A Semigroup Approach to the System with Primary and Secondary Failures

We investigate the solution of a repairable parallel system with primary as well as secondary failures. By using the method of functional analysis, especially, the spectral theory of linear operators and the theory of C0-semigroups, we prove well-posedness of the system and the existence of positive solution of the system. And then we show that the time-dependent solution strongly converges to steady-state solution, thus we obtain the asymptotic stability of the timedependent solution.


Introduction
As science and technology develop, the theory of reliability has infiltrated into the basic sciences, technological sciences, applied sciences, and management sciences.It is well known that repairable parallel systems are the most essential and important systems in reliability theory.In practical applications, repairable parallel systems consisting of three units are often used.Since the strong practical background of such systems, many researchers have studied them extensively under varying assumptions on the failures and repairs; see 1-5 and their references.
The mathematical model of a repairable parallel system with primary as well as secondary failures was first put forward by Gupta; see 1 .This system is consisted of three independent identical units, which are connected in parallel.In the system, one of those units operates, the other two act as warm standby.If the operating unit fails, a warm standby unit is instantaneously switched into operation.The operating unit submits primary failures and secondary failures.The primary failures are the result of a deficiency in a unit while it is operating within the design limits.The secondary failures are the result of causes that stem from a unit operating in a conditions that are outside its design limits.Two important types International Journal of Mathematics and Mathematical Sciences of secondary failures are common cause failures and human error failures.A Common cause failure refers to the situation where multiple units fail due to a single cause such as fire, earthquake, flood, explosion, design flaw, and poor maintenance; see 2, 3 .A human error failure implies a failure of the system due to a mistake made by a human caused by such reasons as inadequate training, improper tools, and working in a poor lighting environment; see 4, 5 .There is one repairman available to repair these units.Once repaired, these units are as good as new.The failure rates of units and system are constant and independent.When the system is operating, the repairman can repair only one unit at a time.If all units fail, the entire system is repaired and checked before beginning further operation of these units.Unlike 4, 5 , the repair times in this system are arbitrarily distributed.
The parallel repairable system with primary and secondary failures can be described by the following equations see 1 : p j x, 0 f j x , j 3, 4, 5,

IC
where f j x ∈ L 1 0, ∞ .The most interesting initial condition is p 0 0 1, p i 0 0, i 1, 2, p j x, 0 0, j 3, 4, 5. IC 0 Here x, t ∈ 0, ∞ × 0, ∞ ; p i t represents the probability that the system is in state i at time t, i 0, 1, 2; p j x, t represents the probability that at time t the failed system is in state j and has an elapsed repair time of x, j 3, 4, 5; λ represents failure rate of an operating unit; λ c i represents common-cause failure rates from state i to state 4, i 0, 1, 2; λ h i represents human-error rates from state i to state 5, i 0, 1, 2; α represents failure rate of standby unit; μ represents constant repair rate if the system is operating; μ j x represents repair-rate when the failed system is in state j and has an elapsed repair time of x for j 3, 4, 5 which satisfies μ j x ≥ 0 j 3, 4, 5 ; λ c i i 0, 1, 2 , λ h i i 0, 1, 2 , λ, μ, and α are positive constants.
In 1 the author analyzed the system using supplementary variable technique and obtained various expressions including the system availability, reliability, and mean time of the failure using the Laplace transform.And then he discovered that the time-dependent availability decreases as time increases for exponential repair-time distribution under the following hypotheses.

1.1
The availability and the reliability depend on the time-dependent solution of the system.In fact, the author used the time-dependent solution in calculating the availability and the reliability.But the author did not discuss the existence of the time-dependent solution and its asymptotic stability, that is, the author did not prove the correctness of the above hypotheses.
It is well known that the above hypotheses do not always hold and it is necessary to prove the correctness.Motivated by this, we will show the well-posedness of the system and study the asymptotic stability of the time-dependent solution in this paper, by using the theory of strongly continuous operator semigroups, from 6-8 .First, we convert the model of the system into an abstract Cauchy problem in a Banach space.Next, we show that the operator corresponding to this model generates a positive contraction C 0 -semigroup.Furthermore, we prove that the system is well-posed and there is a positive solution for given initial value.Finally, we prove that the time-dependent solution converging to its static solution in the sense of the norm through studying the spectrum of the operator and irreducibility of the corresponding semigroup, thus we obtain the asymptotic stability of the time-dependent solution of this system.
In this paper, we require the following assumption for the failure rate μ j x .
Assumption 1.1 general assumption .The function μ j : R → R is measurable and bounded such that lim x → ∞ μ j x exists and

The Problem as an Abstract Cauchy Problem
In this section, we rewrite the underlying problem as an abstract Cauchy problem on a suitable space X, see 6, Definition II.6.1 , also see 7, Definition II.6.1 .As the state space for our problem we choose

2.1
It is obvious that X is a Banach space endowed with the norm p : where p p 0 , p 1 , p 2 , p 3 x , p 4 x , p 5 x t ∈ X.For simplicity, let

2.3
and we denote by ψ j the linear functionals 2.4 Moreover, we define the operators D j on W 1,1 0, ∞ as respectively.To define the appropriate operator A, D A we introduce a "maximal operator" A m , D A m on X given as

2.6
To model the boundary conditions BC we use an abstract approach as in, for example, 9 .For this purpose we consider the "boundary space" and then define "boundary operators" L and Φ.As the operator L we take and the operator Φ ∈ L D A m , ∂X is given by where p p 0 , p 1 , p 2 , p 3 x , p 4 x , p 5 x t ∈ D A m .The operator A, D A on X corresponding to our original problem is then defined as Let p j 0 p j 0, t , j 3, 4, 5, t ≥ 0, then the condition Lp Φp in D A is equivalent to BC .The system of integrodifferential equations R can be written as the following equation:

2.13
For this reason it suffices to study ACP .

Boundary Spectrum
In this section we investigate the boundary spectrum σ A ∩ iR of A. In order to characterise σ A by the spectrum of a scalar 3 × 3-matrix, that is, or on the boundary space ∂X, we apply techniques and results from 10 .We start from the operator A 0 , D A 0 defined by We give the the representation of the resolvent of the operator A 0 needed below to prove the irreducibility of the semigroup generated by the operator A.

3.5
The resolvent operators of the differential operators D j j 3, 4, 5 are given by R γ, D j p x e −γx− x 0 μ j ξ dξ x 0 e γx x 0 μ j ξ dξ p s ds 3.6 Proof.A combination of 11, Proposition 2.1 and 12, Theorem 2.4 yields that the resolvent set of A 0 satisfies ρ A 0 ⊇ S.

3.7
For γ ∈ S we can compute the resolvent of A 0 explicitly applying the formula for the inverse of operator matrices; see 12, Theorem 2.4 .This leads to the representation 3.4 of the resolvent of A 0 .
Clearly, knowing the operator matrix in 3.4 , we can directly compute that it represents the resolvent of A 0 .
The following consequence is useful to compute the boundary spectrum of A.

Corollary 3.2. The imaginary axis belongs to the resolvent set of
The eigenvectors in ker γ − A m can be computed as follows.
Lemma 3.3.For γ ∈ C, one has 14 Proof.If for p ∈ X, 3.11 -3.14 are fulfilled, then we can easily compute that p ∈ ker γ − A m .Conversely, condition 3.9 gives a system of differential equations.Solving these differential equations, we see that 3.11 -3.14 are indeed satisfied.
The domain D A m of the maximal operator A m decomposes, using 10, Lemma 1.2 , as Moreover, since L is surjective, is invertible for each γ ∈ ρ A 0 , see 10, Lemma 1.2 .We denote its inverse by and call it "Dirichlet operator." We can give the explicit form of D γ as follows.
International Journal of Mathematics and Mathematical Sciences Lemma 3.4.For each γ ∈ ρ A 0 , the operator D γ has the form where
International Journal of Mathematics and Mathematical Sciences 11 Remark 3.5.For γ ∈ ρ A 0 the operator ΦD γ can be represented by the 3 × 3-matrix where International Journal of Mathematics and Mathematical Sciences

3.21
The operators D γ and Φ allow to characterise the spectrum σ A and the point spectrum σ p A of A. Before doing so we extend the given operators to the product X × ∂X as in 13, Section 3 .
ii A 0 : Hence, A| X 0 can be identified with the operator A, D A .
The spectrum of A can be characterise by the spectrum of operators on the boundary space ∂X as follows.

3.25
Proof.Let us first show the equivalence

3.26
We can decompose γ − A as We conclude from this that the invertibility of γ − A is equivalent to the invertibility of I − BR γ, A 0 .From one can easily see that I − BR γ, A 0 is invertible if and only if 1 / ∈ σ ΦD γ .This proves 3.26 .Since by our assumption 1 / ∈ σ ΦD γ 0 , it follows that γ 0 ∈ ρ A .Therefore, ρ A is not empty.
Hence we obtain from 6, Proposition IV.2.17 that since A is the part of A in X 0 .This shows ii .
To prove i observe first that A and A have the same point spectrum, that is,

3.31
This shows that γ ∈ σ p A .

International Journal of Mathematics and Mathematical Sciences
Conversely, if we assume that γ ∈ σ p A , then there exists 0 / f ∈ D A m such that we conclude that f ∈ ker γ − A m and thus
Using the Characteristic Equation 3.8 we can show that 0 is in the point spectrum of A.
Lemma 8.8.For the operator A, D A one has 0 ∈ σ p A .
Proof.By the Characteristic Equation 3.8 it suffices to prove that 1 ∈ σ p ΦD 0 .Since

3.35
We can compute the jth column sum j 1, 2, 3 of the 3 × 3-matrix ΦD 0 as follows: International Journal of Mathematics and Mathematical Sciences
Indeed, 0 is even the only spectral value of A on the imaginary axis.

3.47
By Assumption 1.1, we have

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Substituting p j x into 3.39 -3.41 we get the following system of equations:

3.52
The matrix of the coefficient of the above system is denoted by

3.54
This shows that the matrix D is a diagonally dominant matrix, it follows that the determinant of the matrix D is not equal to 0. Therefore, system 3.52 has a unique solution p 0 , p 1 , p 2 .
Combining this with 3.46 we obtain that the equation aiId − A P Ψ has exactly one solution p 0 , p 1 , p 2 , p 3 x , p 4 x , p 5 x ∈ D A , this yields ai ∈ ρ A .

Well-Posedness of the System
The main gaol in this section is to prove the well-posedness of the system.In order to prove this, we will need some lemmas.

Lemma 8.1. A : D A → R A ⊂ X is a closed linear operator and D A is dense in X.
Proof.We will prove the assertion in two steps.
We first prove that A is closed.For any given x , p n 4 x , p

4.3
Then we obtain from Assumption 1.1 that x μ j x j 3, 4, 5.

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This is equivalent to the following system of equations: x f 5 x .

4.6
Integrating both sides of last three equations from 0 to β > 0, we have

4.8
We know from the boundedness of μ j x that

4.10
From the above deduction we have

4.11
This shows that A P 0 t F t , hence A, D A is closed.We now prove that D A is dense in X.We define Then by 14 E is dense in X.If we define there exists a number then H is dense in E. Therefore, in order to prove that D A is dense in X, it suffices to prove that D A is dense in H. Take any p x p 0 , p 1 , p 2 , p 3 x , p 4 x , p 5 x ∈ H, 4.14 then there exist numbers α i such that p i x 0, for all x ∈ 0, α i i 3, 4, 5 ; that is, p i x 0 for x ∈ 0, s , here 0 < s min{α 3 , α 4 , α 5 }.We introduce a function It is easy to verify that ϕ s x ∈ D A .Moreover This shows that D A is dense in H.

Lemma 8.2. A, D A is a dispersive operator.
Proof.For p ∈ D A , we may choose where If we define

By and the boundary conditions on p ∈ D A we obtain that
Ap, φ −a 0 p 0 μp International Journal of Mathematics and Mathematical Sciences

4.21
This shows that A, D A is a dispersive operator.
Proof.Let γ ∈ R, γ > 0, then all the entries of ΦD γ are positive and we have

4.22
We also have

4.24
It follows from this that ΦD γ < 1, and thus also

Asymptotic Stability of the Solution
In this section, we prove the asymptotic stability of the system by using C 0 -semigroup theory.First we express the resolvent of A in terms of the resolvent of A 0 , the Dirichlet operator D γ and the boundary operator Φ, compare with 10 .
Proof.Under our assumption, we see from the Characteristic Equation 3.8 that 1 / ∈ σ ΦD γ and it follows from theProof that γ − A is invertible with inverse Using the explicit representation 3.28 for I − BR γ, A 0 we compute and since A ∼ A| X 0 , it follows that R γ, A R γ .

5.6
The above representation for the resolvent of A 0 shows that it is a positive operator for γ > 0. This property is very useful in the following lemma to prove the irreducibility of the semigroup generated by A. For the notation and terminology concerning positive operators we refer to the books 8, 15 .

Lemma 8.2. The semigroup T t t≥0 generated by A, D A is irreducible.
Proof.We know from 8, Definition C-III 3.1 that the irreducibility of T t t≥0 is equivalent to the existence of γ > 0 such that 0 < p ∈ X implies R γ, A p 0. We now suppose that γ > 0 and 0 < p ∈ X.Then also R γ, A 0 p > 0 and ΦR γ, A 0 p > 0. It follows from theProof of Lemma 8.3 that ΦD γ < 1 for all γ > 0. Hence the inverse of Id ∂X − ΦD γ can be computed via the Neumann series

5.7
We know from the form of ΦD γ that for every i ∈ {1, 2, 3} there exists k ∈ N such that the real number ΦD γ k ΦR γ, A 0 p i > 0. Therefore, and by the form of D γ we have This implies R γ, A p 0, 5.10 and hence T t t≥0 is irreducible.
We now use the information obtained on σ A ∩ iR and on T t t≥0 to prove our main result on the asymptotic behaviour of the solutions of ACP .We first show that the International Journal of Mathematics and Mathematical Sciences semigroup is relatively weakly compact, see 6, Section V.2.b , and then we argue as in 16 Since the order interval −np, np is weakly compact in X, see 15, page 92 , the orbit {T t w : t ≥ 0} is relatively weakly compact in X.So far, we have shown that the orbits of elements w ∈ X p are relatively weakly compact.Since the semigroup T t t≥0 is bounded and X p is dense in X, we know from 6, Lemma V.2.7 that {T t : t ≥ 0} ⊆ L X is relatively weakly compact.By 6, Lemma V.2.7 we obtain that the semigroup T t t≥0 is mean ergodic.
We can now show the convergence of the semigroup to a one-dimensional equilibrium point.
Theorem 8.4.The space X can be decomposed into the direct sum where X 1 fix T t t≥0 ker A is one dimensional and spanned by a strictly positive eigenvector p ∈ ker A of A. In addition, the restriction T t | X 2 t≥0 is strongly stable.
Proof.Since by Lemma 8.3 every p ∈ X has a relatively weakly compact orbit, T t t≥0 is totally ergodic; see 18, Proposition 4.3.12 .This implies that X can be decomposed into

5.20
where ker A fix T t t≥0 and X 1 and X 2 are invariant under T t t≥0 ; see 6, Lemma V.4.4 .
There exists p ∈ ker A such that p > 0; see theProof of Lemma 8.3.Moreover, by the same construction as in theProof of 6, Lemma V.2.20 i , we find p ∈ X such that p > 0 and A p 0. Hence we obtain that dim ker A 1, 5.21 and that p is strictly positive, that is, p 0; see 8, Proposition C-III 3.5 .We now consider the generator A 2 , D A 2 of the restricted semigroup T 2 t t≥0 , where and T 2 t T t | X 2 .Clearly, T 2 t t≥0 is bounded and totally ergodic on X 2 ; that is, e −iat T t t≥0 is mean ergodic for all a ∈ R.This implies that ker A 2 −iat separates ker A 2 −iat for all a ∈ R; see 6, Theorem V.4.5 .By Lemma 8.9 ker A 2 − iat {0}, thus ker A 2 − iat {0} for all a ∈ R. Hence it follows that σ p A 2 ∩iR ∅.Applying the Arendt-Batty-Lyubich-V u Theorem, see 18, Theorem 5.5.5 , we obtain the strong stability of T 2 t t≥0 .
Combining Lemmas 8.8, 8.9, and 8.3 with Theorem 8.4 we obtain the following main result.
Hypothesis 1.The system has a unique positive time-dependent solution p x, t .The time-dependent solution p x, t converges to the steady-state solution p x as time tends to infinity, where p x, t p 0 t , p 1 t , p 2 t , p 3 x, t , p 4 x, t , p 5 x, t , p x p 0 , p 1 , p 2 , p 3 x , p 4 x , p 5 x .
Therefore, 1 / ∈ σ ΦD γ .Using the Characteristic Equation 3.8 we conclude that γ ∈ ρ A for γ ∈ R, γ > 0. The operator A, D A generates a positive contraction C 0 -semigroup T t t≥0 .We now characterize the well-posedness of ACP as follows; see 6, Corollary II.6.9 .For a closed operator A, D A on X the associated abstract Cauchy problem ACP is well-posed if and only if A, D A generates a strongly continuous semigroup on X.From Theorem 8.5 and 6, Proposition II.6.2 we can state our main result.The system R , BC , and IC 0 has a unique positive solution p x, t which satisfies p •, t 1, t ∈ 0, ∞ ., p 2 , p 3 x, t , p 4 x, t , p 5 x, t , then P t satisfies the system of equations , 17 .Denote by fix T t t≥0 :To study the asymptotic behaviour of the semigroup T t t≥0 the following compactness property is useful.The set {T t : t ≥ 0} ⊆ L X is relatively compact for the weak operator topology.In particular, it is mean ergodic, that is, Proof.From 0 ∈ σ p B and 5.12 it follows that there exists 0 / p ∈ fix T t t≥0 .By the positivity of the semigroup we have Suppose that |p| < T t |p|.Since T t t≥0 is a contraction semigroup and the norm on X is strictly monotone, we obtain that we can already assume that p > 0. Since T t t≥0 is irreducible, we obtain from 8, Proposition C-III 3.5 a that p is a quasi-interior point of X which implies that Let n ∈ N and take w ∈ −np, np , that is, −np ≤ w ≤ np.Then −np −nT t p ≤ T t w ≤ nT t p np ∀t ≥ 0.5.18