Research Article On Stress-Strength Reliability with a Time-Dependent Strength

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1 Quality and Reliability Engineering Volume 03, Article ID 4788, 6 pages Research Article On Stress-Strength Reliability with a Time-Dependent Strength Serkan Eryilmaz Department of Industrial Engineering, Atilim University, Incek, Ankara, Turkey Correspondence should be addressed to Serkan Eryilmaz; seryilmaz@atilim.edu.tr Received 5 July 0; Accepted December 0 Academic Editor: Shey-Huei Sheu Copyright 03 Serkan Eryilmaz. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. e study of stress-strength reliability in a time-dependent context needs to model at least one of the stress or strength quantities as dynamic. We study the stress-strength reliability for the case in which the strength of the system is decreasing in time and the stress remains xed over time; that is, the strength of the system is modeled as a stochastic process and the stress is considered to be a usual random variable. We present stochastic ordering results among the lifetimes of the systems which have the same strength but are subjected to different stresses. Multicomponent form of the aforementioned stress-strength interference is also considered. We illustrate the results for the special case when the strength is modeled by a Weibull process.. Introduction Stress-strength models are of special importance in reliability literature and engineering applications. A technical system or unit may be subjected to randomly occurring environmental stresses such as pressure, temperature, and humidity and the survival of the system heavily depends on its resistance. In the simplest form of the stress-strength model, a failure occurs when the strength (or resistance) of the unit drops below the stress. In this case the reliability RR is de ned as the probability that the unit s strength is greater than the stress, that is, RRR PPPPP P PPP, where YY is the random strength of the unit and XX is the random stress placed on it. is reliability has been widely studied under various distributional assumptions on XX and YY. (See, e.g., Johnson [] andkotzetal.[] for an extensive and lucid review of the topic.) In the aforementioned simplest form, stress and strength quantities are considered to be both static. Dynamic modeling of stress-strength interference might offer more realistic applications to real-life reliability studies than static modeling and it enables us to investigate the time-dependent (dynamic) reliability properties of the system. Let XXXXXX and YYYYYY denote the stress that the system is experiencing and strength of the system at time tt, respectively. en the lifetime of the system is represented as TT T TTT {ss sssss(ss) >YY(ss)}. () e most important characteristic in reliability analysis is the reliability function of a system which is de ned as the probability of surviving at time tt, that is, RR (tt) =PP{TT TTT}. () is function is also known as the survival function in the reliability literature and its exact formulation is of special importance in engineering applications. e reliability function for the lifetime random variable given in () is RR (tt) =PP{XX (ss) < YY (ss), 0 sssss}. (3) e function given by (3) has been investigated in several papers. Whitmore [3] computed the function RRRRRR under the assumptions that XXXXXX and YYYYYY are independent Brownian motions. Ebrahimi [4] studied the properties of RRRRRR assuming the strength of the system YYYYYY is decreasing in time. In this paper, we study RRRRRR assuming (i) YYYYYY is decreasing in time, that is, PPPPPPPP ) YYYYY )} = for all tt < tt and (ii) XXXXXX X XX; that is, stress remains xed over time (static). e rst assumption is common in reality because the system s strength could degrade due to aging. In Section, we provide some stochastic ordering results among the lifetimes of the systems which have the same strength but are subjected to different stresses. In Section 3, stress-strength interference is

2 Quality and Reliability Engineering considered for multicomponent systems. Finally, in Section 4 some results are presented for the special case when the strength is modeled by a Weibull process.. Reliability and Ordering Properties Under the assumptions (i), (ii) XX and YYYYYY are independent and the reliability function can be formulated as RR (tt) =PP{XX XXX(tt)} = FF (xx) dddd tt (xx) = GG tt (xx) dddd (xx), where FFFFFF F FFFFF F FFF, GG tt (xxx x xxxxxxxxx x xxx and GG tt (xxx x GG tt (xxx. e following example illustrates the computation of reliability function for the stochastic strength process given with its analytical form. at is, the strength aging deterioration process is expressed as a function of time, and a random variable. Example. Let YYYYYY be de ned by (4) YY (tt) =ee CCCCC, ttttt (5) where CC follows Pareto distribution with c.d.f. FF CC (xxx x x x (μμμμμμ, xxxxx, μμμμ. en the c.d.f. of YYYYYY is GG tt (xx) = μμ tt (ln xx), 0<xxxxx μμμμ. (6) Let XX have a c.d.f. FFFFFF F FF θθ, 0<xxxx, θθθθ. en using (4)we have ee μμμμ RR (tt) = μμ tt 0 (ln xx) θθθθθθθθ dddd =ee μμμμμμ tt μμ ee μμμμ xx θθθθ θθθθ 0 (ln xx) ddddd dd d dd Differentiating RRRRRR w.r.t. tt using the rule of differentiation under the integral sign, the p.d.f. of TT isfoundas ff (tt) = μμ ee μμμμ θθθθ θθ 0 Using (8) in (7)weobtain xx θθθθ (7) ddddd (8) (ln xx) RR (tt) =ee μμμμμμ tttt (tt). (9) Integrating both sides of the last equation over (0, ) we get EE (TT) = 3μμμμ which is the MTTF of the system. (0) e process de ned by (5) can be considered in a more general form given by YY (tt) =ee CCCCCCCCC, tt t tt () where SSSSS S S and SSSSSS is a nondecreasing function. In this case the reliability function and MTTF of the system are foundtobe RR (tt) =ee μμμμμμμμμμ SS (tt) μμ ee μμμμμμμμ xx θθθθ θθ 0 (ln xx) ddddd MTTF = ee μμμμμμμμμμ dddd d SS (TT) EE 0 SS (TT). () For the system de ned in Example itcanbeeasilyseen that an increase in θθ leads to a decrease in MTTF of the system. Since EEEE E EEEEEE E EE, the larger θθ the harsher the stress and hence the smaller the reliability. is can be theoretically established using the concept of stochastic ordering as in the following lines. We investigate the behaviour of the lifetime of the system under different stresses in terms of stochastic ordering. In this context, we consider the following stochastic orderings between the lifetimes. Let XX and YY be two random variables having cumulative distributions FF and GG, densities ff and gg, hazard rates rr FF and rr GG, and reversed hazard rates h FF and h GG, respectively. Note that the hazard and reversed hazard rates are de ned, respectively, as rr FF (tt) = ff (tt) FF (tt), h FF (tt) = ff (tt) FF (tt). (3) Usual Stochastic Ordering. XX is said to be smaller than YY in usual stochastic ordering if PPPPP P PPP P PPPPP P PPP for all xx, or equivalently EEEEEEEEE E EEEEEEEEE for all increasing functions Ψ for which the expectations exist. is relation is denoted by XX st YY. Hazard Rate Ordering. XX is said to be smaller than YY in hazard rate ordering if rr FF (xxx x xx GG (xxx for all xx or GGGGGGGFFFFFF is a nondecreasing function of xx andwewritexx hr YY. Reversed Hazard Rate Ordering. XX is said to be smaller than YY in the reversed hazard rate order if h FF (xxx x x GG (xxx for all xx or GGGGGGGGGGGGG is a nondecreasing function of xx and we write XX rh YY. Likelihood Ratio Ordering. XX is said to be smaller than YY in likelihood ratio ordering (written as XX YY) if ggggggggggggg is nondecreasing function of xx. We have the following implications among these orderings: XX YYY hr XX YY XX rh XX st YYY (4) YY For more details on stochastic orderings refer to Shaked and Shanthikumar [5]. e following concepts will be useful for the next section.

3 Quality and Reliability Engineering 3 e nition. A function kkkkkk kkk issaidtobesign regular of order (SR ) if εε kkkkkk kkk k k and εε kk xx,yy kk xx,yy 0, (5) kk xx,yy kk xx,yy whenever xx < xx, yy < yy, and εε ii {, } for ii i ii i. If the conditions given in e nition hold with εε = and εε =then kk issaidtobetotally positive of order (TP ); and kk issaidtobereverse regular of order (RR ) if they hold with εε =and εε =. Proposition 3. Let TT ii denote the lifetime of the system whose stress-strength pair is (XX ii, YYYYYYY, ii i ii i. en (a) if XX st XX then TT st TT ; rh (b) if XX XX and gggggg ggg g ggggggggggg tt (xxx is RR in (xxx xxx hr then TT TT ; (c) if XX XX and h(xxx xxx x xxxxxxxxxxx tt (xxx is RR in (xxx xxx then TT TT. Proof. e proof of (a) immediately follows because PPPPP ii ttt t ttttt tt (XX ii )], ii i ii i, and GG tt (xxx is increasing in xx. e proofs of (b) and (c) can be obtained as an application of eorems.b.4,.b.5, and.c.7 in Shaked and Shanthikumar [5]. ese results are obtained using basic composition formula of Karlin [6]. Example 4. Consider the process de ned in Example with FF CC (xxx x x x xx xx, xxxxand let XX ii have a c.d.f. FF ii (xxx x xx θθ ii, 0<xxxx, ii i ii i. In this case GG tt (xx) =xx /tt, 0<xxxxx h (xxx xx) = GG tt (xx) = tt xx/tt ln xxx (6) systems consist of several components and the components might have different statistical properties. Multicomponent stress-strength reliability in a static form has been studied in various papers including Bhattacharyya and Johnson [7], Chandra and Owen [8], Johnson [], Pandey et al. [9], Eryilmaz [0], and Eryilmaz []. Assume that a system consists of components and the deteriorating strength of the iith component at time tt is denoted by the process YY ii (ttt, iii. e components are subjected to a common random stress XX. If TT ii denotes the lifetime of the iith component then the joint survival function of TT,TT,,TT isgivenby RR tt,tt,,tt =PPTT >tt,tt >tt,,tt >tt =PPXX XXX tt,xxxxx tt,,xxxxx tt. (8) If the components are independent then we have RR tt,tt,,tt = PP YY ii tt ii >xx dddd (xx). (9) From (9) it follows that the lifetimes of the components are dependent even if the strengths of them are independent. is positive dependence among the lifetimes arises from common environmental stress characterized by XX. ere are many types of positive dependence. e likelihood ratio (or TP ) dependence as the strongest notion of positive dependence is de ned as follows. Let TT, TT have the joint probability density fffff,tt ). en recall from e nition that fffff,tt ) is TP if ff tt() ff tt (),tt(),tt() ff tt() ff tt(),tt(),tt() 0, (0) gg (xxx xx) = GG tt (xx) = tt xx/tttt. For any xx < xx and tt < tt h xx,tt h xx,tt hxx,tt h xx,tt = tt ln xx ln xx xx /tt tt xx /tt xx /tt xx /tt 0 (7) for tt () < tt () and tt () < tt (). e random variables TT and TT are said to be likelihood ratio (or TP ) dependent if their joint density is TP. e following result can be proved using the basic composition formula of Karlin [6] together with ff tt,tt = () xxx xx h () xxx xx dddd (xx), () which implies the RR property of h(xxx xxx. If θθ θθ then XX XX. us by the Proposition 3 we have TT TT. rh Similarly, gggggg ggg is also RR in (xxx xxx. If θθ θθ then XX XX hr andhencebytheproposition 3 we have TT TT. 3. Multicomponent Setup In the previous sections we analyzed stress-strength reliability for a single component system. Most of the engineering where h (iii (xxx xxx x xxxxxxxxxxx ttttt (xxx and GG ttttt (xxx x xxxxx ii (ttt t ttt, ii i ii i. Proposition 5. If h () (xxx xx ) is TP (RR ) in (tt, xxx and h () (xxx xx ) is TP (RR ) in (xxx xx ), then TT and TT are likelihood ratio dependent. Example 6. Let YY ii (ttt t tt CC iitt, tttt, ii i ii i and CC ii be an exponential random variable with c.d.f. FF CCii (xxx x x x xx μμ ii xx, xxxx. Also assume that the common random stress XX has

4 4 Quality and Reliability Engineering c.d.f. FFFFFF F FF θθ, 0<xxxx. In this case the joint survival function of TT and TT isfoundtobe RR tt,tt = Since + θθθθ μμ + θθθθ θθθθ μμ + θθθθ θθ μμ /tt +μμ /tt +θθ, tt,tt 0. () h (iii (xxx xx) = μμ ii tt xxμμ ii /tt ln xx (3) is RR for ii i ii i, TT and TT are likelihood ratio dependent. Consider a system φφ with components which has two possible states; φφφφif the system is functioning and φφφ 0 if the system has failed. Since the state of the system is determined by the states of its components we can write φφ φ φφφφφ,,xx ), where xx ii = if the iith component is functioning and xx ii = 0 if it has failed. e function φφ is called structure function. A system with structure function φφ is coherent if φφ is nondecreasing in each argument, and each component is relevant to the performance of the system. If the components lifetimes are denoted by TT,,TT, then TT T TTTTT,TT,,TT ) represents the lifetime of the system. Let TT,TT,,TT denote the i.i.d. lifetime random variables with continuous distribution. Samaniego [] introduced the signature of a coherent system TT T TTTTT,TT,,TT ) as the vector pp p ppp,pp,,pp ), where =PPφφ TT,TT,,TT =TT (iii, ii i ii i i iii (4) where TT (iii denotes the smallest iith in TT,TT,,TT, showing that =#of orderings for which the iith failure causes system failure (n ). (5) A general formula for the reliability function of any coherent structure consisting of components can be given by using the concept of signature if the components are independent and identical. Samaniego [] (see also [3]) showed that the reliability function of a coherent system TTT φφφφφ,tt,,tt ) can be represented as PP {TT TTT} = PP TT (iii >tt. (6) Navarro et al. [4] (see also Navarro and Rychlik [5]) proved that the representation (6) also holds whenever (TT,TT,,TT ) has an absolutely continuous exchangeable joint distribution. e following theorem provides the reliability function of any coherent structure consisting of components. eorem 7. Let TT ii denote the lifetime of the iith component whose strength is YY ii (ttt, ii i ii ii i i ii,thatis,tt ii = inf {ss s ss s 0,XXXXX ii (ssss. If φφ denotes the structure function of the coherent system with lifetime TT, that is, TT T TTTTT,TT,,TT ) and YY (tttt tt (tttt t t tt (ttt are i.i.d. with c.d.f. GG tt (xxx x xxxxx ii (ttt t xxx, iiithen where PP TT (iii >tt = PP {TT TTT} = jjjjjjjjjj n PP TT (iii >tt, (7) ( ) mm jj n mm EEGG tt (XX) jjjjj. mmmm (8) Proof. Under the assumption that YY (tttt tt (tttt t t tt (ttt are i.i.d. the joint survival function of TT,TT,,TT is RR tt,tt,,tt = GG ttii (xx) dddd (xx). (9) e function given by (33) is a mixture of independent variate d.f. s with equal marginals; that is, the random vector (TT,TT,,TT ) is positive dependent by mixture (PDM). PDM d.f. s are exchangeable. (See, e.g., Shaked [6] for the concept of PDM and associated exchangeability). Since the representation (6) also holds for exchangeable lifetimes we get (7) with PP TT (iii >tt = jj PP XX XXX (tt),,xxxxx jj (tt), jjjjjjjjjj XXXXX jjjj (tt),,xxxxx (tt). (30) e usage of inclusion-exclusion principle for the probability inside the sum gives PP TT (iii >tt = jjjjjjjjjj n jj ( ) mm n mm mmmm PPXX XXX (tt),,xxxxx jjjjj (tt). e proof is now completed by conditioning on XX. 4. Weibull Stress-Strength Model (3) In this section we study the stress-strength reliability for the Weibull process which can be used to model the decreasing strength of a unit. Chiodo and Mazzanti [7] studied stressstrength reliability and its estimation for aged power system components subjected to voltage surges using Weibull process. Let YYYYYY be a Weibull process whose one-dimensional distribution is GG tt (xx) =PP{YY (tt) xx} = exp xx αα (tt) ββ, xx x xx (3)

5 Quality and Reliability Engineering 5 where the shape parameter ββ is assumed to be time independent and the intensity function αααααα is decreasing in time with ααααα α α. Similarly, assume that the stress random variable XX has a Weibull distribution with c.d.f. FF (xx) = exp xx θθ ββ, xx x xx xxx xx x xx (33) Under these assumptions the reliability function is found to be RR (tt) = +(θθθθθ (tt)) ββ, tt t tt (34) e following results can be obtained from Proposition 3. Corollary 8. Let TT ii denote the lifetime of the system whose stress-strength pair is (XX ii, YYYYYYY, ii i iii, where XX ii has a Weibull distribution with scale parameter θθ ii and shape parameter ββ and YYYYYY is a Weibull process whose distribution is given by (3). en, since gggggg ggg g ggggggggggg tt (xxx x (βββββββββββββββββββββ ββββ ee (xxxxxxxxxxββ is RR in (xxx xxx, if θθ θθ then hr TT TT. Corollary 9. Under the same assumptions of Corollary 8, the function xxββ h (xxx xx) = GG tt (xx) = ββ αα ββββ (tt) αα (tt) ee (xxxxxxxxxx ββ (35) is RR in (xxx xx ). Indeed, because αααααα is decreasing, αα (ttt t t, and hence h(xxx xxx x x. For xx < xx and tt < tt, h xx,tt h xx,tt hxx,tt h xx,tt 0 (36) which implies that h(xxx xxx is RR in (xxx xxx. erefore, from Proposition 3, if θθ θθ then TT TT. Remark 0. If αααααα α αααα then RR (tt) = +(θθθθ) ββ, tttt (37) which is the survival function of the log-logistic distribution. at is, the lifetime of the system has log-logistic distribution with scale parameter λλ λ λλλλ and shape parameter ββ. If ββββ thenthemttfofthesystemisfoundtobe EE (TT) = πππππππ sin πππππ. (38) eorem. Let YY ii (ttt be a Weibull process with intensity αα ii (ttt associated with iith component, ii i ii. Assume that the common random stress XX has a Weibull distribution with c.d.f. given by (33). enthejointsurvivalfunctionoftt,,tt is RR tt,,tt = +θθθθθ tt ββ + +θθθθθ tt ββ, (39) and the survival copula associated with (39) belongs to the Claytonfamilyandisgivenby CC uu,,uu = /uu + + /uu. (40) Proof. Using (9)one can write RR tt,,tt =PPXX XXX tt,,xxxxx tt = ee (xxxxx (tt )) ββ ee (xxxxx (tt )) ββ ββ 0 θθ xxββββ ββ ee (xxxxxx dddd θθ = +θθθθθ tt ββ + +θθθθθ tt ββ. (4) y the de nition of survival copula (see, e.g., Nelsen [8, page 3]) CC uu,,uu =RRRR tt,,rr tt =RRαα θθ /ββ,,αα θθ /ββ uu uu = uu + +uu (4) which is known to be a Clayton copula (see, e.g., Nelsen [8, page 5]). us the proof is completed. Remark. For αα ii (ttt t tttt, ii i ii,(39) becomes the survival function of the multivariate log-logistic distribution generated by the Clayton family of copulas. Example 3. Consider the system consisting of components whose deteriorating strengths are modeled by a Weibull process with the common intensity function αααααα. Suppose that these components are subjected to a common random stress XX which has c.d.f. given by (33). en, the lifetimes of thecomponentsareexchangeableandwehave EEGG tt (XX) jjjjj = +jjjjj θθθθθ (tt)) ββ. (43) If, for example, a system has a -out-of-3 structure, that is, the system functions if and only if at least two components function, then since pp p ppp pp pp using eorem 7, the reliability of the system is found to be PP TT () >tt = Acknowledgment 3 +(θθθθθ (tt)) ββ +3(θθθθθ (tt)) ββ. (44) e author thanks the anonymous referee for his/her helpful comments and suggestions. References [] R. A. Johnson, Stress-strength models for reliability, in Handbook of Statistics, P. R. Krishnaiah and C. R. Rao, Eds., vol. 7, pp. 7 54, Elsevier, Amsterdam, North-Holland, 988.

6 6 Quality and Reliability Engineering [] S. Kotz, Y. Lumelskii, and M. Pensky, e Stress-Strength Model and its Generalizations, World Scienti c, River Edge, NJ, USA, 003. [3] G. A. Whitmore, On the reliability of stochastic systems: a comment, Statistics & Probability Letters, vol. 0, no., pp , 990. [4] N. Ebrahimi, Two suggestions of how to de ne a stochastic stress-strength system, Statistics & Probability Letters, vol. 3, no. 6, pp , 985. [5] M. Shaked and J. G. Shanthikumar, Stochastic Orders, Springer Series in Statistics, Springer, New York, NY, USA, 007. [6] S. Karlin, Total Positivity. Vol. I, Stanford University Press, Stanford, Calif, USA, 968. [7] G. K. Bhattacharyya and R. A. Johnson, Estimation of reliability in a multicomponent stress-strength model, the American Statistical Association, vol. 69, pp , 974. [8] S. Chandra and D. B. Owen, On estimating the reliability of a component subject to several different stresses (strengths), Naval Research Logistics Quarterly, vol., pp. 3 39, 975. [9] M. Pandey, M. B. Uddin, and J. Ferdous, Reliability estimation of an s-out-of-k system with non-identical component strengths: the Weibull case, Reliability Engineering and System Safety, vol. 36, no., pp. 09 6, 99. [0] S. Eryilmaz, Consecutive kk-out-of : GG system in stressstrength setup, Communications in Statistics. Simulation and Computation, vol. 37, no. 3 5, pp , 008. [] S. Eryilmaz, Multivariate stress-strength reliability model and its evaluation for coherent structures, Multivariate Analysis, vol. 99, no. 9, pp , 008. [] F. Samaniego, On closure of the IFR class under formation of coherent systems, IEEE Transactions on Reliability, vol. 34, no., pp. 69 7, 985. [3] S. Kochar, H. Mukerjee, and F. J. Samaniego, e signature of a coherent system and its application to comparisons among systems, Naval Research Logistics, vol. 46, no. 5, pp , 999. [4] J. Navarro, J. M. Ruiz, and C. J. Sandoval, A note on comparisons among coherent systems with dependent components using signatures, Statistics & Probability Letters, vol. 7, no., pp , 005. [5] J. Navarro and T. Rychlik, Reliability and expectation bounds for coherent systems with exchangeable components, Journal of Multivariate Analysis, vol. 98, no., pp. 0 3, 007. [6] M. Shaked, A concept of positive dependence for exchangeable random variables, e Aals of Statistics, vol. 5, no. 3, pp , 977. [7] E. Chiodo and G. Mazzanti, Bayesian reliability estimation based on a weibull stress-strength model for aged power system components subjected to voltage surges, IEEE Transactions on Dielectrics and Electrical Insulation, vol. 3, no., pp , 006. [8] R. B. Nelsen, An Introduction to Copulas, Springer Series in Statistics, Springer, New York, NY, USA, nd edition, 006.

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