Abstract

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 -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 time-dependent solution.

1. 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 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]): For , the boundary conditions are prescribed, and we consider the usual initial condition where . The most interesting initial condition is Here ; represents the probability that the system is in state at time , ; represents the probability that at time the failed system is in state and has an elapsed repair time of , ; represents failure rate of an operating unit; represents common-cause failure rates from state to state 4, ; represents human-error rates from state to state 5, ; represents failure rate of standby unit; represents constant repair rate if the system is operating; represents repair-rate when the failed system is in state and has an elapsed repair time of for which satisfies ; , , , , and 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.

Hypothesis 1 . The system has a unique positive time-dependent solution

Hypothesis 2 . The time-dependent solution converges to the steady-state solution as time tends to infinity, where 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 -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 .

Assumption 1.1 (general assumption). The function is measurable and bounded such that exists and

2. The Problem as an Abstract Cauchy Problem

In this section, we rewrite the underlying problem as an abstract Cauchy problem on a suitable space , see [6, Definition II.6.1], also see [7, Definition II.6.1]. As the state space for our problem we choose It is obvious that is a Banach space endowed with the norm where .

For simplicity, let and we denote by the linear functionals Moreover, we define the operators on as respectively. To define the appropriate operator we introduce a “maximal operator” on given as

To model the boundary conditions () we use an abstract approach as in, for example, [9]. For this purpose we consider the “boundary space” and then define “boundary operators” and . As the operator we take and the operator is given by where .

The operator on corresponding to our original problem is then defined as Let , , , then the condition in is equivalent to (). The system of integrodifferential equations () can be written as the following equation: Let , then (2.11) is equivalent to the following operator equation: Thus, the above equations (), (), and () can be equivalently formulated as the abstract Cauchy problem If is the generator of a strongly continuous semigroup and the initial value in () satisfies , then the unique solution of (), (), and () is given by For this reason it suffices to study ().

3. Boundary Spectrum

In this section we investigate the boundary spectrum of . In order to characterise by the spectrum of a scalar -matrix, that is, or on the boundary space , we apply techniques and results from [10]. We start from the operator defined by We give the the representation of the resolvent of the operator needed below to prove the irreducibility of the semigroup generated by the operator .

Lemma 3.1. Let and set Then one has Moreover, if then where The resolvent operators of the differential operators are given by for .

Proof. A combination of [11, Proposition ] and [12, Theorem ] yields that the resolvent set of satisfies For we can compute the resolvent of explicitly applying the formula for the inverse of operator matrices; see [12, Theorem ]. This leads to the representation (3.4) of the resolvent of .
Clearly, knowing the operator matrix in (3.4), we can directly compute that it represents the resolvent of .

The following consequence is useful to compute the boundary spectrum of .

Corollary 3.2. The imaginary axis belongs to the resolvent set of that is,

The eigenvectors in can be computed as follows.

Lemma 3.3. For one has

Proof. If for , (3.11)–(3.14) are fulfilled, then we can easily compute that . 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 of the maximal operator decomposes, using [10, Lemma ], as

Moreover, since is surjective, is invertible for each , see [10, Lemma ]. We denote its inverse by and call it “Dirichlet operator.’’

We can give the explicit form of as follows.

Lemma 3.4. For each the operator has the form where

The operator can be computed explicitly for .

Remark 3.5. For the operator can be represented by the -matrix where

The operators and allow to characterise the spectrum and the point spectrum of . Before doing so we extend the given operators to the product as in [13, Section ].

Definition 3.6. (i) (ii), (iii)(iv), (v),

Remark 3.7. (i) Note that . For the resolvent of is (ii) The part of in is
Hence, can be identified with the operator .

The spectrum of can be characterise by the spectrum of operators on the boundary space as follows.

Characteristic Equation 3.8
Let . Then(i)(ii)If, in addition, there exists such that , then

Proof. Let us first show the equivalence We can decompose as We conclude from this that the invertibility of is equivalent to the invertibility of . From one can easily see that is invertible if and only if . This proves (3.26). Since by our assumption , it follows that . Therefore, is not empty. Hence we obtain from [6, Proposition IV.2.17] that since is the part of in . This shows (ii).
To prove (i) observe first that and have the same point spectrum, that is, Suppose now that . Then there exists such that . Since , we can compute This shows that .
Conversely, if we assume that , then there exists such that . From we conclude that and thus It follows from the decomposition (3.15) that and hence .

Using the Characteristic Equation we can show that is in the point spectrum of

Lemma 8.8. For the operator one has .

Proof. By the Characteristic Equation it suffices to prove that . Since where We can compute the th column sum of the -matrix as follows: This shows that is column stochastic, its transpose is row stochastic, and hence . Since , also holds. Therefore, by the Characteristic Equation we conclude that

Indeed, 0 is even the only spectral value of on the imaginary axis.

Lemma 8.9. Under Assumption 1.1, the spectrum of satisfies

Proof. For any , , we consider the resolvent equation where . This equation is equivalent to the following system of equations: Solving (3.42)–(3.45), we get Since By Assumption 1.1, we have It follows that , Let Then Since hence ,
Substituting into (3.39)–(3.41) we get the following system of equations: The matrix of the coefficient of the above system is denoted by Since This shows that the matrix is a diagonally dominant matrix, it follows that the determinant of the matrix is not equal to Therefore, system (3.52) has a unique solution Combining this with (3.46) we obtain that the equation has exactly one solution this yields .

4. 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. is a closed linear operator and is dense in .

Proof. We will prove the assertion in two steps.
We first prove that is closed. For any given We suppose that where That is, Then we obtain from Assumption 1.1 that Furthermore, This is equivalent to the following system of equations: Integrating both sides of last three equations from to we have This yields We know from the boundedness of that Furthermore, we have It follows from (4.8) that is absolutely continuous and Therefore, and From the above deduction we have This shows that hence is closed.
We now prove that is dense in . We define Then by [14] is dense in If we define then is dense in Therefore, in order to prove that is dense in it suffices to prove that is dense in Take any then there exist numbers such that for all ; that is, for here We introduce a function where It is easy to verify that Moreover This shows that is dense in

Lemma 8.2. is a dispersive operator.

Proof. For we may choose where If we define and for then we have By (4.20) and the boundary conditions on we obtain that This shows that is a dispersive operator.

Lemma 8.3. If , then

Proof. Let , then all the entries of are positive and we have We also have Using (4.22) and (4.23) we can estimate the th column sum as It follows from this that and thus also Therefore, Using the Characteristic Equation we conclude that for ,

From Lemmas 8.1, 8.2, and 8.3 and Phillips theorem (see [8, Theorem C-II 1.2]), we immediately obtain the following result.

Theorem 8.4. The operator generates a positive contraction -semigroup

We now characterize the well-posedness of () as follows; see [6, Corollary II.6.9].

Theorem 8.5. For a closed operator on the associated abstract Cauchy problem () is well-posed if and only if generates a strongly continuous semigroup on X.

From Theorem 8.5 and [6, Proposition II.6.2] we can state our main result.

Theorem 8.6. The system (), (), and () has a unique positive solution which satisfies ,

Proof. From Theorems 8.4, 8.5, and [6, Proposition II.6.2] we obtain that the associated abstract Cauchy problem () has a unique positive time-dependent solution which can be expressed as Let then satisfies the system of equations Since Using (4.27)-(4.28) we compute By (4.26) and (4.29) we obtain Therefore, This shows , for all

5. Asymptotic Stability of the Solution

In this section, we prove the asymptotic stability of the system by using -semigroup theory. First we express the resolvent of in terms of the resolvent of , the Dirichlet operator and the boundary operator , compare with [10].

Lemma 8.1. Let Then

Proof. Under our assumption, we see from the Characteristic Equation that and it follows from theProof that is invertible with inverse Using the explicit representation (3.28) for we compute Define . Then Since and since , it follows that

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

Lemma 8.2. The semigroup generated by is irreducible.

Proof. We know from [8, Definition C-III ] that the irreducibility of is equivalent to the existence of such that implies . We now suppose that and . Then also and It follows from theProof of Lemma 8.3 that for all . Hence the inverse of can be computed via the Neumann series We know from the form of that for every there exists such that the real number . Therefore, and by the form of we have This implies and hence is irreducible.

We now use the information obtained on and on to prove our main result on the asymptotic behaviour of the solutions of (). We first show that the semigroup is relatively weakly compact, see [6, Section V.2.b], and then we argue as in [16, 17]. Denote by According to [6, Corollary IV.(i)] we have the equality

To study the asymptotic behaviour of the semigroup the following compactness property is useful.

Lemma 8.3. The set is relatively compact for the weak operator topology. In particular, it is mean ergodic, that is, exists for all .

Proof. From and (5.12) it follows that there exists By the positivity of the semigroup we have Suppose that Since is a contraction semigroup and the norm on is strictly monotone, we obtain that which is a contradiction. Thus holds, and we can already assume that Since is irreducible, we obtain from [8, Proposition C-III 3.5(a)] that is a quasi-interior point of which implies that is dense in Let and take that is, . Then Since the order interval is weakly compact in , see [15, page 92], the orbit is relatively weakly compact in So far, we have shown that the orbits of elements are relatively weakly compact. Since the semigroup is bounded and is dense in , we know from [6, Lemma V.2.7] that is relatively weakly compact. By [6, Lemma V.2.7] we obtain that the semigroup is mean ergodic.

We can now show the convergence of the semigroup to a one-dimensional equilibrium point.

Theorem 8.4. The space can be decomposed into the direct sum where is one dimensional and spanned by a strictly positive eigenvector of . In addition, the restriction is strongly stable.

Proof. Since by Lemma 8.3 every has a relatively weakly compact orbit, is totally ergodic; see [18, Proposition ]. This implies that can be decomposed into where and and are invariant under ; see [6, Lemma ]. There exists such that ; see theProof of Lemma 8.3. Moreover, by the same construction as in theProof of [6, Lemma (i)], we find such that and Hence we obtain that and that is strictly positive, that is, ; see [8, Proposition C-III 3.5].
We now consider the generator of the restricted semigroup , where and . Clearly, is bounded and totally ergodic on ; that is, is mean ergodic for all . This implies that separates for all ; see [6, Theorem ]. By Lemma 8.9, thus for all . Hence it follows that Applying the Arendt-Batty-Lyubich-V Theorem, see [18, Theorem ], we obtain the strong stability of .

Combining Lemmas 8.8, 8.9, and 8.3 with Theorem 8.4 we obtain the following main result.

Corollary 8.5. There exists , such that for all where , .

Since the semigroup gives the solutions of the original system, we obtain our final result.

Corollary 8.6. The time-dependent solution of the system (), (), and () converges strongly to the steady-state solution as time tends to infinite, that is, , where and as in Corollary 8.5.

6. Conclusions

In this paper, we considered a repairable system involving primary as well as secondary failures. By using the -semigroup theory of bounded linear operator on Banach space, we proved that the corresponding dynamic operator generates positive contractive -semigroup and the system is well-posed. Furthermore, we proved the existence of positive solution of the system. Moreover, we obtained the result on the asymptotic stability of the solution of this system, that is, the convergence to a one-dimensional equilibrium. TheProof is based on the Arendt-Batty-Lyubich-V Theorem [18, Theorem ].

Acknowledgments

The author expresses his gratitude to Professor Rainer Nagel and Dr. Agnes Radl as well as the editor and referee for the constructive comments and valuable suggestions. The author also wishes to thank DAAD for the financial support. This work was supported by Doctor Foundation of Xinjiang University (no. BS080108) and the Natural Science Foundation of China (no. 10861011).