Bayesian Assessment of Availabilities and Unavailabilities of Multistate Monotone Systems

Dept. of Math. University of Oslo Statistical Research Report No 3

ISSN 0806–3842 June 2010

Bayesian Assessment of Availabilities and Unavailabilities of Multistate Monotone Systems

Bent Natvig · Jørund G˚ asemyr · Trond Reitan June 29, 2010

Abstract In the present paper we consider a multistate monotone system of multistate components. Following a Bayesian approach, the ambition is to arrive at the posterior distributions of the system availabilities and unavailabilities to the various levels in a fixed time interval based on both prior information and data on both the components and the system. We argue that a realistic approach is to start out by describing our uncertainty on the component availabilities and unavailabilities to the various levels in a fixed time interval, based on both prior information and data on the components, by the moments up till order mof their marginal distributions. From these moments analytic bounds on the corresponding moments of the system availabilities and unavailabilities to the various levels in a fixed time interval are arrived at. Applying these bounds and prior system information we may then fit prior distributions of the system availabilities and unavailabilities to the various levels in a fixed time interval.

These can in turn be updated by relevant data on the system. This generalizes results given in (Natvig and Eide 1987) considering a binary monotone system of binary components at a fixed point of time. Furthermore, considering a simple network system, we show that the analytic bounds can be slightly improved by straightforward simulation techniques.

Keywords availabilities·Bayesian assessment·multistate monotone systems

·unavailabilities

AMS 2000 Classification 62NO5, 90B25

B. Natvig·J. G˚asemyr

Department of Mathematics, University of Oslo, P.O. Box 1053 Blindern, N–0316, Oslo, Norway e-mail: [email protected]

e-mail: [email protected] T. Reitan

Department of Biology, University of Oslo, P.O. Box 1066 Blindern, N–0316, Oslo, Norway e-mail: [email protected]

1. Basic definitions and ideas

LetS={0,1, . . . , M} be the set of states of the system; theM+ 1 states rep- resenting successive levels of performance ranging from the perfect functioning levelM down to the complete failure level 0. Furthermore, letC={1, . . . , n}

be the set of components and in generalSi,i= 1, . . . , nthe set of states of the ith component. We require {0, M} ⊆Si ⊆S. Hence, the states 0 and M are chosen to represent the endpoints of a performance scale that might be used for both the system and its components. Note that in most applications there is no need for the same detailed description of the components as for the system.

Letxi, i= 1, . . . , n denote the state or performance level of theith compo- nent at a fixed point of time andx= (x₁, . . . , x_n). It is assumed that the state, φ, of the system at the fixed point of time is a deterministic function ofx; i.e.

φ=φ(x). Herextakes values in S₁×S₂× · · · ×S_n and φtakes values in S.

The functionφis called the structure function of the system. We often denote a multistate system by (C, φ).

We start by giving a series of basic definitions.

Definition 1 A system is a multistate monotone system (MMS) iff its structure functionφ satisfies:

(i) φ(x)is non-decreasing in each argument

(ii) φ(0) = 0 and φ(M) =M 0= (0, . . . ,0),M = (M, . . . , M).

Definition 2 The monotone system(A, χ)is a module of the monotone system (C, φ)iff

φ(x) =ψ[χ(x^A),x^Ac],

whereψis a monotone structure function andA⊆C.

Intuitively, a module is a monotone subsystem that acts as if it were just a supercomponent.

Definition 3 A modular decomposition of a monotone system(C, φ)is a set of disjoint modules{(Ak, χk)}^r_k=1 together with an organizing monotone structure functionψ, i.e.

(i) C=∪^r_i=1Ai whereAi∩Aj =∅ i6=j, (ii) φ(x) =ψ[χ1(x^A1), . . . , χr(x^Ar)] =ψ[χ(x)].

Making a modular decomposition of a system is a way of breaking it into a collection of subsystems which can be dealt with more easily.

In the following y<xmeans that y_i ≤x_i fori= 1, . . . , n, andy_i < x_i for somei.

Definition 4 Letφbe the structure function of an MMS and letj∈ {1, . . . , M}.

A vectorxis said to be a path vector to levelj iffφ(x)≥j. The corresponding path set is given by C_φ^j(x) ={i|xi≥1}. A minimal path vector to levelj is a path vectorx such thatφ(y)< j for all y<x. The corresponding path set is also said to be minimal.

Definition 5 Letφbe the structure function of an MMS and letj∈ {1, . . . , M}.

A vectorx is said to be a cut vector to levelj iffφ(x)< j. The corresponding cut set is given by D^j_φ(x) ={i|x_i < M}. A minimal cut vector to level j is a cut vectorxsuch thatφ(y)≥j for ally>x. The corresponding cut set is also said to be minimal.

We now consider the relation between the stochastic performance of the system (C, φ) and the stochastic performances of the components. Letτ be an index set contained in [0,∞).

Definition 6 The performance process of theith component,i= 1, . . . , n is a stochastic process{Xi(t), t∈τ}, where for each fixedt∈τ, Xi(t) is a random variable which takes values inSi. Xi(t)denotes the state of the ith component at time t. The joint performance process for the components {X(t), t ∈τ} = {(X1(t), . . . , X_n(t)), t∈ τ} is the corresponding vector stochastic process. The performance process of an MMS with structure functionφis a stochastic process {φ(X(t)), t∈τ}, where for each fixedt∈τ, φ(X(t))is a random variable which takes values inS. φ(X(t))denotes the system state at time t.

We assume that the sample functions of the performance process of a com- ponent are continuous from the right on τ. It then follows that the sample functions of {φ(X(t)), t ∈ τ} are also continuous from the right on τ. Now consider a time intervalI= [tA, tB]⊂[0,∞) and letτ(I) =τ∩I.

Definition 7 The marginal performance processes {Xi(t), t∈τ}, i= 1, . . . , n are independent in the time intervalI iff, for any integermand{t1, . . . , tm} ⊂ τ(I) the random vectors{X1(t1), . . . , X1(tm)}, . . . ,{Xn(t1), . . . , Xn(tm)} are independent.

Definition 8 The joint performance process for the components {X(t), t∈τ}

is associated in the time intervalIiff, for any integermand{t1, . . . , t_m} ⊂τ(I) the random variables in the array

X₁(t₁). . . X₁(t_m) ...

X_n(t₁). . . X_n(t_m)

are associated.

For an introduction to the theory of associated random variables we refer to (Barlow and Proschan 1975).

Definition 9 Let i= 1, . . . , n, j= 1, . . . , M. The availability,p^j(I)_i , and the unavailability, q^j(I)_i , to level j in the time interval I of the ith component are given by

p^j(I_i ⁾=P[X_i(s)≥j ∀s∈τ(I)] q_i^j(I)=P[X_i(s)< j ∀s∈τ(I)].

The availability,p^j(I)_φ , and the unavailability,q_φ^j(I), to leveljin the time interval I for an MMS with structure functionφare given by

p^j(I_φ ⁾=P[φ(X(s))≥j ∀s∈τ(I)] q^j(I)_φ =P[φ(X(s))< j ∀s∈τ(I)].

Let fori= 1, . . . , n, j= 0, . . . , M,

r^j(I)_i =p^j(I)_i −p^j+1(I)_i =P[ min

s∈τ(I)X_i(s) =j]

r^j(I)_φ =p^j(I)_φ −p^j+1(I)_φ =P[ min

s∈τ(I)φ(X(s)) =j].

Introduce fori= 1, . . . , nthe component availability and unavailability vectors

p^(I_i ⁾=

p^j(I_i ⁾ _j=1,...,M q^(I)_i =

q_i^j(I) _j=1,...,M , then×M component availability and unavailability matrices

P^(I)_φ =

p^j(I_i ⁾ _i=1,...,n

j=1,...,M

Q^(I)_φ =

q_i^j(I) _i=1,...,n

j=1,...,M

and the system availability and unavailability vectors

p^(I_φ⁾=

p^j(I_φ ⁾ _j=1,...,M q^(I)_φ =

q_φ^j(I) _j=1,...,M . Finally, introduce fori= 1, . . . , n the component parameter vectors

r^(I)_i =

r^j(I)_{i j=0,...,M} , then×(M+ 1) parameter matrix

R^(I)_φ =

r_i^j(I) _i=1,...,n

j=0,...,M

and the system parameter vector

r^(I)_φ =

r^j(I)_{φ j=0,...,M} .

WhenI= [t, t], we just drop I from the notation and use reliability and unreli- ability instead of respectively availability and unavailability.

Note that fori= 1, . . . , n

p^j(I)_i +q^j(I_i ⁾≤1 p^j(I)_φ +q^j(I)_φ ≤1. (1) Suppose now that we runK_iindependent experiments for componentireg- istering x^(k)_i (s) ∀s ∈τ(I) in the kth experiment, k = 1, . . . , K_i, i = 1, . . . , n.

Let forj= 1, . . . , M, i= 1, . . . , n

D_i^1j(I)=

Ki

X

k=1

I[x^(k)_i (s)≥j ∀s∈τ(I)] D_i^2j(I)=

Ki

X

k=1

I[x^(k)_i (s)< j ∀s∈τ(I)], and forj= 0, . . . , M, i= 1, . . . , n

D^j(I_i ⁾=

K_i

X

k=1

I[ min

s∈τ(I)x^(k)_i (s) =j].

Let forr= 1,2 D^r(I)_i = (D^r1(I_i ⁾, . . . , D^rM(I)_i ),D^r(I)= (D^r(I)₁ , . . . ,D^r(I_n ⁾). Fur- thermore, letD^(I)_i = (D_i^0(I), . . . , D_i^M(I)) andD^(I)= (D^(I)₁ , . . . ,D^(I)_n ).

Suppose also that we run K independent experiments on the system level registeringφ(x^(k)(s)) ∀s∈τ(I) in thekth experiment,k= 1, . . . , K. Let for j= 1, . . . , M

D_φ^1j(I)=

K

X

k=1

I[φ(x^(k)(s))≥j ∀s∈τ(I)]

D_φ^2j(I)=

K

X

k=1

I[φ(x^(k)(s))< j ∀s∈τ(I)], and forj= 0, . . . , M

D^j(I)_φ =

K

X

k=1

I[ min

s∈τ(I)φ(x^(k)(s)) =j].

Let forr= 1,2 D^r(I)_φ = (D^r1(I)_φ , . . . , D^rM(I)_φ ). Furthermore, letD^(I)_φ =

(D^0(I)_φ , . . . , D^M_φ ^(I)). When I = [t, t], we also drop I from the notation in all these data variables and data vectors.

Assume that the prior distribution of respectively the component availability and unavailability matrices, before running any experiment on the component level,π(P^(I)_φ ) andπ(Q^(I_φ⁾), can be written as

π(P^(I)_φ ) =

n

Y

i=1

πi(p^(I)_i ) π(Q^(I)_φ ) =

n

Y

i=1

πi(q^(I)_i ),

where for i = 1, . . . , n π_i(p^(I)_i ) is the prior marginal distribution of p^(I)_i and πi(q^(I)_i ) is the prior marginal distribution ofq^(I)_i . Hence, we assume that the components have independent prior component availability vectors and inde- pendent prior component unavailability vectors.

Note that before the experiments are carried through D^1j(I)_i is binomially distributed with parameters Ki and p^j(I)_i , and D_i^2j(I) binomially distributed with parameters Ki and q^j(I)_i . We assume that given P^(I)_φ , D^1(I)₁ , . . . ,D^1(I)_n are independent and that givenQ^(I)_φ ,D^2(I)₁ , . . . ,D^2(I)_n are independent. Hence, since we have assumed that the components have independent prior availabil- ity vectors, using Bayes‘ theorem the posterior distribution of the component availability matrix,π(P^(I)_φ |D^1(I)), can be written as

π(P^(I)_φ |D^1(I)) = π(D^1(I)|P^(I)_φ )π(P^(I)_φ ) Rπ(D^1(I)|P^(I)_φ )π(P^(I)_φ )dP^(I)_φ

= Qn

i=1πi(Di1(I)

|p^(I_i ⁾)πi(p^(I)_i ) Qn

i=1

Rπi(Di1(I)

|p^(I)_i )πi(p^(I_i ⁾)dp^(I)_i

=

n

Y

i=1

πi(Di1(I)

|p^(I)_i )πi(p^(I)_i ) πi(D^1(I)_i )

=

n

Y

i=1

πi(p^(I)_i |D^1(I)_i ),

whereπi(p^(I)_i |D^1(I)_i ) is the posterior marginal distribution ofp^(I)_i . Similarly, the posterior distribution of the component unavailability matrix, π(Q^(I)_φ |D^2(I)),

can be written as

π(Q^(I)_φ |D^2(I)) =

n

Y

i=1

πi(q^(I)_i |D^2(I)_i ).

Hence, the posterior component availability vectors are independent givenD^1(I) and the posterior component unavailability vectors are independent givenD^2(I). Now specialize I= [t, t] and assume that the component statesX₁, . . . , X_n are independent given P_φ. Since in this casep_φ is a function of P_φ, the dis- tribution,π(p_φ(Pφ)|D¹), can then be arrived at. Based on prior knowledge on the system level this may be adjusted to the prior distribution of the system reliability vector,π0(p_φ(Pφ)|D¹). Note that before the experiments are carried through D^1j_φ is binomially distributed with parameters K and p^j_φ. Including the dataD¹_φ, we end up with the posterior distribution of the system reliability vector,π(p_φ(Pφ)|D¹,D¹_φ), forj = 1, . . . , M.

When considering the case I= [t, t], we can instead ofPφ as well consider the parameter matrixRφ and assume that the components have independent prior vectorsr_i, i= 1, . . . , n, each having a Dirichlet distribution being the nat- ural conjugate prior. Furthermore, we assume that givenR_φ, D₁, . . . ,D_n are independent. Note that before the experiments are carried throughD_i is multi- nomially distributed with parametersK_iandr_i. Hence, the posterior marginal distribution of r_i given the data D_i, π_i(r_i|D_i), is Dirichlet. Furthermore, we have

π(Rφ|D) =

n

Y

i=1

πi(ri|Di).

Hence, life can be made easy at the component level. Assume that the compo- nent statesX1, . . . , Xn are independent givenRφ. The distribution,

π(rφ(Rφ)|D), is tried to be arrived at. If this is successful, based on prior knowl- edge on the system level, it is adjusted toπ0(rφ(Rφ)|D). This may be possible for simple systems. Note that before the experiments are carried through,Dφis multinomially distributed with parametersK andrφ. Hence, ifπ0(rφ(Rφ)|D), as in a dream, ended up as a Dirichlet distribution, the posterior distribution, π(rφ(Rφ)|D,Dφ), also would be a Dirichlet distribution. Do not forget this was a dream, also based on independent components given Rφ! So life will at least not be easy at the system level even whenI= [t, t].

For an arbitraryIp^(I)_φ is not a function of justP^(I)_φ , andq^(I)_φ not a function of just Q^(I)_φ . Hence, the approach above for the case I = [t, t] can not be extended.

In Section 2 we discuss two different approaches to the computation of pos- terior moments for component availabilities and unavailabilities, the first one generalizing an approach given in (Mastran and Singpurwalla 1978). In Section 3 we start out by describing our uncertainty on the component availabilities and unavailabilities to the various levels in a fixed time interval, based on both prior information and data on the components, by the moments up till order mof their marginal distributions. From these moments analytic bounds on the corresponding moments of the system availabilities and unavailabilities to the various levels in a fixed time interval are arrived at. Applying these bounds and prior system information we may then fit prior distributions of the system availabilities and unavailabilities to the various levels in a fixed time interval.

These can in turn be updated by relevant data on the system. This generalizes results given in (Natvig and Eide 1987) considering a binary monotone system of

binary components at a fixed point of time. In Section 4 we present a straight- forward simulation technique for obtaining bounds that improve the analytic bounds. Considering a simple network system, we show that the former bounds are slightly better than the latter.

2. Moments for posterior component availabili- ties and unavailabilities

Based on the experiences of the previous section we reduce our ambitions.

We start by specifying marginal momentsE{(p^j(I)_i )^s} andE{(q^j(I)_i )^s} for s= 1, . . . , m+K_i, j= 1, . . . , Mofπ_i(p^(I)_i ) andπ_i(q^(I)_i ), i= 1, . . . , n. We will first il- lustrate how these can be updated to give posterior momentsE{(p^j(I)_i )^s|D^1j(I)_i } and E{(q_i^j(I))^s|D_i^2j(I)} for s = 1, . . . , m, j = 1, . . . , M by using Lemma 1 in (Mastran and Singpurwalla 1978). Note that we loose information by condi- tioning onD_i^1j(I) instead ofD^1(I)_i andD^2j(I)_i instead ofD^2(I)_i . However, such improved conditioning does not work with this approach. We have

E{(p^j(I)_i )^s|D_i^1j(I)} ∝ Z 1

0

(p^j(I)_i )^s(p^j(I_i ⁾)^D1j(I)i (1−p^j(I)_i )^Ki−D1j(I)i π_i(p^j(I)_i )dp^j(I)_i

= Z 1

0

(p^j(I)_i )^s+Di1j(I)

Ki−D^1j(I)_i

X

r=0

K_i−D^1j(I_i ⁾ r

(−1)^r(p^j(I)_i )^rπ_i(p^j(I)_i )dp^j(I)_i . Hence,

E{(p^j(I)_i )^s|D^1j(I)_i }=

P^Ki−D^1j(I)_i r=0

K_i−D_i^1j(I) r

(−1)^rE{(p^j(I)_i )^{s+D1j(I)i +r}} P^Ki−D_i^1j(I)

r=0

Ki−D^1j(I)_i r

(−1)^rE{(p^j(I)_i )^{D1j(I)i +r}} .

A similar expression is valid forE{(q_i^j(I))^s|D_i^2j(I)}. The advantage of using this lemma is that it is applicable for general prior distributionsπi(p^j(I)_i ) and πi(q^j(I)_i ). A serious drawback is, however, that to arrive atE{(p^j(I)_i )^s|D^1j(I)_i } andE{(q^j(I)_i )^s|D^2j(I)_i }fors= 1, . . . , m, j= 1, . . . , Mone must specify marginal moments up till orderm+K_i of the corresponding prior distributionsπ_i(p^j(I)_i ) andπi(q^j(I_i ⁾). This may be completely unrealistic unlessKi is small.

A more realistic alternative is given in the following. Assume that the com- ponents have independent prior parameter vectorsr^(I)_i , i= 1, . . . , n, each having a Dirichlet distribution being the natural conjugate prior. Note that before the experiments are carried throughD^(I)_i is multinomially distributed with param- etersKi andr^(I)_i . Hence, the posterior marginal distribution ofr^(I)_i given the dataD^(I)_i , πi(r^(I)_i |D^(I)_i ), is Dirichlet. We now have

p^j(I)_i =

M

X

`=j

r^`(I)_i D^1j(I)_i =

M

X

`=j

D_i^`(I). (2)

Hence, the posterior marginal distribution ofp^j(I)_i given the dataD^1j(I)_i is beta.

Accordingly, we loose no information by conditioning onD_i^1j(I)instead ofD^1(I)_i .

We now assume that the prior distribution πi(p^j(I)_i ) is beta with parameters a^j(I)_i and b^j(I)_i . It then follows that π_i(p^j(I)_i |D_i^1j(I)) is beta with parameters a^j(I)_i +D^1j(I)_i andb^j(I)_i +Ki−D^1j(I_i ⁾. We have

E{(p^j(I)_i )^s|D_i^1j(I)}= Z 1

0

(p^j(I)_i )^s Γ(a^j(I)_i +b^j(I_i ⁾+Ki)

Γ(a^j(I)_i +D^1j(I)_i )Γ(b^j(I)_i +Ki−D^1j(I_i ⁾) (p^j(I)_i )^{aj(I)i +D1j(I)i −1}(1−p^j(I)_i )^{bj(I)i +Ki−D1j(I)i −1}dp^j(I)_i

= Γ(a^j(I)_i +b^j(I)_i +Ki)Γ(a^j(I)_i +D_i^1j(I)+s) Γ(a^j(I)_i +D_i^1j(I))Γ(a^j(I)_i +b^j(I)_i +Ki+s) Z 1

0

Γ(a^j(I)_i +b^j(I_i ⁾+Ki+s)

Γ(a^j(I)_i +D_i^1j(I)+s)Γ(b^j(I)_i +Ki−D_i^1j(I)) (p^j(I)_i )^{aj(I)i +D1j(I)i +s−1}(1−p^j(I)_i )^{bj(I)i +Ki−D1j(I)i −1}dp^j(I_i ⁾

= Γ(a^j(I)_i +b^j(I)_i +K_i)Γ(a^j(I)_i +D_i^1j(I)+s) Γ(a^j(I)_i +D_i^1j(I))Γ(a^j(I)_i +b^j(I)_i +K_i+s) ,

the integral being equal to 1 since we are integrating up a beta density with parameters a^j(I)_i +D^1j(I_i ⁾+s andb^j(I)_i +K_i−D^1j(I)_i . A similar expression is valid forE{(q_i^j(I))^s|D_i^2j(I)}.

3. Bounds for moments for system availabilities and unavailabilities

From the marginal momentsE{(p^{`(I)</sup><sub>i</sub> )<sup>s</sup>|D<sub>i</sub><sup>1`(I)}} and E{(q_i^{`(I)</sup>)<sup>s</sup>|D<sup>2`(I)}_i }, we de- rive lower bounds on the marginal momentsE{(p^j(I)_φ )^s|D^1(I)}and upper bounds on the marginal momentsE{(p^j(I)_φ )^s|D^2(I)}fors= 1, . . . , m, `= 1, . . . , M, j = 1, . . . , M. Similarly, we derive lower bounds on the marginal moments

E{(q_φ^j(I))^s|D^2(I)}and upper bounds on the marginal moments

E{(q_φ^j(I))^s|D^1(I)}. Note that we now do not necessarily need the marginal performance processes of the components to be independent in I. From these bounds and prior knowledge on the system level we may fitπ0(p^(I)_φ ) andπ0(q^(I)_φ ).

This may finally be updated to giveπ(p^(I)_φ |D^1(I)_φ ) andπ(q^(I)_φ |D^2(I)_φ ).

What we will concentrate on is how to establish the bounds on the marginal moments of system availabilities and unavailabilities from the marginal moments of component availabilities and unavailabilities. To simplify notation we drop the reference to the data (D^1(I),D^2(I)) from experiments on the component level.

Let us just for a while return to the caseI= [t, t] and assume that the component statesX₁, . . . , X_n are independent givenR_φ. Then we get

p^j_φ(R_φ) =X x

I[φ(x)≥j]

n

Y

i=1

r_i^xi.

Hence, since we assume that the components have independent prior vectorsr_i fori= 1, . . . , n, generalizing a result in (Natvig and Eide 1987), we get

E{p^j_φ(R_φ)}=X x

I[φ(x)≥j]

n

Y

i=1

E{r_i^xi}=p^j_φ(E{Rφ}), where

E{R_φ}=

E{r_i^j} _i=1,...,n

j=0,...,M

.

Accordingly, one can arrive at an exact expression for E{p^j_φ(R_φ)} for not too large systems. The point is, however, that there seems to be no easy way to extend the approach above to give exact expressions for higher order moments ofp^j_φ(Rφ). Hence, even whenI = [t, t] and component states are independent givenR_φ, one needs bounds on higher order moments ofp^j_φ(R_φ).

We need the following theorem proved in (Natvig and Eide 1987).

Theorem 1. If Y1, . . . , Yn are associated random variables such that 0≤Yi ≤ 1, i= 1, . . . , n, then for α >0

E{(

n

Y

i=1

Yi)^α} ≥

n

Y

i=1

E{(Yi)^α} (3)

E{

n

a

i=1

Y_i}=E{1−

n

Y

i=1

(1−Y_i)} ≤

n

a

i=1

E{Y_i}. (4)

In the special case of independent random variables Y₁ and Y₂ with 0 ≤Y_i ≤ 1, i= 1,2, we have

E{(

2

a

i=1

Yi)²} ≥

2

a

i=1

E{(Yi)²}. (5)

Proof: For the caseY₁, . . . , Y_n binary and α= 1, Equations (3) and (4) are proved in Theorem 3.1, page 32 of (Barlow and Proschan 1975). The proof, however, also works when 0 ≤ Y_i ≤1, i = 1, . . . , n. Using this fact and that non-decreasing functions of associated random variables are associated we get

E{(

n

Y

i=1

Y_i)^α}=E{

n

Y

i=1

(Y_i)^α} ≥

n

Y

i=1

E{(Y_i)^α},

and Equation (3) is proved. Equation (4) is proved in the same way. Finally, Equation (5) follows since

(

2

a

i=1

Y_i)²=

2

a

i=1

Y_i²+ 2(Y₁²−Y₁)(Y₂²−Y₂)≥

2

a

i=1

Y_i², and thatY₁and Y₂ are assumed to be independent.

Equation (5) reveals the unpleasant fact that a symmetry in Equations (3) and (4) seems only possible forα= 1, whenY1, . . . , Yn are not binary.

In the following, considering an MMS (C, φ), for j ∈ {1, . . . , M} let y^j_k = (y^j_1k, . . . , y^j_nk), k = 1, . . . , n^j_φ be its minimal path vectors to level j and z^j_k = (z_1k^j , . . . , z_nk^j ),k= 1, . . . , m^j_φbe its minimal cut vectors to level j and

C_φ^j(y^j_k), k= 1, . . . , n^j_φ andD_φ^j(z^j_k), k= 1, . . . , m^j_φ the corresponding minimal path and cut sets to levelj.

The three following theorems are taken from (Natvig 2011).

Theorem 2. Let (C, φ)be an MMS and let forj= 1, . . . , M

`^‘j(I)_φ (P^(I_φ⁾) = max

1≤k≤n^j_φ

Y

i∈C^j_φ(y^j_k)

p^y

j ik(I) i

`¯_φ^‘j(I)(Q^(I)_φ ) = max

1≤k≤m^j_φ

Y

i∈D_φ^j(z^j_k)

q^z

j ik+1(I)

i .

If the joint performance process of the system‘s components is associated inI, or the marginal performance processes of the components are independent inI, then

`^‘j(I)_φ (P^(I)_φ )≤p^j(I)_φ ≤ inf

t∈τ(I)

1−`¯_φ^‘j([t,t])(Q^([t,t])_φ )

≤1−`¯_φ^‘j(I)(Q^(I)_φ ) (6)

`¯_φ^‘j(I)(Q^(I)_φ )≤q_φ^j(I)≤ inf

t∈τ(I)

1−`^‘j([t,t])_φ (P^j([t,t])_φ )

≤1−`^‘j(I)_φ (P^j(I)_φ ). (7) Theorem 3. Let (C, φ)be an MMS and let forj= 1, . . . , M

`^∗∗j(I)_φ (P^(I)_φ ) =

m^j_φ

Y

k=1

a

i∈D_φ^j(z^j_k)

p^z

j ik+1(I) i

`¯^∗∗j(I)_φ (Q^(I)_φ ) =

n^j_φ

Y

k=1

a

i∈C_φ^j(y^j_k)

q^y

j ik(I)

i .

If the marginal performance processes of the components are independent inI, then

`^∗∗j(I)_φ (P^(I)_φ )≤p^j(I)_φ ≤ inf

t∈τ(I)

1−`¯^{∗∗j([t,t])}_φ (Q^([t,t])_φ )

≤1−`¯^∗∗j(I)_φ (Q^(I_φ⁾) (8)

`¯^∗∗j(I)_φ (Q^(I)_φ )≤q^j(I)_φ ≤ inf

t∈τ(I)

1−`^{∗∗j([t,t])}_φ (P^([t,t])_φ )

≤1−`^∗∗j(I)_φ (P^(I_φ⁾). (9) Theorem 4. Let (C, φ) be an MMS with modular decomposition given by Def- inition 3 and let forj= 1, . . . , M

B_φ^∗j(I)(P^(I)_φ ) = max

j≤k≤M[max[`<sup>‘k(I)</sup><sub>φ</sub> (P<sup>(I)</sup><sub>φ</sub> ), `^∗∗k(I)_φ (P^(I)_φ )]]

B¯_φ^∗j(I)(Q^(I)_φ ) = max

1≤k≤j[max[¯`<sub>φ</sub><sup>‘k(I)</sup>(Q<sup>(I)</sup><sub>φ</sub> ),`¯^∗∗k(I_φ ⁾(Q^(I_φ⁾)].

Introduce the followingr×M module availability and unavailability matrices P^(I)_ψ =

p^j(I)_χ

k k=1,...,r

j=1,...,M

Q^(I)_ψ = q^j(I)_χ

k k=1,...,r

j=1,...,M

, (10)

and correspondingly define the followingr×M matricesB^∗(I)_ψ (P^(I)_φ ),B¯^∗(I)_ψ (Q^(I)_φ ).

Assume the marginal performance processes of the components to be independent in the time intervalI. Then for j= 1, . . . , M

B_ψ^∗j(I)(B^∗(I)_ψ (P^(I)_φ ))≤B_ψ^∗j(I)(P^(I)_ψ )≤p^j(I)_φ

≤ inf

t∈τ(I)

1−B¯_ψ^∗j([t,t])(Q^([t,t])_ψ )

≤1−B¯^∗j(I)_ψ (Q^(I)_ψ )

≤1−B¯_ψ^∗j(I)( ¯B^∗(I)_ψ (Q^(I)_φ ). (11)

B¯_ψ^∗j(I)( ¯B^∗(I)_ψ (Q^(I)_φ ))≤B¯_ψ^∗j(I)(Q^(I_ψ⁾)≤q_φ^j(I)

≤ inf

t∈τ(I)

1−B_ψ^∗j([t,t])(P^([t,t])_ψ )

≤1−B^∗j(I)_ψ (P^(I)_ψ )

≤1−B^∗j(I)_ψ (B^∗(I)_ψ (P^(I)_φ )). (12) We are now ready to establish the bounds for the moments of system avail- abilities and unavailabilities. Introduce the m×n×M arrays of component availability and unavailability moments

E{(P^(I)_φ )^m}=

E{(p^j(I)_i )^s} _s=1,...,m

i=1,...,n

j=1,...,M

(13)

E{(Q^(I)_φ )^m}=

E{(q^j(I)_i )^s} _s=1,...,m

i=1,...,n

j=1,...,M

. (14)

Theorem 5. Let (C, φ) be an MMS. Assume that respectively the component availability vectors p^(I)_i i = 1, . . . , n and the component unavailability vectors q^(I)_i i= 1, . . . , n are independent. Let

`^‘j(I)m_φ (E{(P^(I)_φ )^m}) = max

1≤k≤n^j_φ

Y

i∈C_φ^j(y^j_k)

E{(p^y

j ik(I) i )^m}

u^‘j(I)m_φ (E{(Q^(I)_φ )^m}) = min

1≤k≤m^j_φ m

X

r=o

m r

(−1)^r Y

i∈D^j_φ(z^j_k)

E{(q^z

j ik+1([I])

i )^r}

`¯_φ^‘j(I)m(E{(Q^(I)_φ )^m}) = max

1≤k≤m^j_φ

Y

i∈D^j_φ(z^j_k)

E{(q^z

j ik+1(I)

i )^m}

¯

u_φ^‘j(I)m(E{(P^(I)_φ )^m}) = min

1≤k≤n^j_φ m

X

r=o

m r

(−1)^r Y

i∈C^j_φ(y^j_k)

E{(p^y_i^{ikj ([I])})^r}.

If the joint performance process of the system‘s components is associated inI, or the marginal performance processes of the components are independent inI, then form= 1,2, . . .

`^‘j(I)m_φ (E{(P^(I)_φ )^m})≤E{(p^j(I)_φ )^m}

E{(p^j(I_φ ⁾)^m} ≤u^‘j(I)m_φ (E{(Q^(I_φ⁾)^m}) (15)

`¯_φ^‘j(I)m(E{(Q^(I)_φ )^m})≤E{(q_φ^j(I))^m}

E{(q^j(I)_φ )^m} ≤u¯_φ^‘j(I)m(E{(P^(I)_φ )^m}). (16) Proof: From Equation (6) we have

E{(p^j(I)_φ )^m} ≥E{ max

1≤k≤n^j_φ

Y

i∈C_φ^j(y^j_k)

(p^y

j ik(I)

i )^m} ≥E{ Y

i∈C_φ^j(y^j_k)

(p^y

j ik(I) i )^m}

= Y

i∈C_φ^j(y^j_k)

E{(p^y

j ik(I)

i )^m}, 1≤k≤n^j_φ,

having used the independence of the component availability vectors. Since the inequality holds for all 1≤k≤n^j_φ, the lower bound of Equation (15) follows.

Similarly from Equation (6)

E{(p^j(I)_φ )^m} ≤E{( min

1≤k≤m^j_φ

[1− Y

i∈D^j_φ(z^j_k)

q^z

j ik+1([I])

i ])^m}

≤ min

1≤k≤m^j_φ

E{(1− Y

i∈D^j_φ(z^j_k)

q^z

j ik+1([I])

i )^m}

= min

1≤k≤m^j_φ

E{

m

X

r=o

m r

(−1)^r Y

i∈D_φ^j(z^j_k)

(q^z

j ik+1([I])

i )^r}

= min

1≤k≤m^j_φ m

X

r=o

m r

(−1)^r Y

i∈D^j_φ(z^j_k)

E{(q^z

j ik+1([I])

i )^r},

having used the independence of the component unavailability vectors. Hence, the upper bound of Equation (15) is proved. The bounds of Equation (16) follow completely similarly from Equation (7).

Note that respectivelyp^j([t_i ^1,t1]) andp^j([t_i ^2,t2]), andq_i^j([t1,t1])andq^j([t_i ^2,t2])are dependent fort1∈τ(I),t2∈τ(I),t16=t2. Hence,

E(p^j_i|D^1(I))6=E(p^j_i|D¹) E(q^j_i|D^2(I))6=E(q_i^j|D²).

This means that we cannot apply the best upper bounds in Equations (6) and (7).

Theorem 6. Let (C, φ) be an MMS. Assume that respectively the component availability vectors p^(I)_i i = 1, . . . , n and the component unavailability vectors q^(I)_i i= 1, . . . , n are independent. Let

`^∗∗j(I)m_φ (E{(P^(I)_φ )^m}) =

m^j_φ

Y

k=1 m

X

r=o

m r

(−1)^r

Y

i∈D^j_φ(z^j_k) r

X

s=o

r s

(−1)^sE{(p^z

j ik+1(I)

i )^s}

u^∗∗j(I)1_φ (E{(Q^(I_φ⁾)¹}) =

n^j_φ

a

k=1

Y

i∈C_φ^j(y^j_k)

(1−E{q^y_i^{ikj ([I])}})

`¯_φ^∗∗j(I)m(E{(Q^(I)_φ )^m}) =

n^j_φ

Y

k=1 m

X

r=o

m r

(−1)^r Y

i∈C^j_φ(y^j_k) r

X

s=o

r s

(−1)^sE{(q^y

j ik(I) i )^s}

¯

u_φ^∗∗j(I)1(E{(P^(I)_φ )¹}) =

m^j_φ

a

k=1

Y

i∈D^j_φ(z^j_k)

(1−E{p^z

j ik+1([I])

i }).

If the marginal performance processes of the components are independent inI, then form= 1,2, . . .

`^∗∗j(I)m_φ (E{(P^(I)_φ )^m})≤E{(p^j(I)_φ )^m} (17) E{p^j(I)_φ } ≤u^∗∗j(I)1_φ (E{(Q^(I)_φ )¹}) (18)

`¯_φ^∗∗j(I)m(E{(Q^(I)_φ )^m})≤E{(q_φ^j(I))^m} (19)

E{q^j(I)_φ } ≤u¯_φ^∗∗j(I)1(E{(P^(I)_φ )¹}). (20) Proof: From Equation (8) we have

E{(p^j(I)_φ )^m} ≥E{(

m^j_φ

Y

k=1

a

i∈D^j_φ(z^j_k)

p^z

j ik+1(I)

i )^m}

≥

m^j_φ

Y

k=1

E{( a

i∈D^j_φ(z^j_k)

p^z

j ik+1(I)

i )^m},

having applied Equation (3). The random variables a

i∈D^j_φ(z^j_k)

p^z

j ik+1(I)

i , k= 1, . . . , m^j_φ,

are associated since independent random variables and non-decreasing functions of associated random variables are asssociated, having used the independence of the component availability vectors. Continuing the derivation we get

=

m^j_φ

Y

k=1

E{

m

X

r=o

m r

(−1)^r Y

i∈D^j_φ(z^j_k) r

X

s=o

r s

(−1)^s(p^z

j ik+1(I)

i )^s}