OPPORTUNISTIC MAINTENANCE FOR MULTI-COMPONENT SHOCK MODELS

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2 OPPORTUNISTIC MAINTENANCE FOR MULTI-COMPONENT SHOCK MODELS Lirong Cui 1 School of Management & Economics Beijing Institute of Technology Beijing , P.R. China lirongcui@bit.edu.cn Haijun Li 2 Department of Mathematics Washington State University Pullman, WA 99164, U.S.A. lih@math.wsu.edu July Supported by the NSF of China grant Supported in part by the NSF grant DMI

3 Abstract This paper is concerned with opportunistic maintenance on a multi-component cumulative damage shock model with stochastically dependent components. A component fails when its cumulative damage exceeds a given threshold, and any such a failure creates a maintenance opportunity, and triggers a simultaneous repair on all the components, including the non-failed ones, such that damages accumulated at various components are reduced to certain degrees. Utilizing the coupling method, stochastic maintenance comparisons on failure occurrences under different model parameters are obtained. Some positive dependence properties of this multi-component shock model are also presented. Key words and phrases: Opportunistic maintenance, multi-component cumulative damage shock model, stochastic comparison, positive dependence, coupling method.

4 1 Introduction A system of components working in a random environment are subjected to wear and damage over time and may fail unexpectedly. The components are replaced or repaired upon failure, and such unpleasant events of failure are at the same time also considered in practice as opportunities for preventive maintenance on other components. The goal of this paper is to introduce a general opportunistic maintenance policy for a multi-component cumulative damage shock model and study its structural properties using stochastic comparison methods. Opportunistic maintenance basically refers to the scheme in which preventive maintenance is carried out at opportunities, either by choice or based on the physical condition of the system. In this paper, we focus on the situation in which the opportunities for preventive maintenance are generated by the failure epochs of individual components. At each failure epoch, the failed components are correctively repaired and other components that are still operational are also preventively serviced so that all the components are maintained and restored to certain conditions. An advantage of this opportunistic maintenance is that corrective repair combined with preventive repair can be used to save set-up costs. Note that by combining both types of repair, one may not know in advance which repair actions should be taken, and thus sacrifices the plannable feature of preventive maintenance. However, there are many situations in which opportunistic maintenance is effective. For example, when corrective repair on some components requires dismantling of the entire system, a corrective repair on these components combined with preventive repair on other or neighboring components might be worthwhile. Another instance is when a certain corrective repair on failed components can be delayed until the next scheduled preventive maintenance. Opportunistic maintenance has been first studied in Radner and Jorgenson 1963, and in McCall Since then, many extensions of opportunistic maintenance have been introduced and studied in the literature. Berg (1976) studies a system with two identical components with exponential distributed lifetimes, for which the non-failed component as well as the failed component are both replaced by a new one if the age of the non-failed component exceeds a threshold. Zheng and Fard (1991) examine an opportunistic maintenance policy based on failure rate tolerance for a system with k different types of components. Pham and Wang (2000) propose two new (τ, T ) opportunistic maintenance policies for a k-out-of-n system. These and other opportunistic maintenance models have been summarized in Dekker, van der Schouten and Wildeman (1997) and in Wang (2002). All these models, however, address the optimization issues for components operating independently. The thrust of the maintenance model introduced in this paper is the general opportunistic 1

5 repairs implemented at correlated failures for a system of components that are stochastically dependent. To compare the effectness of various policies, the maintenance comparisons on the failure counts under basic age, block and other policies have been studied in the literature. It is wellknown (see Barlow and Proschan 1981) that the number of failure of components with NBU (New Better Than Used) lifetimes under age or block replacement policy is stochastically smaller than the number of component failures without any preventive maintenance. Block, Langberg and Savits (1990a) extend the maintenance comparison to general block policies, and obtain stronger results that compare the processes of failure counts under different policies. Li and Shaked (2003) examine various maintenance comparisons of failure counts for imperfect repair models with preventive maintenance. Shaked and Shanthikumar (1989) introduce several maintenance policies for multivariate repairable systems operating in random environments, and study multivariate stochastic comparisons on the counters of failure at various components for the models with minimal repairs at failure and block replacements at predetermined time instants. Li and Xu (2004b) develop a multivariate imperfect repair model with coordinated random group replacements, and establish dependence comparisons on the failure counts at various components. The basic idea for maintenance comparisons of repairable systems is that if component lifetimes possess certain positive aging properties, then it should be beneficial to implement a maintenance policy so that the failure occurrences can be reduced. The details on maintenance comparisons and related techniques can be found in the survey papers by Block, Langberg and Savits (1990b) and by Kijima, Li and Shaked (2000) and the references therein. In this paper, we introduce a shock model for a system of components subject to correlated damages of shocks that arrive at the system according to a point process. A component fails when its cumulative damage exceeds a given threshold, and any such a failure creates a maintenance opportunity, and triggers a simultaneous repair on all the components, including the failed ones, such that damages accumulated at various components are reduced to a certain degree. This cumulative damage shock model ables us to develop a coupling method, using which, we study the structural properties of opportunistic maintenance, and compare stochastically the failure counts for this model under different processes of shock arrivals and repair degrees. This research is motivated from the analysis on opportunistic maintenance of a system of components operating in a random environment, where component lifetimes are dependent, and repair degrees at various components are also correlated. These failure and economic correlations, in addition to stochastic dependence introduced by the opportunistic mainte- 2

6 nance scheme, complicate the dependence structure of failures at various components. A failure simultaneously occurred at some components creates an opportunity of preventive repair for other components. As such, if some components fail more frequently, then the other components could receive more preventive repairs, and thus may be less likely to fail. Despite this negative dependence, our results show that the number of failure occurrences among all the components is positively correlated over time if the damages and repair degrees at various components are positively correlated in the sense of association. Our results also show that a correlated shock model with opportunistic maintenance experiences less failures stochastically at various components than a similar shock model without such a maintenance. The paper is organized as follows. Section 2 introduces our model and its performance measures of interest in this study. Section 3 establishes the main result that compares stochastically the processes of failure instants under different model parameters, and also show that opportunistic maintenance reduces the failure occurrences at various components in our shock model. Finally, Section 4 concludes the paper with a temporal association property for the system failure epochs. Throughout this paper, the term increasing and decreasing mean non-decreasing and non-increasing respectively, and the measurability of sets and functions as well as the existence of expectations are often assumed without explicit mention. Any inequality between two vectors with finite or infinite dimensions means the inequalities component-wise. A product space of partially ordered sets is equipped with the component-wise partial ordering. 2 A Multi-component Shock Model with Opportunistic Maintenance In this section, we introduce a cumulative damage shock model for a system of components with a general opportunistic repair policy, and also discuss the difference of our model with the other repair models in the literature. Consider a system of m components that are subjected to random shocks, whose occurrences are governed by a non-explosive point process τ = {τ n, n 1} that has no multiplicities. The counting process of the shocks is given by A = {A(t), t 0}, where A(t) = max{n : τ n t}. Any arriving shock simultaneously inflicts random damages on all the components in the system. Suppose that for each 1 j m, the damage of the nth shock to component j is 3

7 X n,j, and then X n = (X n,1,..., X n,m ), n 1, is the vector of correlated random damages of the nth shock brought to various components in the system. Suppose that X n, n = 1, 2,..., are non-negative bounded random vectors, that is, there exist positive real numbers x j, 1 j m, such that for 1 j m, n 1, 0 X n,j x j, almost surely. (2.1) In other words, we focus on the situations such as normal wear and fatigue, and exclude huge damages resulted from disasters. Damages accumulate additively at various components, and component j, 1 j m, fails when its cumulative damage up to time t exceeds for the first time its design threshold, say d j > 0. Note that several components may fail at the same time. Suppose that the first failure occurs at components j J 1 {1,..., m}; that is, all the components j J 1 fail simultaneously. Thus the first failure time is given by where T 1,J1 = inf{t > 0 : D (1) j (t) > d j, j J 1 }, (2.2) D (1) A(t) j (t) = X n,j for t 0. n=1 At time T 1,J1, the failed components are correctively repaired, and the other components are preventively repaired, so that the damage levels of all components are reduced. Specifically, let Y 1 = (Y 1,1,..., Y 1,m ) be the vector of repair degrees at various components, with 0 < y 1,j Y 1,j 1, where 0 < y 1,j 1 is a real number, 1 j m. At T 1,J1, the combination of corrective and preventive maintenance reduces the cumulative damage level of component j to (1 Y 1,j )D (1) j (T 1,J1 ), 1 j m. All the repair times are negligible. After time epoch T 1,J1, the damage accumulation at component j, 1 j m, continues. The second failure occurs at components j J 2 {1,..., m}, and the failure time is modeled by where T 2,J2 = inf{t T 1,J1 : D (2) j (t) > d j, j J 2 }, (2.3) D (2) j (t) = (1 Y 1,j )D (1) j (T 1,J1 ) + A(t) n=a(t 1,J1 )+1 X n,j for t T 1,J1. In general, at the kth failure time T k,jk, the failed and operational components are all repaired instantaneously, and the cumulative damage level of component j is reduced to (1 Y k,j )D (k) j (T k,jk ), 1 j m. After that, the damage accumulations at various components continue. The epoch of the (k + 1)th failure occurred at components j J k+1 {1,..., m} is then defined by T (k+1),jk+1 = inf{t T k,jk : D (k+1) j (t) > d j, j J k+1 }, n 0, (2.4) 4

8 where D (k+1) j (t) = (1 Y k,j )D (k) j (T k,jk ) + A(t) n=a(t k,jk )+1 X n,j for t T k,jk. (2.5) Here and in the sequel, i n=i X n,j is understood as zero if i < i. Thus, prior to the repair at T k,jk, the damage accumulated at component j is D (k) j (T k,jk ), and after the repair at T k,jk, the damage at component j reduces to D (k+1) j (T k,jk ). After T k,jk, the damage accumulation and failure instances continue. Let Y = {Y n, n 1} be a stochastic process with Y n = (Y n,1,..., Y n,m ), with 0 < y n,j Y n,j 1, 1 j m, (2.6) where 0 < y n,j 1, n 1, 1 j s, are real numbers. Here Y n,j represents the degree of the random repair on component j at the nth failure epoch. We assume that A, X and Y are mutually independent. We also assume that all the repairs on failed components are successful; that is, we assume that x j d j + x j y n,j, 1 j m, n 1. (2.7) Clearly, (2.6) and (2.7) imply that after the repair at any failure epoch T n,jn, 1 j m, D (n+1) j (T n,jn ) = (1 Y n,j )D (n) j (T n,jn ) (1 y n,j )(d j + x j ) d j, almost surely. Therefore, a corrective repair on any failed component always restores that failed component back to working condition. In other words, we do not allow that failed components are left unattended for a period of time. Note that x j d j +x j is increasing in x j, and becomes very small when x j is substantially smaller than d j. Thus, this assumption (2.7) is especially applicable in the situations such as wear and fatigue. Let N(t, A, X, Y) denote the number of failure epochs for the system up to time t, namely, N(t, A, X, Y) = max{n : T n,jn t}. (2.8) Also let N j (t, A, X, Y) denote the number of failures occurred at component j up to time t, namely, N j (t, A, X, Y) = max{n : T j n t}, 1 j m, (2.9) where T j n is the nth failure time of component j, 1 j m. Note that every failure occurrence may consist of a set of simultaneous failures, and as such, m N(t, A, X, Y) N j (t, A, X, Y). j=1 5

9 The performance measures of interest for our model with the shock arrival process A = {A(t), t 0}, the shock damage process X = {X n, n 1} and repair degree process Y = {Y n, n 1} are following two counting processes. N(A, X, Y) = {N(t, A, X, Y), t 0}, and, (2.10) N(A, X, Y) = {(N 1 (t, A, X, Y),..., N m (t, A, X, Y)), t 0}. (2.11) Li and Xu (2001) discuss a cumulative damage shock model with no maintenance and obtain various dependence comparison results on the first failure times of the system components. Kijima (1991) study a univariate cumulative damage shock model with block preventive maintenance, where each planned maintenance reduces the damage level by 100(1 b)%, 0 b 1. Li and Xu (2004a) introduce a condition-based group maintenance policy for multi-component cumulative damage shock models and extend Kijima s model to the multivariate case. In contrast, our cumulative damage shock model introduced here combines corrective and preventive repairs, and implements opportunistic repairs on all the components at failure. It is also worth mentioning that Çinlar and Özekici (1987) introduce a replacement model for multi-component devices operating in random environments, in which component j, 1 j m, fails when its intrinsic age (random cumulative hazard) runs over its intrinsic lifelength. Shaked and Shanthikumar (1989) study several general multi-component replacement systems (some with preventive block replacements) operating in random environments. They assumed that the state of the component is unobservable and obtained structural properties using correlated random cumulative hazard processes of the components. However, we here assume that the damage level of each component is observable and preventive maintenance actions on the components are dependent upon the failure of other components. 3 Stochastic Comparison of Shock Models with Opportunistic Maintenance In this section, we use the coupling technique to obtain some a strong comparison result for shock models with opportunistic maintenance, which will be used later to analyze its temporal dependence properties. We also show that the failure counts at various components for the model with opportunistic repairs are stochastically less than that for the model without any opportunistic repair. 6

10 Let E be a partially ordered Polish space (that is, a complete, separable metric space) with a closed partial ordering. An E-valued random variable X is said to be stochastically larger than another E-valued random variable Y (denoted by X st Y ) if Ef(X) Ef(Y ) for all real-valued increasing functionals f on E. It can be shown that X st Y if and only if one can construct two random variables X and Y on the same probability space such that X and X have the same distribution, and Y and Y have the same distribution, and X Y, almost surely. (3.1) We refer the reader to Shaked and Shanthikumar (1994) for details on this and other properties on stochastic orders. Let D([0, )) be the Polish space of all right-continuous functions that have left-hand limits, equipped with the following partial order, f g if f(t) g(t), for all 0 t <, where f, g D([0, )). The space [D([0, ))] m is also Polish with the component-wise partial order. By a counting process we mean a stochastic process whose sample paths are non-negative, right-continuous step functions, starting at 0 and only increasing by jumps of size 1. Definition 3.1 Let N = {(N 1 (t),..., N m (t)), t 0} and N = {(N 1(t),..., N m(t)), t 0} be two multivariate counting processes. N is said to be stochastically larger than N, denoted by N st N, if Ef(N) Ef(N ) for all real-valued increasing functionals f: [D([0, ))] m R. Note that the ordering in Definition 3.1 can also be characterized by means of stochastic ordering of finite dimensional distributions of N and N. It follows from (3.1) that N st N if and only if one can construct M = {(M 1 (t),..., M m (t)), t 0}, M = {(M 1(t),..., M m(t)), t 0}, on the same probability space such that M and N have the same distribution, and M and N have the same distribution, and (M 1 (t),..., M m (t)) (M 1(t),..., M m(t)), for all t 0, almost surely. (3.2) For a detailed discussion on stochastic comparisons of counting processes, we refer the reader to Kijima, Li and Shaked (2000) and the references therein. Our first result states that increasing repair capacity and decreasing shock damage reduce the number of system failure occurrences. 7

11 Theorem 3.2 Let N(A, X, Y) and N(A, X, Ȳ) be two counting processes of failure instants for two shock models with the same shock arrival process A, but different shock damage processes X = {X n, n 1} and X = { X n, n 1}, and different repair degree processes Y = {Y n, n 1} and Ȳ = {Ȳ n, n 1}, respectively. If X st X, and Y st Ȳ then N(A, X, Y) st N(A, X, Ȳ). Proof. Since X st X, we have X st X. Also because Y st Ȳ, and X and Y are independent, and X and Ȳ are independent, we obtain that ( X, Y) st ( X, Ȳ). Because of (3.1), there exist two stochastic processes ( X, Y ) = {( X n, Y n), n 1} and ( X, Ȳ ) = {( X n, Ȳ n), n 1}, on the same probability space, such that ( X, Y ) and ( X, Y) have the same distribution, and ( X, Ȳ ) and ( X, Ȳ) have the same distribution, and ( X n, Y n) ( X n, Ȳ n), almost surely, for all n 1. That is, X n X n, Y n Ȳ n, almost surely, for all n 1. We can also construct a common shock arrival process A on the same probability space. Let S ( S) denote the shock model with shock arrival process A, random damage process X ( X ), and random repair degree process Y (Ȳ ). Let T n,jn and T n, Jn be, respectively, the nth failure epoch in S and S, at which several components may fail simultaneously. Let N(A, X, Y ) = {N(t, A, X, Y ), t 0} and N(A, X, Ȳ ) = {N(t, A, X, Ȳ ), t 0}, where N(t, A, X, Y ) = max{n : T n,jn t} and N(t, A, X, Ȳ ) = max{n : T n, Jn t}. (3.3) Following our construction, it is evident that N(A, X, Y) and N(A, X, Y ) have the same distribution and N(A, X, Ȳ) and N(A, X, Ȳ ) have the same distribution. N(A, X, Y ) and N(A, X, Ȳ ) are constructed in the same probability space. follows from (3.2) that it suffices to show that In addition, Then, it N(t, A, X, Y ) N(t, A, X, Ȳ ), for all t 0, almost surely, (3.4) or equivalently, by (3.3), we need to show for each n, T n,jn T n, Jn, almost surely. (3.5) Let D (k) (k) j (t) ( D j (t)) be the cumulative damage on the jth component of S ( S) at time t such that T k 1,Jk 1 t T k,jk ( T k 1, Jk 1 t T k, Jk ), 1 j m, k 1. We now establish, 8

12 via induction, (3.5) and the following: If T k 1, Jk 1 < T k,jk, for some k n, then, prior to the repair at T k,jk in S, D (k) j (T k,jk ) D (k) j (T k,jk ), for any 1 j m. (3.6) Initial Step: Since S receives more damage than S at any (common) shock arrival epoch, S experiences an earlier failure. Thus, T 1,J1 T 1, J1, where J 1, J 1 {1,..., m}. Note that T 0, J0 = 0 < T 1,J1, but we obviously have D (1) (1) j (T 1,J1 ) D j (T 1,J1 ) for any 1 j m, prior to the repair in S at the first failure time epoch. Inductive Step: Suppose that (3.5) and (3.6) hold for n 1 We need to prove them for n+1. There are two cases. 1. If T n+1,jn+1 T n, Jn, then clearly, T n+1,jn+1 T n+1, Jn Suppose now that T n+1,jn+1 > T n, Jn. From the inductive hypothesis (3.6), there exists a largest index 0 k n, such that, prior to the repair in S at T k,jk > T k 1, Jk 1, D (k) j (T k,jk ) D (k) j (T k,jk ), for any 1 j m. (3.7) From T k,jk to T n, Jn, S and S experience the same number (n k + 1) of failure occurrences (Figure 1). Arrange all these failure instants in an increasing order, T k,jk = S 1 S 2... S 2(n k+1) 1 S 2(n k+1) = T n, Jn. Let Z l ( Z l ) be the amount of damage S ( S) received from the environment over time interval (S l, S l+1 ], l = 1,..., 2(n k + 1) 1. Because of the coupling construction, S receives more damage than S at any (common) shock arrival epoch, Z l Z l, l = 1,..., 2(n k + 1) 1. Each damage Z l ( Z l ), reduced somewhat at repairs after S l, is added to the cumulative damage at T n, Jn. We now argue that after the repair in S at T n, Jn, the cumulative damage at any component in S is smaller than that in S. For this, we consider the following cases for S l < S l+1. T k,jk T n+1,jn+1 S S T k 1, Jk 1 Tn, Jn Tn+1, Jn+1 Figure 1: Case 2: Time points of interest 9

13 (a) Suppose that S l = T p,jp, k p n. Let q = min{i : T i, Ji T p,jp }. Since T k,jk > T k 1, Jk 1, we must have k q. Then, from the inductive hypothesis (3.5), q p. Thus, the contribution of Zl toward the cumulative damage on component j in S at T n, Jn, after the repair at T n, Jn, is given by n (1 Ȳ i,j) Z l, 1 j m. i=q While the contribution of Z l toward the cumulative damage in S at T n, Jn by n i=p+1 (1 Y i,j)z l, 1 j m. Since Y i,j Ȳ i,j for 1 j m and i 1, and Z l Z l, we have is given n (1 Ȳ i,j) l n (1 Y i,j)z l, 1 j m. i=q i=p+1 Thus, the contribution of Z l toward the cumulative damage on component j in S at T n, Jn, after the repair at T n, Jn, is smaller than the contribution of Z l toward the cumulative damage on component j in S at T n, Jn. (b) Suppose that S l = T q, Jq, k q n. Let p = min{i : T i,ji > T q, Jq }. Then, from the inductive hypothesis (3.5), q + 1 p. Thus, the contribution of Z l toward the cumulative damage in S at T n, Jn, after the repair at T n, Jn, is given by n i=q+1 (1 Ȳ i,j) Z l, 1 j m. While the contribution of Z l toward the cumulative damage in S at T n, Jn by n (1 Y i,j)z l, 1 j m. i=p Since Y i,j Ȳ i,j for 1 j m and i 1, and Z l Z l, we have n i=q+1 (1 Ȳ i,j) Z l n (1 Y i,j)z l, 1 j m. i=p is given 10

14 In either case, for any l = 1,..., 2(n k + 1) 1, the contribution of Zl toward the cumulative damage in S at T n, Jn, after the repair at T n, Jn, is smaller than the contribution of Z l toward the cumulative damage in S at T n, Jn. In addition, because of (3.7) and Y i,j Ȳ i,j for 1 j m and i 1, we have n (1 Ȳ i,j) i=k (k) D j (T k,jk ) n (1 Y i,j)d (k) j (T k,jk ), 1 j m. i=k Thus the contribution of D(k) j (T k,jk ) at T k,jk toward the cumulative damage on component j in S at T n, Jn, after the repair at T n, Jn, is smaller than the contribution of D (k) j (T k,jk ) at T k,jk toward the cumulative damage on component j in S at T n, Jn. Therefore, after the repair at T n, Jn, D (n+1) j ( T n, Jn ) D (n+1) j ( T n, Jn ), 1 j m. Since S receives more damage than S at any (common) shock arrival epoch, then for any t such that T n, Jn t min{t n+1,jn+1, T n+1, Jn+1 }, D(n+1) j (t) D (n+1) j (t), 1 j m. Thus, T n+1,jn+1 = inf{t T n,jn : D (n+1) j (t) > d j, j J n+1 } inf{t T n, Jn : In addition, D (n+1) j (T n+1,jn+1 ) The induction concludes the proof. D (n+1) j (t) > d j, j J n+1 } = T n+1, Jn+1. D (n+1) j (T n+1,jn+1 ), for any 1 j m. Theorem 3.2 can be used to obtain some bounds for the failure count N(t, A, X, Y), as the following example shows. Example 3.3 Consider a cumulative damage shock model S, in which, the shocks arrive at the system according to a Poisson process A with rate λ. The shock damage process X = {X n, n 1} consists of independent and identically distributed (i.i.d.) random vectors. The repair degree process Y = {Y n, n 1} satisfies the assumption (2.7). To find a lower bound for E(N(t, A, X, Y)), we introduce another cumulative damage shock model S, in which the shock arrival process is Poisson process A, the shock damage process is X, but the repair degree process Ȳ = {Ȳn, n 1} is given by Ȳ n,j = 1, for all n 1, 1 j m. That is, such a shock model replaces all the components upon any component failure. Since Y st Ȳ, then, by Theorem 3.2, we have for any t, N(t, A, X, Y) st N(t, A, X, Ȳ). 11

15 Note that N(A, X, Ȳ) = {N(t, A, X, Ȳ), t 0} is a renewal process with inter-failure times that have the same distribution as the following first passage time T = inf{t 0 : A(t) n=1 X n,j > d j, for some j}. (3.8) Let W(t) = A(t) n=1 X n, t 0. Since A(t) is a Poisson process, and X n s are i.i.d., then W = {W(t), t 0} is a Markov process starting at 0, with the following properties. 1. The sample paths of W are almost surely increasing. 2. The discrete-time Markov chain { n i=1 X i, n 1} has a stochastically monotone transition kernel, that is, for any increasing function f : R m R, is increasing in x. n n 1 E[f( X i ) X i = x] = Ef(x + X n ), i=1 i=1 It follows from Shaked and Shanthikumar (1987) that T defined in (3.8) has an IFRA property; that is, its average cumulative hazard rate is increasing. This IFRA property and a result from Barlow and Proschan (1981, page 171) imply that t E(T ) Thus, we obtain a lower bound, E(N(t, A, X, Ȳ)) t E(N(t, A, X, Y)) t E(T ) 1. E(T ) 1. Last and Szekli (1998) study a repairable system with general imperfect repairs, and obtain the comparison results for this model with IFR (Increasing Failure Rate) component lifetimes. However, under the Markovian assumption as in Example 3.3, the time to failure in our opportunistic maintenance model is only IFRA, which is weaker than IFR. Thus, the results in Last and Szekli (1998) can not be applied to the case in Theorem 3.2. In general, it is not true that increasing repair capacities at various components, as in Theorem 3.2, would reduce the vector of failure counts in the stochastic sense. However, since the component lifetimes in a cumulative damage shock model possess a natural multivariate positive aging property, it should be beneficial to implement opportunistic maintenance for multi-component cumulative damage shock models. In fact, we are able to show, in some cases, that the vector of failure counts in a system without opportunistic maintenance is 12

16 stochastically larger than the vector of failure counts in the system equipped with opportunistic maintenance. Consider a shock model in which Y n,j = 1 for all 1 j m, and n 1. For simplicity, let N j (t, A, X) denote the number of failure occurred at component j up to time t in this opportunistic replacement system S with shock arrival process A and shock damage process X. Let N j (t, A, X), 1 j m, denote the number of failure occurred at component j up to time t in a system S with the same shock arrival process A and same shock damage process X, but no opportunistic replacement. Note that this latter shock model S only replaces failed components with new ones upon failure. Theorem 3.4 Let N(A, X) = {(N 1 (t, A, X),..., N m (t, A, X)), t 0}, and N(A, X) = {( N 1 (t, A, X),..., N m (t, A, X)), t 0}. Then N(A, X) st N(A, X). Proof. We construct a common shock arrival process A and a common shock damage process X in the same probability space. Without loss of generality, let S ( S) denote the above-mentioned shock model with shock arrival process A, random damage process X, and opportunistic replacement (no opportunistic replacement). Let Tn j and T n j be, respectively, the nth failure epoch of component j in S and S, 1 j m. Let M j (t) = max{n : Tn j t} and M j (t) = max{n : T n j t}. (3.9) Thus, processes M = {(M 1 (t),..., M m (t)), t 0} and M = {( M 1 (t),..., M m (t)), t 0} are constructed in the same probability space. Following our construction, it is evident that N(A, X) and N(A, X) have the same distributions as M and M respectively. Then, it follows from (3.2) that it suffices to show that (M 1 (t),..., M m (t)) ( M 1 (t),..., M m (t)), for all t 0, almost surely, or equivalently, by (3.9), we need to show for each n, 1 j m, Tn j T n, j almost surely. (3.10) 13

17 We prove (3.10) via induction. Let D j (t) ( D j (t)) be the cumulative damage on the jth component of S ( S) at time t 0. Note that after the replacement of component j at any failure, the damage level reduces to zero. Initial Step: Suppose that all the components j J {1,..., m} fail simultaneously first. From our construction, we have T j 1 = T j 1, for all j J. We now show that T1 i T 1, i for i {1,..., m} J. At the first failure epoch, S replaces all the components with new ones, whereas S performs no opportunistic replacement for the operational components. Thus, after the replacement at the first failure epoch, D i (T j 1 ) = 0 D i ( T j 1 ) for any j J, and i {1,..., m} J. Since the both systems receive the same amount of damage at any shock epoch from the environment, we have D i (t) D i (t) for any 0 t min{t1, i T 1}, i i {1,..., m} J. Thus, T1 i = inf{t : D i (t) > d i } inf{t : D i (t) > d i } = T 1, i for i {1,..., m} J. Inductive Step: Suppose that (3.10) holds for n, and we now argue that it is also true for n + 1. For any 1 j m, if T j n+1 Tn j then T j n+1 Tn+1. j If T j n+1 > Tn, j then we have, after the replacement at Tn, j D j (Tn) j = 0 D j (Tn). j Since the both systems receive the same amount of damage at any shock epoch from the environment, we have D j (t) D j (t) for any Tn j t min{tn+1, j T n+1}. j Thus, T j n+1 = inf{t > Tn j : D j (t) > d j } inf{t > Tn j : D j (t) > d j } = T n+1. j for 1 j m. The induction concludes the proof. 4 Positive Dependence Properties of System Failures As we mentioned before, stochastic dependence introduced by opportunistic maintenance complicates the dependence nature of failure counts at various components. We are able to show in this section that, under certain conditions, the process of system failure occurrences is positively associated over time. We also present a positive dependence property of the failure counts at various components. 14

18 Let E be a partially ordered Polish space with a closed partial ordering. An E-valued random variable X is said to be positively associated if for any two functions f, g that are both increasing (or both decreasing) with respect to, E[f(X)g(X)] [Ef(X)][Eg(X)] or Cov(f(X), g(x)) 0. (4.1) Association of a random vector (X 1,..., X m ) implies the positively orthant dependence of (X 1,..., X m ), that is, m m E( f j (X j )) Ef j (X j ), (4.2) j=1 j=1 for any non-negative functions f 1,..., f m that are all increasing or all decreasing. This positively orthant dependence implies that for any x 1,..., x m, m P (X 1 > x 1,..., X m > x m ) P (X i > x i ), i=1 m P (X 1 x 1,..., X m x m ) P (X i x i ). i=1 The association was first introduced by Esary, Proschan and Walkup (1967), and by Fortuin, Kastelyn and Ginibre (1971), and its general theory was developed by Lindqvist (1988). The association enjoys many desirable properties and has been applied to various areas in probability and statistics. The following basic properties, taken from these papers, can be easily verified. Theorem 4.1 Let E 1 and E 2 be partially ordered Polish spaces. Let X be an E 1 -valued random variable and Y be an E 2 -valued random variable. 1. If X is associated, (Y X = x) is associated for all x, and E[f(Y ) X = x] is increasing (or decreasing) in x for all increasing functional f, then Y is associated. 2. If X is associated and f : E 1 E 2 is increasing (or decreasing), then f(x) is associated. 3. If X is associated, Y is associated, and X and Y are independent, then (X, Y ) is also associated. 4. Let Z = {Z n, n 1} is a discrete-time process of E 1 -valued random variables. Z is associated if and only if Z is associated in time, that is, for any n, (Z 1,..., Z n ) is associated. 15

19 It follows from Theorem 4.1 that in order to obtain the association property, we first need to establish some monotonicity properties of the shock model with opportunistic maintenance. Theorem 3.2 describes the monotonicity properties of our opportunistic maintenance model with respect to the shock damage and repair degree, and our next result gives the monotonicity property with respect to the shock arrival process A = {A(t), t 0}. Let τ = {τ n, n 1} denote the sequence of the shock arrival times, that is, A(t) = max{n : τ n t}. (4.3) A sequence {a n, n 1} of real numbers is said to be increasing if a n a n+1, for all n 1. Lemma 4.2 Let {a n } and {ā n } be two increasing sequences of non-negative real numbers. Consider N({a n }, x, y) = [N(A, X, Y) τ n = a n, X n = x n, Y n = y n, n 1], N({a n }, x, y) = [N(A, X, Y) τ n = a n, X n = x n, Y n = y n, n 1], where x = {x n, n 1}, y = {y n, n 1} are two sequences of non-negative real vectors. If a n ā n for all n 1, then 1. N({a n }, x, y) N({ā n }, x, y). 2. N({a n }, x, y) N({ā n }, x, y). Proof. We only prove (1), and (2) can be similarly proved. Let S ( S) denote the shock model with deterministic shock arrival process {a n } ({ā n }), damage process x, and repair degree process y. Let T n,jn and T n, Jn be, respectively, the nth failure epoch in S and S, at which several components may fail simultaneously. Let N(t, {a n }, x, y) = max{n : T n,jn t} and N(t, {ā n }, x, y) = max{n : T n, Jn t}. (4.4) Note that T n,jn and T n, Jn are all deterministic and any failure in both systems can only occur at the shock arrival epochs. Suppose that the nth failure in S occurs at a kn, for some k n, we have, T n,jn = a kn, n 1. Since S receives the same amount of shock damage at a n as that S receives at ā n, and S implements the opportunistic repair of same degree at a n as that S implements at ā n, the nth failure in S must occur at ā kn. Thus for any n 1, T n,jn = a kn ā kn = T n, Jn. This and (4.4) imply that N(t, {a n }, x, y) N(t, {ā n }, x, y) for any t 0, 16

20 and thus (1) holds. Lemma 4.2 immediately implies the following theorem, which describes the impact of the shock arrival process A on the failure occurrences of our opportunistic maintenance model. Theorem 4.3 Let N(A, X, Y) and N(Ā, X, Y) be two counting processes of failure instants for two shock models with the same shock damage process X, and same repair degree process Y, but different shock arrival processes A = {A(t), t 0} and Ā = {Ā(t), t 0}, respectively. If A st Ā, then 1. N(A, X, Y) st N(Ā, X, Y). 2. N(A, X, Y) st N(Ā, X, Y). Proof. Again, we only prove (1), and (2) can be similarly proved using Lemma 4.2 (2). Let τ = {τ n, n 1} ( τ = { τ n, n 1}) denote the sequence of the shock arrival times of A (Ā). If A st Ā, then τ st τ. It follows from Lemma 4.2 (1) that N({a n }, x, y) is decreasing in {a n } with respect to the component-wise ordering, then we have, N(A, x, y) st N(Ā, x, y). Then (1) follows from unconditioning on X and Y. Consider in the remainder of this paper a shock model with opportunistic maintenance, in which the damage process X = {X n, n 1} is a sequence of independent and identically distributed random vectors, the repair degree process Y = {Y n, n 1} is also a sequence of independent and identically distributed random vectors, and shock arrival process A is a renewal process. Theorem 4.4 If both X n and Y n are associated, then N(A, X, Y) is associated in time; that is, for any 0 t 1 t 2... t n <, (N(t 1, A, X, Y),..., N(t n, A, X, Y)) is associated. Proof. Let τ = {τ n, n 1} be the sequence of shock arrival times of A. Let M(τ, X, Y) = (N(t 1, A, X, Y),..., N(t n, A, X, Y)), and let M({a n }, x, y) denote M(τ, X, Y) given that τ = {a n }, X = x, Y = y. Association of M(τ, X, Y) follows from three steps. 17

21 1. Since A is a renewal process, then τ n = n k=1 Z k where {Z n, n 1} is a sequence of independent and identically distributed random variables. From Theorem 4.1 (2) and (3), we obtain that (τ 1,..., τ n ) is associated for any n 1. It then follows from Theorem 4.1 (4) that τ is associated. From Lemma 4.2, M({a n }, x, y) is decreasing in {a n } with respect to the component-wise order. It follows from Theorem 4.1 (2) that M(τ, x, y) is associated. 2. Since X n is associated for all n, then it follows from Theorem 4.1 (3) and (4) that X is associated. From Theorem 3.2, M(τ, x, y) is stochastically increasing in x. Thus, by (1) we just proved above and Theorem 4.1 (1), we have M(τ, X, y) is associated. 3. Since Y n is associated for all n, then it follows from Theorem 4.1 (3) and (4) that Y is associated. From Theorem 3.2, M(τ, X, y) is stochastically decreasing in y. Thus, by (2) we just proved above and Theorem 4.1 (1), we have M(τ, X, Y) is associated. Using Lemma 4.2 (2), we can also establish a positive dependence property of failure counts at various components. Theorem 4.5 If both X n and Y n are vectors of independent random variables, then the vector of failure counts at various components satisfies the positive orthant dependence property, that is, for any t 0, m m E( f j (N j (t, A, X, Y))) Ef j (N j (t, A, X, Y)) (4.5) j=1 j=1 for any non-negative functions f 1,..., f m that are all increasing or all decreasing. Proof. Let τ = {τ n, n 1} be the sequence of shock arrival times of A. Let (N 1 (t, {a n }, x, y),..., N m (t, {a n }, x, y)) = [(N 1 (t, A, X, Y),..., N m (t, A, X, Y)) τ n = a n, X n = x n, Y n = y n, n 1]. It follows from Lemma 4.2 that (N 1 (t, {a n }, x, y),..., N m (t, {a n }, x, y)) is decreasing in {a n } with respect to the component-wise ordering. From the proof of Theorem 4.4, the renewal process τ is associated, and thus, by Theorem 4.1 (2), (N 1 (t, A, x, y),..., N m (t, A, x, y)) is associated, which implies that m m E( f j (N j (t, A, x, y))) Ef j (N j (t, A, x, y)) (4.6) j=1 j=1 18

22 for any non-negative functions f 1,..., f m that are all increasing or all decreasing. Let X n = (X n,1,..., X n,m ) and Y n = (Y n,1,..., Y n,m ) for n 1. Also let, for 1 j m, Z j = {X n,j, Y n,j, n 1}. Note that X n s are i.i.d, and Y n s are also i.i.d.. Since, for each n, both X n and Y n are vectors of independent random variables, we have Z 1,..., Z m are mutually independent. Let P Zj denote the probability measure induced by Z j, 1 j m, it then follows from (4.6) that m E( f j (N j (t, A, X, Y))) j=1 m j=1 [ Ef j (N j (t, A, x, y))p Zj (dz j )], where z j is a realization of Z j, 1 j m. Note that N j (t, A, X, Y) only depends A and Z j, 1 j m, we obtain that for 1 j m, Ef j (N j (t, A, x, y))p Zj (dz j ) = Ef j (N j (t, A, X, Y)), and thus (4.5) holds. In general, Theorem 3.2 can not be extended to the vector of failure counts at various components, and thus it is not clear whether (4.5) still holds if both X n and Y n are associated vectors. 5 Conclusions In this paper, we introduce a new multi-component cumulative damage shock model to study opportunistic maintenance for a system of stochastically dependent components. The components natural positive aging properties and the coupling method able us to show that if the shock damages are stochastically smaller, and the repair degrees are stochastically larger, then the number of failure occurrences is stochastically smaller. We also show that a shock model with opportunistic maintenance experiences less failures stochastically at various components than a similar shock model without such a maintenance. This finding is consistent with the general understanding that preventive maintenance should be beneficial if component lifetimes possess certain positive aging property. Furthermore, we show that the number of failure occurrences is positively correlated over time if the damages and repair degrees at various components are positively correlated in the sense of association. We also establish a spatial dependence property for the failure counts of various components. The results we obtained in this paper shed new light on understanding the complex structural behavior of opportunistic maintenance for the components operating in a common random environment. 19

23 References [1] Barlow, R. E. and F. Proschan (1981). Statistical Theory of Reliability and Life Testing, To Begin With, Silver Spring, MD. [2] Berg, M. (1976). Optimal replacement policies for two unit machines with increasing running costs. Stochastic Processes and their Applications [3] Block, H. W., Langberg, N., and Savits, T. H. (1990a). Maintenance comparisons: Block policies. Journal of Applied Probability 27, [4] Block, H. W., N. Langberg and T. H. Savits (1990b). Stochastic comparisons of maintenance policies. In Topics in Statistical Dependence (Eds. H. W. Block, A. R. Sampson, and T. H. Savits), IMS Lecture Notes-Monograph Series [5] Çinlar, E. and S. Özekici (1987). Reliability of complex devices in random environments. Probability in the Engineering and Informational Sciences 1, [6] Dekker, R., R. Wildeman and F. A. Van der Duyn Schouten (1997). A review of multicomponent maintenance models with economic dependence. Mathematical Methods of Operations Research [7] Esary, J. D. F., F. Proschan and D. W. Walkup (1967). Association of random variables, with applications. Ann. Math. Statist [8] Kijima, M. (1991). A cumulative damage shock model with imperfect preventive maintenance. Naval Research Logistics [9] Kijima, M., H. Li and M. Shaked (2000). Stochastic processes in reliability, Handbook of Statistics, Vol. 19, D. N. Shanbhag and C. R. Rao, eds [10] Fortuin, C.M., P.W. Kastelyn and J. Ginibre (1971). Correlation inequalities on some partially ordered sets. Commu. Math. Phys [11] Last, G. and Szekli, R. (1998). Stochastic comparison of repairable systems by coupling. Journal of Applied Probability 35, [12] Li, H. and M. Shaked (2003). Imperfect repair models with preventive maintenance. Journal of Applied Probability [13] Li, H. and S. H. Xu (2001). Stochastic bounds and dependence properties of survival times in a multi-component shock model. Journal of Multivariate Analysis

24 [14] Li, H. and S. H. Xu (2004a). A multivariate cumulative damage shock model with preventive maintenance. Technical Report, Washington State University. [15] Li, H. and S. H. Xu (2004b). On the coordinated random group replacement policy in multivariate repairable systems. Operations Research [16] Lindqvist, H. (1988). Association of probability measures on partially ordered spaces. Journal of Multivariate Analysis 26, [17] McCall, J. J. (1963). Operating characteristics of opportunistic replacement and inspection policies. Management Science, [18] Pham, H. and H. Wang (2000). Optimal (τ, T ) opportunistic maintenance of a k-outof-n: G system with imperfect PM and partial failure. Naval Research Logistics [19] Radner, R. and D. W. Jorgenson (1963). Opportunistic replacement of a single part in the presence of several monitored parts. Management Science, [20] Shaked, M. and J. G. Shanthikumar (1987). IFRA properties of some Markov processes with general state space. Mathematics of Operations Research [21] Shaked, M. and J. G. Shanthikumar (1989). Some replacement policies in a random environment. Prob. in the Eng. and Infor. Sciences [22] Shaked, M. and J. G. Shanthikumar (1994). Stochastic Orders and Their Applications, Academic Press. [23] Wang, H. (2002). A survey of maintenance policies of deteriorating systems. European Journal of Operational Research, [24] Zheng, X. and N. Fard (1991). A maintenance policy for repairable systems based on opportunistic failure rate tolerance. IEEE Transactions on Reliability