A MULTIVARIATE CUMULATIVE DAMAGE SHOCK

MODEL WITH BLOCK PREVENTIVE MAINTENANCE 1

Haijun Li

Department of Mathematics

Washington State University

Pullman, WA 99164

lih@math.wsu.edu

Susan H. Xu

Department of Supply Chain and Information Systems

Smeal College of Business Administration

The Pennsylvania State University

University Park, PA 16802

shx@psu.edu

July 2004

1Supported in part by the NSF grant DMI 9812994.

Abstract

This paper is concerned with the dependence structure of a multi-component, cumulative damage

shock model, in which a system of components is subject to correlated shock damages and group

repairs. A component fails when its cumulative damage exceeds a given threshold, and is then

replaced by a new component. We incorporate a block maintenance schedule that simultaneously

performs imperfect repairs at various components according to a predetermined time table.

Using the coupling method, we show that if the multivariate shock damage and imperfect repair

processes are, respectively, stochastically larger and smaller, then the multivariate failure process

(i.e., the failure counts of various components accumulated over time) are stochastically larger.

This result allows us to show that if both correlated shock damages and group imperfect repairs

at various components are more orthant dependent, then the failure counts of various components

are also more orthant dependent. An application to a multi-component system subject to

simultaneous shock damages and simultaneous repairs is also given.

Key words and phrases: Multi-component cumulative damage shock model, system mainte-

nance, conditional-based imperfect repair, coupling, stochastic comparison, orthant dependence,

association, thinning.

1 Introduction

This paper is concerned with the dependence structure of a multi-component, cumulative damage

shock model, in which a system of components is subject to correlated shock damages and

conditional-based group repairs.

Many components or systems age over time, and may fail during their operations. Various

preventive maintenance policies are protocols followed in order to reduce unexpected, costly

incidents of component or system failures. In the single-component setting, the age replacement

policy is perhaps the simplest preventive maintenance policies where a component is replaced

at failure (unplanned replacement) or replaced when its age reaches some fixed value (planned

replacement). The block replacement policy is another widely used preventive maintenance policy

in the single component system where the component is replaced at failure and also at fixed

times zn, 0 < z1 < z2 < . . ., limn→∞ zn = ∞ (planned replacement). Apart from the basic

age and block replacement policies, many other replacement and repair policies for the single

component system have been discussed in the literature. One repair policy is known as the

minimal repair policy, where a component, upon failure, is only minimally repaired; that is,

the component is brought back to the working condition, but it is only as good then as it was

just before it failed. Brown and Proschan (1983) introduce the imperfect repair policy, under

which a component, upon failure, is replaced with probability α and minimally repaired with

probability 1 − α. Kijima (1991) studies a univariate cumulative damage shock model with an

imperfect maintenance policy under which each planned maintenance reduces the damage level

by 100 × (1 − b)%, 0 ≤ b ≤ 1. Last and Szekli (1998) introduce a general repair model using

Kijima’s notion of virtual age (Kijima 1989), and their model is flexible enough to comprise as

special cases many replacement or repair models with or without preventive maintenance. A

recent comprehensive survey on replacement and repair policies for the single-component system

can be found in Wang (2002) and the references therein.

Some group maintenance policies for systems of independent components are also discussed

in the literature. Okumoto and Elsayed (1983) propose a corrective block replacement rule that

replaces all the failed components at every fixed time interval. Sheu (1991) investigates the

preventive block replacement rule, which replaces all the components at every fixed time units,

with immediate failure treatments. Archibald and Dekker (1996) incorporate an age-criterion

within the block replacement rule in the multi-component system, where a failed component is

immediately replaced and a preventive maintenance action is taken every fixed time interval at

which any component whose age exceeds a fixed threshold is replaced. Gertsbakh (1984) shows

that under certain assumptions, the optimal replacement policy for the multi-component system

1

is to replace all failed components as soon as the number of failed components reaches some

prescribed number m. Dekker, Van der Meer, Plasmeijer and Wildeman (1995) modify this m-

failure group replacement rule with an age-criterion, which keeps failed components unattended

for a certain time until m components are failed, and replaces then all the failed components

together with the non-failed components whose age has passed a critical threshold age. A survey

paper focusing on multi-component maintenance models with economic dependence can be found

in Dekker, Van der Duyn Schouten and Wildeman (1997).

To compare the effectiveness of various maintenance policies, stochastic comparisons on the

failure counts under age, block and other policies have been studied in the literature. It is

well-known (see Barlow and Proschan 1981, and Block, Langberg and Savits 1990a) that the

failure counts of components with NBU (New Better Then Used) lifetimes under either the age or

block replacement policy is stochastically smaller than its counterpart in the system without any

preventive maintenance. Shaked and Shanthikumar (1989) consider the multivariate repairable

system with minimal repairs at failures and block replacements at predetermined time instants,

and study the multivariate stochastic comparison on the failure counts at various components.

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. However, most of the studies in this area focus on stochastic comparison of

failure counts under different policies. For a multi-component system with correlated damages

(failure dependence) and coordinated group repairs (economic dependence), the failure counts

of various components are often highly correlated, and a dependence analysis on failure counts

can help us to coordinate repair efforts among the components to improve the overall system

performance under fixed repair capacities at various components. The focus of this paper is the

dependence comparison of failure counts for a multi-component cumulative damage shock model

with general block repairs.

Consider a system of components subject to random shocks over time, whose occurrences

are governed by a point process. An arriving shock simultaneously inflicts correlated damages

on all the components in the system. Damages accumulate additively at various components

and can be observed at any time, and a component fails when its cumulative damage exceeds

a given control limit. Every component that fails is immediately replaced with new one. We

incorporate the Block Imperfect Repair (BI) policy into our cumulative damage shock model.

BI prescribes, at each predetermined maintenance epoch, to simultaneously reduce the observed

damage level of a component to a degree based on its current damage level and a random repair

factor. The random repair factors of various components at each maintenance epoch can be

2

dependent due to the common operating environment. Our objective is to understand how

the dependence nature of correlated shock damages and correlated repair efforts caused by the

common environment affects the dependence structure and performance of the system.

Toward this end, we first study the stochastic structure of the system. We develop a coupling

technique and show that if the multivariate shock damage process and the multivariate imperfect

repair process are, respectively, stochastically larger and smaller, then the multivariate failure

count process is stochastically larger. Based on this, we then show that if shock damages and

simultaneous imperfect repairs are more dependent in the sense of orthant dependence order (see

Section 4 for definition), then the failure counts at various components are also more dependent

in the sense of orthant dependence order. With the Poisson shock arrival process, we further

show that if both simultaneous shock damages and simultaneous imperfect repairs are positively

orthant dependent, then the failure counts of various component are also positively orthant

dependent.

The multi-component repairable system studied by Shaked and Shanthikumar (1989) as-

sumes that the state of the component is unobservable. They obtain stochastic comparison

results using correlated random cumulative hazard processes of the components. In contrast,

we assume that the damage level of each component is observable and our imperfect repairs

are dependent upon the observed damage levels of components and environment factors. Our

‘conditional-based’ maintenance advocates the philosophy that any block preventive repair ac-

tion should also depend on the component’s current condition. More importantly, our study

focuses on understanding the impact of dependence structure on system performance. Among

other things, our results imply that while the system performance can be stochastically improved

by increasing overall repair capacity (e.g., by stochastically increasing the repair factors), it can

be also enhanced by effectively coordinating the existing repair capacities at various components

(e.g., by increasing the dependence strength of the repair factors).

For brevity, throughout the paper we shall use the notation A|B|C to classify various uni-

variate and multivariate repairable systems (the reader is cautioned not to confuse our notation

with these conventionally employed in queueing models). Here the symbol in position A takes

on letters U and M , where U represents the Univariate repairable system and M the Multi-

variate repairable system. The symbol in position B denotes the corrective maintenance policy

for unplanned repairs, where M denotes for Minimal repairs, I for Imperfect repairs, and R for

Replacements (perfect repairs). Finally, the symbol in position C denotes the preventive policy

used for planned maintenance actions, where B represents the Block replacement schedule, A

represents the Age replacement policy, and ∅ means that the maintenance schedule is not incor-

3

porated. For example, the model of Shaked and Shanthikumar (1989) can be classified as the

M |M |B model, and our model, to be described in detail in the next section, can be classified as

the M |R|BI model, where BI stands for the block imperfect repair.

The paper is organized as follows. Section 2 introduces the M |R|BI model. Section 3

develops our coupling method and stochastic comparison results for the M |R|BI model. Section

4 details dependence comparison results for the the M |R|BI model, with an application to the

system of components with simultaneous shock damages and repairs. Throughout this paper,

the terms ‘increasing’ and ‘decreasing’ mean ‘non-decreasing’ and ‘non-increasing’, respectively.

The existence of expectations and measurability of functions and sets are assumed without

explicit mention.

2 A Multivariate Cumulative Damage Shock Model with Block

Imperfect Repairs

Consider a system of m components that are subjected to random shocks, whose occurrences are

governed by a point process τ = {τn, n ≥ 0}, where τn is the arrival time of the nth shock. 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 Xnj , and

then Xn = (Xn1, . . . , Xnm), n ≥ 1, is the vector of correlated random damages of the nth shock

brought to various components in the system. Suppose that Xn, n = 1, 2, . . ., are non-negative

random vectors, and also independent of the shock arrival process τ . Damages accumulate

additively at various components. Let Dj(t) be the cumulative damage of component j at time

t, 1 ≤ j ≤ m, t ≥ 0. At the nth shock arrival time τn, the cumulative damage of component j

increases from level Dj(τ−n ) before the shock to level Dj(τn) = Dj(τ−n ) + Xnj after the shock.

Clearly, the shock process (τ,X), where X = {Xn, n ≥ 1}, inflicts positive damage values to the

cumulative damage process {Dj(t), t ≥ 0}.

To keep the system in a good condition, some preventive maintenance is periodically sched-

uled. The Block Imperfect Repair Policy (BI) is a maintenance policy which, at each of prede-

termined maintenance epochs, simultaneously repairs each component and restores its damage

level to a certain degree. Specifically, let Z = {zn, n ≥ 1} be a schedule of maintenance times,

satisfying 0 < z1 < z2 < . . . with limn→∞ zn = ∞, and Y = {Yn, n ≥ 1} a stochastic process

with Yn = (Yn1, . . . , Ynm), where Ynj represents the impact of the random repair environment

on component j at time zn. We assume Y is independent of τ and X. At the nth repair time

zn, we reduce the cumulative damage of component j from level Dj(zn) to level gj(Dj(zn), Ynj),

4

j = 1, 2, . . . ,m, and after that the damage accumulation at component j continues. We assume

that gj(x, y) is increasing in x and decreasing in y, with gj(x, 0) = x and gj(0, y) = 0. As such,

the maintenance process (Z,Y) adds negative damage values to the cumulative damage process

{Dj(t), t ≥ 0}.

We assume that the first failure time of component j, denoted by T1j , occurs when its

cumulative damage level Dj(t) exceeds for the first time its design threshold, say dj > 0, 1 ≤j ≤ m. Therefore, T1j is given by

T1j = inf{t > 0 : Dj(t) > dj}, 1 ≤ j ≤ m. (2.1)

At time T1j , the failed component j is instantaneously replaced by a new similar component

(or perfectly repaired); that is, at time T1j , the value of the cumulative damage of component

j is set to 0: Dj(T1j) = 0. After time T1j , the (positive and negative) damage accumulation at

component j continues. The second failure time of component j occurs at time

T2j = inf{t > T1j : Dj(t) > dj}, 1 ≤ j ≤ m. (2.2)

In general, at each failure time Tnj , n ≥ 1, the failed component j is replaced by a new similar

component instantaneously and the damage level is set at Dj(Tnj) = 0 after the replacement.

The (n+ 1)st failure time is then defined by

T(n+1)j = inf{t > Tnj : Dj(t) > dj}, n ≥ 1, 1 ≤ j ≤ m. (2.3)

Let Nj(t, τ,X,Y) denote the number of failures of component j up to time t, namely,

Nj(t, τ,X,Y) = max{n : Tnj ≤ t}, 1 ≤ j ≤ m. (2.4)

The performance measure for the M |R|BI model is defined as the multivariate counting process

N(τ,X,Y) = {(N1(t, τ,X,Y), . . . , Nm(t, τ,X,Y)), t ≥ 0}. (2.5)

To gain understanding of our model, let us first consider some special forms of the mainte-

nance function gj(Dj(zn), Yn,j). Suppose

gj(Dj(zn), Ynj) = (1− Ynj)Dj(zn), (2.6)

then 0 ≤ Ynj ≤ 1 represents the degree of preventive repair performed on component j at time

zn. If Ynj = 1 for all 1 ≤ j ≤ m and n ≥ 1, then M |R|BI reduces to a multivariate cumulative

damage shock model M |R|R with preventive block replacement as proposed by Sheu (1991). If

Ynj = 0 for all 1 ≤ j ≤ m and n ≥ 1, then M |R|BI reduces to a multivariate cumulative damage

5

shock model M |R|∅ with no preventive maintenance. Li and Xu (2001a) study the M |R|∅ shock

model and obtain various dependence comparison results on the first failure times of components.

If Ynj = bj , where bj is a constant, thenM |R|BI becomes the multivariate extension of the U |R|Imodel studied by Kijima (1991), where each planned maintenance reduces the damage level by

100(1 − b)%, 0 ≤ b ≤ 1. If Ynj = 1 with probability αj and Ynj = 0 with probability 1 − αj ,

0 ≤ αj ≤ 1, j = 1, . . . ,m, then it becomes a multivariate extension of the univariate cumulative

damage shock model U |R|I (Brown and Proschan (1983)).

Notice that in the above special cases, the maintenance actions are independent of Dj(zn),

the damage level of component j when the maintenance is performed. The unique feature of

our imperfect repair function gj(Dj(zn), Ynj) is that it is conditional-based, i.e., it depends on

the observed damage level and the random repair factor Ynj which represents the impact of the

random repair environment on component j at time zn. For instance,

gj(Dj(zn), Ynj) = min{Dj(zn), fj(Ynj)}

or

gj(Dj(zn), Ynj) = max{Dj(zn)− fj(Ynj), 0}.

In the first case, fj(Ynj) is a decreasing function of Ynj and can be understood as a (random)

control limit such that component j is imperfectly repaired to the control limit if its damage

level exceeds the limit, or not repaired at all otherwise. In the second case, fj(Ynj) is an

increasing function and represents the repair resource allocated to component j to reduce its

damage level. Our block imperfect repair policy is in the same spirit of incorporating an age-

criterion in the preventive block replacement rule as suggested by Archibald and Dekker (1994),

except we incorporate a state-criterion in our preventive block repairs. It is worth emphasizing

that the random process Y, which is independent of the damage process D, is an important

and unique construct of our maintenance policy. We allow Yn to have dependent components,

whereby we can model the scenario that maintenance efforts across different components are

simultaneously affected by some common random environmental factors, such as spare parts,

repair equipment, and repair crew, among others. For example, suppose Y is the total random

repair capacity at time zn. If this capacity is evenly distributed among all m components,

then Ynj = Ym , j = 1, . . . ,m. Our model can realistically reflect various random phenomena in

the actual implementation of a maintenance policy, and thus allows us to make a probabilistic

assessment of the performance of a maintenance policy. Our block imperfect repair policy is an

extension of the coordinated random group replacement (CRGR) policy introduced by Li and

Xu (2004), who study the effect of dependency of CRGR on the component failure process.

6

3 Stochastic Orders of Failure Counts in the M |R|BI Model

In this section, we use the inductive coupling technique to obtain some stochastic comparison

results of the M |R|BI model, which will be used later to analyze its dependence structure.

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

of 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 Suppose that N = {(N1(t), . . . , Nm(t)), t ≥ 0} and N′ = {(N ′1(t), . . . , N

′m(t)), t

≥ 0} are 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 on

[D([0,∞))]m.

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 = {(M1(t), . . . ,Mm(t)), t ≥ 0} and 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

(M1(t), . . . ,Mm(t)) ≥ (M ′1(t), . . . ,M

′m(t)), for all t ≥ 0, almost surely. (3.2)

7

Here and in the sequel, any inequality of two vectors with finite or infinite dimension means

inequalities component-wise.

As we shall see later, dependence comparison ofM |R|BI stems from the structural properties

of τ , X and Y. We now establish these properties. We show in the next theorem that, with

other things being equal, the multivariate counting process N(τ,X,Y), defined in (2.5), becomes

stochastically smaller if damage process X becomes stochastically smaller, and the repair factor

process Y becomes stochastically larger. Let S and S ′ denote theM |R|BI models with processes

(τ,X,Y) and (τ,X′,Y′), respectively.

Theorem 3.2 Let N(τ,X,Y) and N(τ,X′,Y′) be two multivariate counting processes arising

from systems S and S ′, respectively. Let processes τ , X and Y (τ , X′ and Y′) be independent.

If X ≥st X′, and Y ≤st Y′, then

N(τ,X,Y) ≥st N(τ,X′,Y′).

Proof. Since X and Y are independent, and X′ and Y′ are independent, then X ≥st X′

and Y ≤st Y′ imply that (−X,Y) ≤st (−X′,Y′). Following (3.1), we assume, without loss

of generality, that the damage-repair processes (X,Y) = {Xn,Yn, n ≥ 1} and (X′,Y′) =

{X′,Y′n, n ≥ 1} are constructed on the same probability space and Xn ≥ X′

n, and Yn ≤ Y′n, for

all n ≥ 1, most surely. We also assume that the arrival process τ is common in both systems,

and also constructed in the same probability space. Let Tnj and T ′nj be the nth failure time of

component j in S and S ′, respectively, for 1 ≤ j ≤ m. Let

Nj(t, τ,X,Y) = max{n : Tnj ≤ t} and Nj(t, τ,X′,Y′) = max{n : T ′nj ≤ t}. (3.3)

Then, it follows from (3.2) that it suffices to show that for all 1 ≤ j ≤ m, and t ≥ 0,

Nj(t, τ,X,Y) ≥ Nj(t, τ,X′,Y′), almost surely, (3.4)

or equivalently, by (3.3), we need to show for each n and j,

Tnj ≤ T ′nj , almost surely. (3.5)

Toward this end, we first observe that in the case that there is no failure between maintenance

epochs zn and zn+1, then, prior to the repairs at these maintenance epochs,

Dj(zn) ≥ D′j(zn) =⇒ Dj(zn+1) ≥ D′

j(zn+1), (3.6)

where Dj(t) and D′j(t) denote the damages accumulated at component j by time t in S and S ′,

respectively. This can be argued as follows. Immediately after the nth maintenance time zn, the

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damages at component j in S and S ′ reduce to gj(Dj(zn), Ynj) and gj(D′j(zn), Y ′nj), respectively.

Since Yn ≤ Y′n, almost surely, and Dj(zn) ≥ D′

j(zn), and gj is increasing in its first argument

and decreasing in its second argument, we have gj(Dj(zn), Ynj) ≥ gj(D′j(zn), Y ′nj). In addition,

after zn, S receives more shock damage than S ′ does at any shock arrival epoch, and thus (3.6)

holds true. We now use induction on n to show (3.5).

Initial Step: If T ′1j ≤ z1, then from the above construction, and the fact that Xn ≥ X′n for all

n ≥ 1, we have

T1j = inf{τk > 0 :k∑

n=1

Xnj > dj} ≤ inf{τk > 0 :k∑

n=1

X ′nj > dj} = T ′1j .

Now suppose T ′1j > z1. Then there must exist l ≥ 1 such that zl < T ′1j < zl+1. We consider two

cases.

Case 1. If T1j ≤ zl, then T1j ≤ zl < T ′1j , and the statement holds for n = 1.

Case 2. Suppose that T1j > zl. Since, prior to the repair at z1, Dj(z1) ≥ D′j(z1), and no

failures occur until time zl, we use (3.6) several times and obtain Dj(zl) ≥ D′j(zl), prior

to the repair at zl. After we perform maintenance at time zl, the damage levels in S and

S ′ satisfy

gj(Dj(zl), Ylj) ≥ gj(D′j(zl), Y

′lj).

After zl, S receives more damage than S ′ does at any shock arrival epoch, and so Dj(t) ≥D′

j(t) for zl ≤ t ≤ min{T1j , T′1j}. Thus,

T1j = inf{t > 0 : Dj(t) > dj} ≤ inf{t > 0 : D′j(t) > dj} = T ′1j .

This verifies our claim for n = 1.

Inductive Step: Suppose that Tnj ≤ T ′nj . We prove (3.5) for n+ 1. First, if T(n+1)j ≤ T ′nj then

T(n+1)j ≤ T ′(n+1)j . Suppose that T(n+1)j > T ′nj . Thus, after the replacement at T ′nj in S ′,

Dj(T ′nj) ≥ D′j(T

′nj) = 0. (3.7)

Let zan = min{zk : zk > T ′nj} be the earliest maintenance time after T ′nj . Also let zbn = max{zk :

zk < T ′(n+1)j} be the latest maintenance time prior to T ′(n+1)j . We again consider two cases.

Case 1. zan > zbn : This means that there is no maintenance epoch between T ′nj and T ′(n+1)j .

From (3.7) and the coupling construction, we haveDj(t) ≥ D′j(t) for T ′nj ≤ t ≤ min{T(n+1)j ,

T ′(n+1)j}. Thus,

T(n+1)j = inf{t > T ′nj : Dj(t) > dj} ≤ inf{t > T ′nj : D′j(t) > dj} = T ′(n+1)j .

9

Case 2. zan ≤ zbn : This means that there is at least one maintenance epoch between T ′nj and

T ′(n+1)j . If T(n+1)j < zbn , then T(n+1)j < T ′(n+1)j holds trivially. Suppose that T(n+1)j > zbn .

Because of (3.7), Dj(zan) ≥ D′j(zan), prior to the repair at zan . Then use (3.6) several

times and we have Dj(zbn) ≥ D′j(zbn), prior to the repair at zbn . Therefore,

gj(Dj(zbn), Ybnj) ≥ gj(D′j(zbn), Y ′bnj).

After zbn , S receives more shock damage than S ′ does at any shock arrival epoch, and so

Dj(t) ≥ D′j(t) for zbn ≤ t ≤ min{T(n+1)j , T

′(n+1)j}. Thus,

T(n+1)j = inf{t > zbn : Dj(t) > dj} ≤ inf{t > zbn : D′j(t) > dj} = T ′(n+1)j .

Therefore, T(n+1)j ≤ T ′(n+1)j and (3.5 ) holds for any n and j.

Intuitively, Theorem 3.2 means that enhancing maintenance efforts (e.g., by increasing the

capacity of repair facilities) pays in the multivariate cumulative damage shock model. This

finding is consistent with the general understanding that preventive maintenance is beneficial if

component lifetimes possess certain positive aging property. Shaked and Shanthikumar (1987)

show that the first passage time of an increasing, stochastically monotone Markov process is

IFRA (Increasing Failure Rate in Average). It follows from this result that the component

lifetimes in our M |R|BI model, under the Markovian assumption on the shock arrival and

damage processes, are IFRA. Using the notion of virtual age, Last and Szekli (1998) obtain

the comparison results for a general U |I|∅ model with IFR (Increasing Failure Rate, which is

stronger than IFRA) or DFR component lifetimes.

In the M |R|BI model, any component failure results in a replacement (complete repair).

It is evident that our coupling method used in the proof of Theorem 3.2 depends on complete

repairs at failure, and cannot be extended to the model with imperfect repairs at failure.

The following corollary of Theorem 3.2 describes the impact of shock arrival process τ on the

failure counts of the M |R|BI model. An increasing sequence al = {aln, n ≥ 1} of real numbers

is said to be a thinning of another increasing sequence a = {an, n ≥ 1} if

al ⊆ a. (3.8)

Here, the nth entry in al maps to the l(n)th entry in a, that is, aln = al(n), n = 1, 2, . . ., for a

strictly increasing function l on {1, 2, . . . , }.

Corollary 3.3 Let N(a,X,Y) and N(al,X,Y) be the multivariate counting processes arising

from two M |R|BI models with deterministic shock arrival times τ = a and τ l = al, respectively,

10

where al is a thinning of a as defined by (3.8). If shock damage vectors Xn are independent and

identically distributed (i.i.d.), Then

N(a,X,Y) ≥st N(al,X,Y). (3.9)

Proof. Consider two M |R|BI models N(a,X,Y) and N(a,X′,Y) with the same deterministic

shock arrival process a, the same repair factor process Y, but different damage processes X =

{Xn, n ≥ 1} and X′ = {X′n, n ≥ 1}, respectively, where

X′k =

Xl(n) if k = l(n) for some n.,

0 otherwise.

Obviously, X ≥st X′. By Theorem 3.2, we have N(a,X,Y) ≥st N(a,X′,Y). Since Xn’s are

i.i.d., N(a,X′,Y) and N(al,X,Y) have the same distribution. This proves our claim.

Note that it is easy to construct a counter-example to show that N(a,X,Y) is not stochas-

tically increasing in a with respect to the usual component-wise ordering.

4 Dependence Comparison of Failure Counts in the M |R|BI

Model

In this section, we study the dependence structure of the component failure process in our

M |R|BI model via dependence comparison methods. Our analysis is based on the monotone

properties of the component failure process developed in the previous section. We utilize the

orthant dependence comparison.

Definition 4.1 Let Z = (Z1, . . . , Zm) and Z′ = (Z ′1, . . . , Z′m) be two Rm -valued random vec-

tors. Z is said to be more orthant dependent than Z′, denoted by Z ≥od Z′, if

P (Z > z) ≥ P (Z′ > z) and P (Z ≤ z) ≥ P (Z′ ≤ z), for all z ∈ Rm.

Note that if Z ≥od Z′, then Zj and Z ′j must have the same distribution, 1 ≤ j ≤ m. There-

fore, the orthant dependence order compares the strength of dependence of random vectors.

The examples and applications of the orthant dependence order can be found in Tong (1980,

1990). The orthant dependence order enjoys some nice properties. In particular, the following

preservation property will be used in this section.

Lemma 4.2 If Z ≥od Z′, and if fj : R → R, j = 1, . . . ,m, are all increasing or all decreasing,

then

(f1(Z1), . . . , fm(Zm)) ≥od (f1(Z ′1), . . . , fm(Z ′m)).

11

For expositional simplicity, hereafter we shall assume that the damage process X = {Xn, n ≥1} is a sequence of independent random vectors, and the repair factor process Y = {Yn, n ≥ 1}is a sequence of independent random vectors (see Remark 4.4 for relaxation of this assumption).

For convenience, we denote X·j = {Xnj , n = 1, 2, . . .} as the jth component damage process of

X, and let x·j be a realization of X·j , j = 1, . . . ,m. Let Y·j and y·j be similarly defined for Y.

We can write the vector of failure counts N(t, τ,X,Y) = (Nj(t, τ,X·j ,Y·j), 1 ≤ j ≤ m). For

a fixed schedule, let S and S ′ denote the M |R|BI models with input processes (τ,X,Y) and

(τ,X′,Y′), respectively.

Theorem 4.3 Let N(t, τ,X,Y) and N(t, τ,X′,Y′) be the two failure count vectors up to time

t, arising from S and S ′, respectively. If Xn ≤od X′n and Yn ≤od Y′

n, n ≥ 1, then

N(t, τ,X,Y) ≤od N(t, τ,X′,Y′). (4.1)

Proof. Without loss of generality, we assume that (X,X′) and (Y,Y′) are independent, and

(X,Y) and (X′,Y′) are independent. We first prove that

N(t, τ,X,Y) ≤od N(t, τ,X,Y′). (4.2)

Let k = max{n : zn ≤ t} < ∞ be the number of maintenance epochs before time t. Note that

N(t, τ,X,Y) depends on Y only through (Y1, . . . ,Yk) with Yi = (Yi1, . . . , Yim), i = 1, 2, . . . , k.

We shall prove (4.2) by induction on k.

If k = 1, then conditioning on τ = a, X = x and Y1 = y1, it follows from Theorem 3.2 that

Nj(t,a,x·j , {y1j}) is decreasing in y1j for 1 ≤ j ≤ m. Therefore, Lemma 4.2 and unconditioning

on τ and X imply that the theorem holds for k = 1.

Assume that the theorem holds for k = n. We prove (4.2) for k = n + 1. Conditioning on

τ = a and X = x,

N(t,a,x, {Y1, . . . ,Yn+1}) = (Nj(t,a,x·j , {Y1j , . . . , Y(n+1)j}), 1 ≤ j ≤ m)

≤od (Nj(t,a,x·j , {Y ′1j , . . . , Y′n,j , Y(n+1)j}), 1 ≤ j ≤ m)

≤od (Nj(t,a,x·j , {Y ′1j , . . . , Y′n,j , Y

′(n+1)j}), 1 ≤ j ≤ m)

= N(t,a,x, {Y′1, . . . ,Y

′n+1}), (4.3)

where the first inequality follows from the induction hypothesis and the assumption that Y =

{Yn, n ≥ 1} forms a sequence of independent random vectors, and the second inequality follows

from Theorem 3.2, which states that N(t,a,x,y) is decreasing in y, and Lemma 4.2. Uncondi-

tioning on τ = a and X = x yields (4.2).

12

Similarly, let K = max{k : τk ≤ t} be the random number of shocks arrived before t.

We assume that the point process τ is non-explosive; that is, for any given t, K is finite with

probability 1. Clearly, N(t, τ,X,Y) depends on X only through (X1, . . . ,XK) with Xi =

(Xi1, . . . , Xim), i = 1, 2, . . . ,K. Conditioning on K = n, we can show, using a similar induction

technique as before, that

N(t,a, {X1, . . . ,Xn},y) ≤od N(t,a, {X′1, . . . ,X

′n},y), n = 1, 2, . . . ,

given that τ = a and Y = y. Unconditioning on τ and Y, we obtain that

N(t, τ,X,Y) ≤od N(t, τ,X′,Y). (4.4)

Finally, combining (4.2) and (4.4) yields (4.1).

Remark 4.4 1. Theorem 4.3 can be extended to the supermodular order, a dependence

notion that is stronger than the orthant order.

2. Theorem 4.3 still holds if the damage and repair processes follow the multivariate separable

process (Li and Xu 2001b, 2004). Roughly speaking, the multivariate separable process is

a multidimensional stochastic process {Zn, n ≥ 0} generated by a sequence of independent

random vectors {εn, n ≥ 0}, which, in our problem context, can be considered as the

common random environment the system experiences over time, with the property that

each component process {Znj , n ≥ 0} depends only on its own history and the jth element

of the environmental process, {εnj , n ≥ 0}, j = 1, . . . ,m.

3. If we treat Y as a policy decision, where increasing the orthant dependence strength of

Y means to increase coordination of the repair efforts across different components and

keep each marginal repair effort the same, then Theorem 4.3 implies, by coordinating the

repair efforts, we can increase the joint probability that the component failure counts are

below any given vector. This demonstrates that, while increasing overall repair capacity

(i.e., stochastically increasing Y) can improve system performance, enhancing coordination

among repair efforts (i.e., increasing the orthant dependence order of Y) can also improve

certain performance measures. Of course, increasing coordination may also have an adverse

effect on other system performance measures, for example, it increases the joint probability

that the component failure counts exceed any given vector, as seen from (4.1).

Theorem 4.3 also allows us to understand how the dependence properties of X and Y affect

N(t, τ,X,Y) when the shock arrival process τ is a Poisson process.

13

Definition 4.5 Let Z = (Z1, . . . , Zm) be an Rm-valued random vector. Also let ZI = (ZI1 , . . . ,

ZIm) be a random vector such that Zj and ZI

j have the same distribution for each j and

ZI1 , . . . , Z

Im are independent. We say Z = (Z1, . . . , Zm) is positively orthant dependent (POD) if

Z ≥od ZI .

Obviously, Z is POD if and only if for any non-negative functions fj , j = 1, . . . ,m, that are all

increasing or all decreasing,

Em∏

j=1

fj(Zj) ≥m∏

j=1

Efj(Zj). (4.5)

We will call ZI the independent counterpart of Z. Intuitively, POD is a positive dependence

concept because it means that the components of Z are more likely to simultaneously take on

large or small values, compared with its independent counterpart.

As we mentioned earlier, N(τ,X,Y) is not stochastically monotone in τ = {τn, n ≥ 1} with

respect to the usual component-wise ordering. Thus, in order to discuss the positive dependence

property of the M |R|BI model, we have to utilize the monotonicity of N(a,X,Y) with respect

to the thinning order of a, and a positive dependence property, known as association, of the

Poisson shock arrival process with respect to the thinning order.

Let E be a partially ordered Polish space with a closed partial ordering ≤. An E-valued

random variable Z 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(Z)g(Z)] ≥ [Ef(Z)][Eg(Z)] or Cov(f(Z), g(Z)) ≥ 0. (4.6)

Association of a random vector (Z1, . . . , Zm) trivially implies the positively orthant dependence

of (Z1, . . . , Zm). Thus, association is a stronger positive dependence notion than POD. The

association of random vectors was first introduced by Esary, Proschan and Walkup (1967), and

a general theory on association of E-valued random element was developed by Lindqvist (1988).

The association enjoys many desirable properties, and the following basic properties, taken from

Lindqvist (1988), can be easily verified.

Theorem 4.6 Let E1 and E2 be partially ordered Polish spaces. Let X be an E1-valued random

variable and Y be an E2-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 ∈ E1 for all increasing functional f : E2 → R, then Y is associated.

2. If X is associated and f : E1 → E2 is increasing (or decreasing), then f(X) is associated.

14

3. If X is associated, Y is associated, and X and Y are independent, then (X,Y ) is also

associated.

A function φ : R∞ → R is said to be increasing with respect to thinning if for any two

increasing sequences a = {an} and al = {aln} satisfying (3.8),

φ(al) ≤ φ(a) if al ⊆ a. (4.7)

A point process τ = {τn, n ≥ 1}, where τn is the nth arrival time, is said to be associated with

respect to the thinning order if for any two functions φ and ψ that are both increasing (or both

decreasing) with respect to the thinning order (4.7),

E[φ(τ)ψ(τ)] ≥ E[φ(τ)]E[ψ(τ)]. (4.8)

It was shown in Burton and Franzosa (1990, Theorem 4.7 and Example 5.1) that a Poisson

process τ is associated with respect to the thinning order.

Theorem 4.7 Consider the M |R|BI model with the Poisson shock arrival process τ and main-

tenance schedule Z. Let both the damage process X and the repair factor process Y be the

sequences of i.i.d. random vectors. If both Xn and Yn are POD, then the vector of failure

counts N(t, τ,X,Y) is also POD, that is,

N(t, τ,X,Y) ≥od NI(t, τ,X,Y).

Proof. Since both Xn and Yn are POD, we have Xn ≥od XIn and Yn ≥od YI

n, where XIn and

YIn are independent counterparts of Xn and Yn, respectively. From Theorem 4.3, we have

N(t, τ,X,Y) ≥od N(t, τ,XI ,YI),

where XI = {XIn, n ≥ 1}, and YI = {YI

n, n ≥ 1}. Now, we need to show that N(t, τ,XI ,YI) is

POD.

We first prove that, conditioning on YI = yI , N(t, τ,XI ,yI) is associated in the sense of

(4.6). For this, consider the following three facts.

1. The Poisson arrival process τ is associated with respect to the thinning order.

2. Given that τ = a, where a = {an}, N(t,a,XI ,yI) depends on XI only through (XI1, . . . ,X

Ik)

with k = max{n : an ≤ t}. Note that XIn is the independent counterpart of Xn, and

hence associated. From Theorem 4.6 (3), we have (XI1, . . . ,X

Ik) is associated. It follows

from Theorem 3.2 that N(t,a,xI ,yI) is increasing in (xI1, . . . ,x

Ik) with respect to the

component-wise order, then, by Theorem 4.6 (2), N(t,a,XI ,yI) is associated.

15

3. It follows from Corollary 3.3 that given that τ = a, N(t,a,XI ,yI) is stochastically in-

creasing in a with respect to the thinning order (3.8). That is, for any increasing function

f : Rm → R, Ef(N(t,a,XI ,yI)) is increasing in a with respect to the thinning order.

From Theorem 4.6 (1), we obtain that N(t, τ,XI ,yI) is associated, and hence POD. We have

E[m∏

j=1

fj(Nj(t, τ,XI·j ,y

I·j))] ≥

m∏j=1

E[fj(Nj(t, τ,XI·j ,y

I·j))],

for non-negative functions fj , 1 ≤ j ≤ m, that are all increasing or all decreasing. Unconditioning

on YI , and noting that YI·j , 1 ≤ j ≤ m, are independent, we have,

E[m∏

j=1

fj(Nj(t, τ,XI·j ,Y

I·j))] ≥

m∏j=1

E[fj(Nj(t, τ,XI·j ,Y

I·j))].

That is, N(t, τ,XI ,YI) is positively orthant dependent.

Remark 4.8 1. Theorem 4.7 can be generalized to certain shock arrival processes. For

example, Theorem 4.7 holds if τ = a is deterministic. In general, our theorem holds for

point process τ that is associated with respect to the thinning order.

2. Theorem 4.3 and Theorem 4.7 allow us the generate the product-form bounds for the joint

distribution and survival functions of N(t, τ,X,Y), primary interest for most practical

problems. For example, consider Xn =st (X,X, . . . ,X) and Yn =st (Y, Y, . . . , Y ), here

and in the sequel, ‘=st’ denotes the equality in distribution. Then the joint failure process

N(t, τ,X,Y) essentially reduces to the single component failure process whose distribution

is easy to obtain. Let X′n = (X ′

n1, . . . , X′nm) and Y′ = (Y ′n1, . . . , Y

′nm) be any POD vectors

such that X ′nj =st X and Y ′nj =st Y for all j. Thus,

Xn ≥od X′n, Yn ≥od Y′

n.

Let nk be the smallest integer in a vector n. Then, from Theorem 4.3, we have an upper

bound distribution function for N(t, τ,X′,Y′) as

P (N(t, τ,X,Y) ≤ n) = P (Nk(t, τ,X, Y ) ≤ nk) ≥ P (N(t, τ,X′,Y′) ≤ n).

For a Poisson arrival process τ , it follows from Theorem 4.7 that a lower bound of

N(t, τ,X′,Y′) is given by

P (N(t, τ,X′,Y′) ≤ n) ≥m∏

j=1

P (Nj(t, τ,X, Y ) ≤ nj).

The upper and lower bounds of the survival function of N(t, τ,X′,Y′) can be similarly

obtained.

16

We conclude the paper with an example for the system subject to simultaneous component

shock damages. Consider an M |R|BI system S that has a Poisson shock arrival process τ .

Suppose that shock damage vectors Xn =st X are i.i.d., where X has the following structure:

Components 1 to k1 are subject to the shock with identical damage X1, components k1 + 1

to k1 + k2 are subject to the shock with identical damage X2, and, in general, components∑j−1i=1 ki + 1 to

∑ji=1 ki are subject to the shock with identical damage Xj , j = 1, 2, . . . , s, and∑s

i=1 ki = m , where X1, . . . , Xs are i.i.d. nonnegative random variables. Similarly, suppose

that repair factor vectors Yn =st Y are i.i.d. and satisfy: Components∑j−1

i=1 li +1 to∑j

i=1 li are

subject to the repair with identical factor Yj , j = 1, 2, . . . , u, and∑u

i=1 li = m, where Y1, . . . , Yu

are i.i.d. nonnegative random variables.

We now introduce the concept of sequential packing and use it to construct system S ′ from

system S. Let (B1, . . . , Br) be a partition of {1, . . . , s} such that every element of Bi is smaller

than every element of Bi+1. For any k = (k1, . . . , ks), define,

k′i =

∑

j∈Bikj , i = 1, . . . , r,

0, i = r + 1, . . . , s.

We say k′ = (k′1, . . . , k′s) is a sequential packing of k, and denote this as k ≤pck k′.

Let k′ and l′ be two vectors of non-negative integers with s and u dimensions respectively.

Suppose k ≤pck k′ and l ≤pck l′. We construct S ′ that is the same as S, except that (a) its i.i.d.

shock damage vectors X′n =st X′ satisfy: Components

∑j−1i=1 k

′i + 1 to

∑ji=1 k

′i are subject to the

shock with identical damage Xj , j = 1, 2, . . . , r, where∑r

i=1 k′i = m; (b) its i.i.d. repair factor

vectors Y′n =st Y′ satisfy: Components

∑j−1i=1 l

′i + 1 to

∑ji=1 l

′i are subject to the repair with

identical factor Yj , j = 1, 2, . . . , r′ and∑r′

i=1 l′i = m. It follows from Theorem 3.4 of Shaked and

Shanthikumar (1997) that if k ≤pck k′, and l ≤pck l′, then

X ≤od X′ and Y ≤od Y′.

From this and Theorem 4.3, we have that, if k ≤pck k′, and l ≤pck l′, then the vector of failure

counts in S ′, denoted by (Nj(t,k′, l′), 1 ≤ j ≤ m), is more positively orthant dependent than the

vector of failure counts in S, denoted by (Nj(t,k, l), 1 ≤ j ≤ m). It follows immediately that

max1≤j≤m

{Nj(t,k, l)} ≥st max1≤j≤m

{Nj(t,k′, l′)}.

This means that, with the failure count for each component being stochastically identical in

both systems, the system with more identical shock damages and more identical repairs has,

stochastically, the smaller maximal count of component failures.

17

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