferece calculatio or compoet failure data statistics is performed firstly to obtai compoet reliabilities. The, the compoet reliability idexes are used to calculate system reliability accordig to the system fuctioal logic, such as es, parallel, or k-out-of- redudat, etc. Nevertheless, system reliability depeds ot oly o the reliabilities of the idividual compoets but also o the s-depedece amog compoet failures, uless the loads imposed to the idividual compoets are statistically idepedet of each other, or all the loads are determiistic (a typical situatio is that all the compoets are subjected to the same costat load). I the situatio that both the load ad stregth are radom variables, the degree of failure depedece depeds o the relative magitude of load dispersio to compoet stregth dispersio [1]. For a compoet, the ucertaity of failure or survival may come from the ucertaity of load, the ucertaity of stregth, or the both. That makes o differece for compoet reliability idex. However, for systems, the ucertaity of load meas the existece of s-depedece amog compoet failures, while the ucertaity of compoet stregth meas wholly differet [1]. Goble et al [2] also illustrated such tred accordig to Mote Carlo simulatio results. It illustrated that the probability of commo cause failure is substatially reduced whe the radom stresses subjected to the idividual compoets are idepedet. It also showed that commo cause beta factor is clearly related to the variatio i the stregth distributio if all uits i a redudat architecture had idetical stregth, they would always fail simultaeously. It does imply that idetical uits that respod i the same way to a commo stress will be more likely to have commo cause failures. Sice the differet roles of load dispersio ad compoet stregth dispersio ca ot be distiguished i the course of covetioal compoet reliability calculatio, oe ca ot expect to express the reliability of s-depedet system by meas of compoet reliability idex straightforward. I fact, whe calculatig compoet reliability by either load-stregth iterferece aalysis or statistical estimatio of failure evet data, the

3 Load-Stregth Order Statistics Iterferece Models 25 parameter of load dispersio ad that of compoet stregth dispersio are equally itegrated ito the calculatio, ad importat iformatio for s-depedet system reliability estimatio is lost [3]. For example, a valve system failure evet data iclude failure evets of various failure multiples for a 10-redudacy valve system. Amog total 34 demads to the system, there are 5 sigle failures, 2 double failures, 1 triple failures, ad 26 successful demads. I other words, there are 12 ( ) compoet failures amog the 340 (3410) demads to the compoets. Obviously, these data cotai both idividual compoet failure iformatio ad system s-depedet failure iformatio. Accordig to these evet data, the compoet failure probability is estimated as 12/340= (per demad), where the meaigful items are the umbers of compoets failed ad the total umbers of the compoets tested. For the compoet failure probability idex (per demad), it makes o differece whether the failed compoets come from sigle failure evet or multiple failure evet. As a subsequece, the compoet failure probability idex (per demad) ca ot provide the details about the associated failure evets ay more, i.e., it does ot cotai ay iformatio about how may failed compoets come from sigle failure evet, ad how may failed compoets come from multiple failure evet, which is idispesable for system reliability estimatio. I oe word, the compoet reliability idex does ot cotai failure depedece iformatio. Therefore, the covetioal es, parallel or k-out-of- redudat system reliability models are ot applicable to s-depedet systems. Alteratively, a variety of CCF models have bee developed [4-10] ad applied i the field of uclear idustry, oil & chemical idustry, rocket lauchig, ad mechaical egieerig. Nevertheless, it is still cosidered that the mechaisms of s-depedece amog compoets are very complicated, carryig out quatitative descriptio about the degree of such s-depedece is very difficult, ad assumptios have to be made [9]. Obvious gap still exists as to practical usage of depedet system reliability model, especially for high redudacy systems [11, 12]. Notatio x y X i X (k) f(x) F(x) g (k) (x) g (i)(j) (x,z) A B i X R s compoet stregth radom variable load radom variable stregths of the ith compoet k-order statistic of compoet stregth umber of compoets i a system probability desity fuctio of compoet stregth radom variable x cumulative distributio fuctio of compoet stregth x probability desity fuctio of the k-th order statistic X (k) joit probability desity fuctio of (X (i),x (j) ), 1 ij evet of es system failure evet of the i-th weakest compoet failure system stregth es system reliability

4 26 L. Xie, J. Zhou, Y. Wag ad X. Wag R 1st h(y) R para s R th P k/ k R / s R kth N M F N (x) F M (x) Rid para Rid k R / id r y 0 a i,b i probability of X (1) > y probability desity fuctio of load y parallel system reliability probability of X () > y failure probability of k-out-of- evet k-out-of-(f) system reliability probability of X (k) > y miimum of compoet stregth of o-i.i.d. compoets maximum of compoet stregth of o-i.i.d. compoets cumulative distributio fuctio of the miimum statistic cumulative distributio fuctio of the maximum statistic reliability of s-idepedet es system reliability of s-idepedet parallel system reliability of s-idepedet k-out-of-(f) system failure multiple uified load radom variable coefficiets expectatio stadard deviatio 2. Order Statistics ad System Reliability Modelig Depedig o the respective compoet or system property ad associated operatig eviromet, the failure attributes of differet compoets or systems are cosiderably differet. For compoets, there are electroic compoets with expoetially distributed life ad costat failure rate, ad mechaical compoets with the Weibull distributed life ad variable failure rate. For systems, there are s-idepedet systems i which compoet failures are mutually idepedet ad s-depedet systems i which commo cause failure plays a importat role. O the other had, despite the differet attributes or behaviors, failure is othig but the result of the iteractio betwee load ad stregth. As a pair of coflictig factors i failure problem, load ad stregth are the two basic goverig variables for reliability aalysis. Through a thorough aalysis to their characteristics, ad the relatioship betwee the two variables as well, a whole picture about compoet ad/or system failure ca be clearly depicted. Especially, by meas of the cocept of order statistics, system reliability models ca be developed without ay special cosideratio o whether the compoet failures are idepedet of each other or ot. Both the system comprisig of compoets with idepedet ad idetically distributed ( i.i.d.) stregth ad the system comprisig of compoets with idepedet ad o-idetically distributed stregth are cosidered. Cosiderig a system of which the compoet stregths are i.i.d. variables. Let X 1,

5 Load-Stregth Order Statistics Iterferece Models 27 X 2,, X be the stregths of the compoets respectively. X (k) (k=1~), the k-order statistic i a sample of size, stads for the stregth of the k-th weakest compoet of the samples. With probability desity fuctio (pdf) f(x) ad cumulative distributio fuctio (cdf) F(x) of the compoet stregth radom variable x, the probability desity fuctio g (k) (x) of the k-th order statistic X (k) has the followig form [13]: k k g k x F x F x f x k k (1) Especially, g x F x f x (2) g x F x f x (3) Furthermore, the joit probability desity fuctio of (X (i),x (j) ) (1 i < j ) is: i j i j gi j xz Fx Fz Fx Fz f x f z (4) i j i j System failure starts from compoet failure, ad weaker compoets always fail before stroger compoets do. By icorporatig order statistics cocept ito the well kow load-stregth iterferece model, system reliability model ca be developed without special cosideratio or ay assumptio o failure depedece. Series System As a es system, ay compoet failure ca lead to system failure. System might fail due to sigle compoet failure or multiple compoets simultaeous failures. Sice compoet failure always begis from the weakest oe, es system failure always implicates the weakest compoet failure, or vice versa, o matter other compoets fail or ot. Let evet A deote es system failure ad evet B i (i=1, 2,, ) the i-th weakest compoet failure, the A = B 1 B 2 B i B = B 1 (5) That is to say, the evet of es system failure is equivalet to that of the weakest compoet (correspodig to the miimum order statistic of stregth) failure. I case of all the i.i.d. compoets are subjected to the same radom load, the stregth of the es system is determied by the miimum order statistic of the compoet stregths. The relatioship betwee the sample of system stregth ad the sample set of the compoet stregths is X = mi X 1, X 2,, X (6) Sice the stregth of a es system ca be exactly represeted by the miimum order statistic of its compoet stregths, es system reliability ca be defied as the probability that the miimum order statistic of compoet stregth exceeds the applied

6 28 L. Xie, J. Zhou, Y. Wag ad X. Wag load o the compoets. Such a probability ca be calculated by meas of load-stregth iterferece model. Differet from the covetioal load-stregth iterferece model for compoet reliability calculatio, the stregth used here is system stregth rather tha compoet stregth. I other word, es system reliability is modeled based o load - miimum order statistic of compoet stregth iterferece relatioship, i.e. st Rs R hy y g x dx d y (7) where, Rs stads for es system reliability, R 1st stads for the probability that the miimum order statistic of compoet stregths exceeds load, h(y) is the pdf of load y, ad g (1) (x) is the pdf of the first (miimum) order statistic of compoet stregth x. Sice g x F x f x, so that hy i.e., y g x dx d y hy y F x f x dx dy d F x hy y dx dy dx hy F y dy hy y f x dx dy Rs hy y f x dx d y This equatio is the same as the es system reliability model proposed by meas of the so called system-level load-stregth iterferece aalysis [3]. That is to say, the order statistic model of es system reliability is equivalet to the system-level load-stregth iterferece model. Parallel System Parallel system fails whe ad oly whe all its compoets fail. I other word, parallel system does ot fail util the strogest compoet fails. By the defiitio of the maximum order statistic, the reliability of parallel system ca also be calculated through load - stregth order statistic iterferece aalysis. For a parallel system comprisig of i.i.d. compoets ad all the compoets beig subjected to the same radom load, its reliability equals to the probability that the maximum (-th) order statistic of compoet stregths is greater tha the load: para th Rs R h y y g x dx d y (8) para where, R s stads for parallel system reliability, R th stads for the probability that the maximum order statistic exceeds the applied load.

7 Load-Stregth Order Statistics Iterferece Models 29 Sice g x F x f x, so that hy Thus, y g x dx d y hy y F x f x dx dy d F x hy y dx dy dx hy F y dy y hy f x dx dy para y Rs hy f x dx d y It is proved that the order statistics model for parallel system reliability accords also with the system-level load-stregth iterferece model proposed by Xie et al [3]. k-out-of- (F) System Similarly, by the joit probability desity fuctio of two order statistics, the failure probability of k-out-of- evet ca be expressed as k y P h y y g k j x z dxdz d y (j=k+1) (9) The reliability of k-out-of-(f) system equals to k kth Rs R hy y gk x dxdy (10) k R / s where, stads for k-out-of-(f) system reliability, R kth stads for the probability that the k-th order stregth statistic of compoet stregths exceeds the associated load. 3. System Reliability Models for o i.i.d. Compoets For a system of which the compoet stregths are idepedet ad o-idetically distributed variables, let x 1, x 2,, x stad for the stregths of the compoets respectively, ad F i (x) the distributio fuctio of x i (i=1~). With the otatios of N=mi{X 1, X 2,, X } ad M=max{X 1, X 2,, X }, the distributio fuctios of the miimum statistic ad the maximum statistic are respectively FN x Fi x (11) i FM x Fi x (12) i By meas of the iterferece aalysis betwee the miimum stregth statistic ad the applied load, es system reliability, which is equal to the probability that the miimum stregth statistic exceeds the applied load, ca be expressed as: Rs hy FN y dy (13)

8 30 L. Xie, J. Zhou, Y. Wag ad X. Wag Parallel system reliability, which is equal to the probability that the maximum statistic exceeds the applied load, ca be expressed by meas of the iterferece relatioship betwee the maximum stregth statistic ad the applied load: para Rs hy FM y dy (14) 4. Covetioal System Reliability Models Traditioally, system reliability is calculated through compoet reliability, which is equal to the probability of compoet stregth x exceedig the associate load y, ad ormally calculated by meas of the well kow load-stregth iterferece model: R h y y f x dx d y (15) where, h(y) is the probability desity fuctio of load y, ad f(x) is the probability desity fuctio of compoet stregth x. With the assumptio of idepedet failure, covetioal es, parallel, ad k-out-of-(f) system reliability models take o the forms show i Eq.16, Eq.17, ad Eq.18, respectively. id R R (All compoets have the same reliability R) (16) Rid Ri i ( R i is the reliability of the i-th compoet) para Rid R (All compoets have the same reliability R) (17) para Rid i (1 - R i ( R i is the reliability of the i-th compoet) k k r r Rid r ( 1 - R) R r (All compoets have the same reliability R) (18) where, para R id, id R, ad k R stad for the reliabilities of s-idepedet es, parallel, ad k-out-of-(f) system respectively, id r r r is the combiatio of r elemets selected from a set of elemets without regard to the order of selectio. Sice these models were developed uder the assumptio of idepedet failure, they do ot have the capability of reflectig failure depedece which is quite commo for the majority of systems. The degree of the differece betwee the covetioal idepedet system reliability model ad the order statistic model depeds o the degree of the s-depedece

9 Load-Stregth Order Statistics Iterferece Models 31 amog compoet failures, which is determied by the relative magitude of load dispersio to compoet stregth dispersio [3]. 5. Reliability Models for Systems with Compoets Subjected to Differet Loads Geerally, a system might comprise various compoets, ad the loads applied to the idividual compoets are differet, too. I the case that the load applied to each compoet is idepedet of each other, the covetioal system reliability models such as the equatios (16-18) will work well. Otherwise, depedet system reliability models should be developed to take the place of these equatios. As a example, let us cosider a typical situatio i which all the loads are liearly correlated radom variables, i.e., y i =a i y 0 +b i, where y i is the load applied o compoet i ad y 0 is a uified load radom variable, a i ad b i are costats. For such a loadig coditio, system reliability models ca be developed by meas of uificatio treatmet to the liearly correlated radom loads. Suppose the load applied to the i-th compoet follows the ormal distributio with expectatio i ad stadard deviatio i, i.e., y i ~N( i, i ), it is easy to get the relatioship betwee y i ad a stadard ormally distributed variable y 0 (y 0 ~N(0,1)): y 0 = (y i - i ) / i (19) or y i = i y 0 + i (20) Evidetly, the followig trasformatio holds true (referrig to Fig.1) ad the right side is a exteded load-stregth iterferece model. i h y y f x dx d y h y y f x dx d y (21) where, h 0 (y) is the pdf of the stadard ormal distributio. i Uified load distributio True load distributio Stregth distributio Fig.1: Illustratio of Load Uificatio ad Exteded Load-Stregth Iterferece Relatioship

10 32 L. Xie, J. Zhou, Y. Wag ad X. Wag I such a situatio, es system reliability ad parallel system reliability ca be preseted respectively as: Rs h0 y Fi i y i d y (22) i para Rs h0 y Fi i y i d y (23) i 6. Model Verificatio For purpose of verificatio ad compariso, the covetioal s-idepedet system reliability models, the system-level load-stregth iterferece models [3], as well as the order statistics models are applied simultaeously to calculate the reliability of systems with differet load distributios ad/or differet compoet stregth distributios. All the radom variables cocered are assumed to be ormally distributed. For the -compoet es system ( = 1~100), i the coditio of load expectatio l = 400, load stadard deviatio l = 60, ad compoet stregth expectatio s = 600, compoet stregth stadard deviatio s = 60, the reliabilities calculated by the differet models are show i Fig.2. Obviously, the order statistics model gives out the same results as the system-level load-stregth iterferece model does, while the covetioal s-idepedet system model provides very coservative results (i.e., the covetioal model uder-estimates es system reliability). For the -compoet parallel system ( = 1~10), i the coditio of l = 400, l =120, ad s = 600, s = 120, the reliabilities calculated by the differet models are show i Fig.3. It ca be easily see that the order statistics model still gives out the same results as the system-level load-stregth iterferece model does, while the covetioal idepedet system reliability model over-estimates parallel system reliability. Order statistics model Depedet system model Idepedet system model System size Fig.2: Series System Reliabilities Calculated by Differet Models

11 Load-Stregth Order Statistics Iterferece Models 33 Order statistics model Depedet system model Idepedet system model System size Fig.3: Parallel System Reliabilities Calculated by Differet Models For k-out-of- (F) system ( k = 3, = 5~30), i the coditio of l = 300, l = 80, ad s = 600, s = 60, the reliabilities calculated by the differet models are show i Fig.4. Similarly, we ca see that the order statistics model gives out the same results as the system-level load-stregth iterferece model does, while the covetioal idepedet model over-estimates the system reliability. For a k-out-of- (F) system (k = 3, = 15), i the coditio of l = 300, l = 30~120, ad s = 600, s = 60, the reliabilities calculated by the differet models are show i Fig.5. Agai, the order statistics model gives out the same results as the system-level load-stregth iterferece model does, the covetioal idepedet system reliability model over-estimates the system reliability. The differeces betwee the two kids of models icrease drastically with the icrease i load dispersio. Order statistics model Depedet system model Idepedet system model System size Fig.4: k-out-of- (F) System Reliabilities Calculated by Differet Models

12 34 L. Xie, J. Zhou, Y. Wag ad X. Wag Order statistics model Depedet system model Idepedet system model Load stardad deviatio Fig.5: k-out-of- (F) System Reliabilities Calculated by Differet Models For the k-out-of- (F) system (k = 3, = 15), i the coditio of l = 300, l = 80, ad s = 600, s = 40~160, the reliabilities calculated by the differet models are show i Fig.6. Oce more, the order statistics model gives out the same results as the system-level load-stregth iterferece model does, the covetioal idepedet system model over-estimates the system reliability. Order statistics model Depedet system model Idepedet system model Elemet stregth stadard deviatio Fig.6: k-out-of- (F) System Reliabilities Calculated by Differet Models 7. Coclusios Load-stregth iterferece aalysis is a fudametal method for reliability calculatio, ad the order statistics of compoet stregth are the right variables to represet system stregth. The iterferece aalyses betwee load ad the respective order statistics of compoet stregth provide a approach to modelig system reliability, by which failure depedece ca be reflected.

13 Load-Stregth Order Statistics Iterferece Models 35 Accordig to the cocept of order statistics, es system failure is equivalet to the failure of the weakest compoet i statistical sese, i.e., the failure of the compoet with the stregth correspodet to the first order statistic. Parallel system failure ca be cosidered as a process of which the weakest compoet (correspodet to the first order statistic) fails ahead of all the others, the secod weakest compoet (correspodet to the secod order statistic) fails successively, ad so o. Naturally, the strogest compoet (correspodet to the last order statistic) fails last. Cosequetly, the reliability of a es system is equal to the probability that the stregth of the compoet correspodet to the first order statistic exceeds the load, the reliability of a parallel system is equal to the probability that the stregth of the compoet correspodet to the last order statistic exceeds the load, ad the reliability of a k-out-of-(f) system is equal to the probability that the stregth of the compoet correspodet to the k th order statistic exceeds the load. Sice o assumptio of s-idepedet failure is made, the proposed system reliability models are able to reflect the effect of depedece amog compoet failures. I other words, although o commo cause failure is specially emphasized, its effect is aturally icluded i the load - stregth order statistics iterferece models. Ackowledgmet The research work is supported by the Natioal Sciece Foudatio of Chia with the grat No The authors are grateful to Dr. J. Lig of Daimler Chrysler Corporatio for his help i improvig the Eglish presetatio. Refereces [1] Xie L.Y., A Kowledge Based Multi-Dimesio Discrete CCF Model, Nuclear Eg. ad Desig, vol.183, pp ,1998 [2] Goble W.M., Brombacher A.C., ad Bukowski J.V. et al, Usig Stress-Strai Simulatios to Characterize Commo Cause, Probabilistic Safety Assessmet ad Maagemet (ed. A. Mosleh ad R.A. Bari), Spriger, New York, pp , 1998 [3] Xie L.Y. Zhou J.Y. ad Hao C.Z., System-Level Load-Stregth Iterferece Based Reliability Modelig of k-out-of- Depedet System, Reliability Eg. ad System Safety, vol.84, pp , 2004 [4] Bukowski J.V. ad Goble W. M., Verifyig Commo-Cause-Failure Reductio Rules for Fault Tolerat Systems via Simulatio Usig a Stress-Stregth Failure Model, ISA Trasactios, vol.40, pp , 2001 [5] Döre P., Basic Aspects of Stochastic Reliability Aalysis for Redudacy Systems, Reliability, Eg. ad System Safety, vol.24, pp , 1989 [6] Hauptmas U., The Multi-Class Biomial Failure Rate Model, Reliability Eg. ad System Safety, vol.53, pp.85-96, 1996 [7] Hughes R. P., A New Approach to Commo Cause Failure, Reliability Eg., vol.17, pp , 1987 [8] Makamo T. ad Kosoe M., Depedet Failure Modelig i Highly Redudat Structure - Applicatio to BWR Safety Valves, Reliability Eg. ad System Safety,

14 36 L. Xie, J. Zhou, Y. Wag ad X. Wag vol.35, pp , 1992 [9] Vaurio J.K., Commo-Cause Failure Models, Data, Quatificatio, IEEE Trasactios o Reliability, vol.48, pp , 1999 [10] Zhag T. ad Horigome M., Availability ad Reliability of System with Depedet Compoets ad Time-Varyig Failure ad Repair Rates, IEEE Trasactios o Reliability, vol.50, pp , 2001 [11] Mosleh A., Parry G.W., ad Zikria A.F., A Approach to the Aalysis of Commo Cause Failure Data for Plat-Specific Applicatio, Nuclear Eg. ad Desig, vol.150, pp.25-47, 1994 [12] Kvam P.H., ad Miller J.G., Commo Cause Failure Predictio Usig Data Mappig, Reliability Egieerig ad System Safety, vol.76, pp , 2002 [13] Keett R.S. ad Zacks S., Mode Idustrial Statistics: Desig ad Cotrol of Quality ad Reliability, Duxbury Press, 1999 Liyag Xie received his BS (1982) i Mechaical Maufacturig, MS (1985) & PhD (1988) i Mechaical Fatigue ad Reliability from Northeaster Uiversity, Sheyag, Chia. His research iterest icludes structural fatigue, system reliability ad probabilistic risk assessmet.