Well-posedness and Stability of the Repairable System with Three Units and Vacation
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Xiaoshuang Han
Abstract
The stability of the repairable system with three units and vacation was investigated by two different methods in this note. The repairable system is described by a set of ordinary differential equation coupled with partial differential equations with initial values and integral boundaries. To apply the theory of positive operator semigroups to discuss the repairable system, the system equations were transformed into an abstract Cauchy problem on some Banach lattice. The system equations have a unique non-negative dynamic solution and positive steady-state solution and dynamic solution strongly converges to steady-state solution were shown on the basis of the detailed spectral analysis of the system operator. Furthermore, the Cesáro mean ergodicity of the semigroup T(t) generated by the system operator was also shown through the irreducibility of the semigroup.
1 Introduction
Repairable system is not only an important class of systems discussed in reliability theory but also one of main objects studied in reliability mathematics. It consists of some components under the supervision of one or more repairmen. If a component fails at any time, it is immediately sent to the repair facility for repair.
The repairable system with three units is frequently encountered in practice. Many authors have discussed this repairable system where the repairmen always remain idle until the failed units present. For instance, Song and Deng analyzed the reliability of a three-unit system in a changing environment in [1]. Li et al. studied a repairable system with three units and two different repair facilities, and derived the explicit expressions of the state probabilities of the system and the steady-state reliability characteristics of the system in [2]. Kovalenko investigated a three-components system consisting of one master control element and two slave elements with priority serving by a single repair facility, and obtained the readiness factor and the average up-time in [3]. Hu et al. discussed a three-unit system with n failure modes and priority, and obtained explicit expressions of the steady-state probabilities of the system in [4].
The vacation conceptions (single vacation, multiple vacation, and delay vacation[5]) were first introduced into the repairable system by Su and Shi in [6]. The three-unit repairable system with vacations were discussed by Hu and his collaborators in recent years. Hu et al discussed the three-unit system with multiple vacations and priority in [7] and [8]. Thethree-unit system with single vacations was established and its reliability was then studied in [9]. They defined the repairable system and derived integro-differential equations, which describe the system precisely. By transforming the system equations into the first-order ordinary differential equations, they obtained the explicit formulas for the steady-state solution, the steady-state availability and mean up-time of the system.
However, the above authors investigated the repairable systems from the reliability theory only and mainly devoted to obtaining the quantities related to reliability of the system. They focused not at all on the problems of uniqueness, existence and asymptotic stability of the dynamic solutions to the system equations. We shall concern these questions of the repairable system with three units and single vacation in this paper. Indeed, the well-posedness and stability of the repairable systems have attracted the attention of many researchers since the end of last century. The corresponding properties of various repairable systems without vacation were obtained by the C0 semigroups theory of functional analysis in [10–19].
Though several different methods given by the authors of [10–19] were given to deal with the well-posedness and stability of the repairable systems, it is not easy to obtain the well-posedness and stability of the repairable system with vacation due to the complexity of the system equations. Analyzing the methods of [10–19] in length, we find out that the spectral analysis of the system operator plays important role in those methods. So we begin this paper with the spectral analysis. On the basis of the detailed spectral analysis of the system operator, we show that the C0 semigroup T(t) generated by the system operator is positive, contractive and irreducible. Thus the well-posedness of the system is obtained, that is the system equations have a unique non-negative dynamic solution. Moreover, we show the system equations have positive steady-state solution and dynamic solution strongly converges to steady-state solution by two different methods. One way follows from the extension theorem of Lyubich and Phong in [20]. Another way is based on the irreducibility of the semigroups.
The structure of this paper is organized as follows. In section 2, we will introduce the repairable system with three units and vacation. The state space and system operator are also introduced in this section. In section 3, several lemmas and corollaries are given to obtain the spectral distributions of the system operator. In section 4, we analyze the spectral distribution of the system operator. In the section 5, main results of this paper are obtained.
2 System equations and system operator
Following [9], the system considered here consists of three dissimilar units (called units 1, 2 and 3) and one repairman with single vacation. Initially, the system with three new units begins to operate. The repairman does not take a vacation and remain idle until the first failed unit appears. A repaired unit is as good as a new one. The failure time distributions are exponential, and the repair time distributions are general: λk and μk(x) are the failure rates and repair rate functions of unit k (k = 1,2,3). The vacation time distribution is assumed to be general as well, η(x) is the vacation rate function. By utilizing the similar method of [9], we are able to obtain the integro-differential equations
with boundary conditions:
and the only nonzero initial condition is p0(0) = 1.
The state space is chosen as
Obviously, X is a Banach space with the norm
in which
To introduce the system operator, we need the identities
Thus, from the system equations of (1)∼(24), the system operator A defined in the state space X is given by
From the above definitions of the state space and the operator A, the equations (1)∼(24) is equivalent to the following abstract Cauchy problem on the Banach space X.
3 Lemmas and corollaries
The main results are obtained under the following hypotheses.
General Hypothesesμi(x) (i = 1, 2, 3) and η(x) are bounded functions satisfying
It is easy to obtain the following lemmas and corollaries by the general hypotheses. The proofs of these lemmas and corollaries are given in the appendix of this note.
Lemma 1
For the repair rate functions μi(x) (i = 1,2,3) and vacation rate function η(x), we have
Lemma 2
If let
Lemma 3
If z ∈ ℂ, Rez > 0 or z = ai, a ≠ 0, then the following inequalities hold
We are able to derive the following corollaries from the above lemmas.
corollary 1
If z is the same as above, set
corollary 2
Let c = min{c1,c2,c3,c4} and the matrix T be the same as in corollary 1. If f(z) were the determinant of the matrix T, then the function f(z) have finite zeros in the region {z ∈ C|0 > Re z > −c}.
Furthermore, we need two other lemmas to analyze the spectral distributions of system operator.
Lemma 4
If z ∈ {z ∈ C|0 > Re z > −c}, for each given
Lemma 5
If let
4 Spectral analysis of the system operator
In this section, we shall give the detailed spectral analysis of the system operator with the help of the lemmas and corollaries listed in the previous section. Moreover, several useful corollaries are also obtained.
Theorem 1
Zero is the eigenvalue of system operator A whose geometric multiplicity is one. Furthermore, system operator A has a positive eigenfunction corresponding to eigenvalue zero.
Proof
If the equation
Since the desired nonzero solution
Solving the equations (37)∼(39) with the help of (28), we have
Substitute the functions above into the corresponding equations (35)∼(36), it follows from lemma 1 that
Secondly, it is easy to see
Substitute p7(x) into the equations (41)∼(43) and then solve them with the help of the identity (30), we obtain
Similarly, it follows from the equations (44)∼(46) that
At last, solving the equations (47)∼(49) leads to
Substitute the above functions pj(x) (j = 1, 2, 3, 7, ⋯, 16) into the identities (25)∼(27), (29), (31) and (33), we will obtain a homogeneous linear equations on p0, p1(0), p2(0), p3(0), p7(0), p14(0), p15(0) and p16(0)
in which the notations ai(i = 1,⋯,8) and bj(j = 1,2) are given in lemma 2. If let
then the coefficients matrix of the equations is the matrix B. Since the sum of every column of B is zero by the arguments of lemma 2, the determinant of the matrix B is zero. Thus, there exists nonzero solution to the equations. This means the nonzero solution to the equation
which show the geometric multiplicity of the eigenvalue zero is one. Thus the first statement of the theorem is right. The second statement is easy to be verified by setting p7(0) = 1. The proof of the theorem is complete.
Moreover, it is not hard to know the system operator A is densely defined and closed by direct verification. We do not want to give the proofs of both properties because the verification is trivial. For more information on the denseness and closeness of the similar operator, you can refer to [22]. With the help of these basic properties, we can present the following spectral distribution of the system operator.
Theorem 2
The spectral set σ(A) of the system operator A lies in left half-plane {z ∈ ℂ|Rez < 0} and zero is the unique point spectrum on the imaginary axis.
Proof
To show the arguments of the theorem 2, if let ρ(A) be the resolvent set of the operator A, then it is sufficient to show T := {z ∈ ℂ|Rez > 0 or z = ia, a ≠ 0} ⊂ ρ(A). To this end, for each given
Firstly, solving the equations above by the functions defined in lemma 4, we have
Substitute the above functions into corresponding boundary conditions (25)∼(27), (29), (31) and the first equation, inhomogeneous linear equations on p0, p1(0), p2(0), p3(0), p7(0), p14(0), p15(0), p16(0)
are obtained with the help of the notations in corollary 2 and lemma 5. The coefficient matrix of the equations (73)∼(80) is just the matrix T given in corollary 2, which tells us the matrix T is a diagonally dominant matrix about column, so it is nonsingular (see page 184 of [21]). According to lemma 5, el (l = 0, 1, ⋯, 7) belong to the complex number set ℂ. These statements imply the unique solution to the equations (73)∼(80) exists. Since pk(x)(k = 1,2,3,7,⋯,16) and pj(x,y)(j = 4,5,6) are uniquely determined by p1(0),p2(0),p3(0),p7(0), p14(0),p15(0),p16(0), the unique solution to the equation (zI − A)
Recall that the spectral bound s(T) of an unbounded operator T is the quantity sup{Reλ : λ ∈ σ(T)}. Thus, theorem 1 and theorem 2 imply the spectral bound s(A) of the system operator A is zero. Moreover, the proof of the theorem 2 implies the following corollary is true.
corollary 3
There exists a positive constant δ such that {z|z ∈ ℂ, Rez > −δ, z ≠ 0} ⊂ ρ(A).
Proof
According to the proof of the theorem 2, we know, for z ∈ ℂ, Rez > −c, z ∈ ρ(A) if and only if f(z) ≠ 0. Notice that f(z) is given in proof of corollary 2 and it shows the analytic function f(z) has finite zeros in the strip {z ∈ ℂ| −c < Rez < 0}. If let z1, z2, ⋯, zn be the zeros of f(z) and δ = −max{Rezi|1 ≤ i ≤ n}, then we would have {z|z ∈ ℂ, Rez > −δ, z ≠ 0} ⊂ ρ(A).
To give the next corollary, we need some notations on the state space X. For
and call
To indicate that 0 ≤
In fact, the above notations and concepts are the terminologies of Banach lattice. You can refer to [23–25] for detailed information. Thus, we have the following corollary by proof of the theorem 2.
corollary 4
If
Proof
According to the proof of the theorem 2, R(λ,A)
then the Cramer’s rule is applied to the equations (73)∼(80) results in
is given in proof of the corollary 2. Finally, for λ > s(A) = 0, according to the results of corollary 2 we know 0 < gk < 1(k = 7,8,9,12), 0 <
Since one of the right terms of the above identity must be positive and the other terms are nonnegative, |T5| is greater than 0. Thus, p7(0) > 0 due to
5 The well-posedness and stability of the system
By Theorem II 6.7 and Definition II 6.8 of [25], we know that the well-posedness of the system (32) is equivalent to the operator A generating a C0 semigroup T(t) on X. In fact, the C0 semigroup T(t) generated by the operator A is even positive, contractive and irreducible since we have the following results.
Theorem 3
The operator A is a dispersive operator in X.
Proof
We denote by X+ and [
in which sign(a) is the sign of the number a. Then
It follows from the boundary conditions (25)∼(31) that
If substitute them into the original inequality, we obtain (A
Now we can obtain one of the main results of this note.
Theorem 4
The operator A generates some positive C0semigroups T(t), which is contractive and irreducible, on Banach lattice X. Therefore, the unique non-negative solution
Proof
According to Philips Theorem in the positive semigroup theory (see [23]), A generates a positive C0-semigroup of contractions if and only if A is a dispersive and the operator I − A is surjective. Since theorem 2 has asserted that 1 belongs to the resolvent set of the operator A and theorem 3 show that A is a dispersive, we obtain the semigroup T(t) generated by system operator A is positive and contractive. Thus the equations (1)∼(24) have a unique solution by well known result in the theory of the C0-semigroup and it could be expressed as T(t)
Furthermore, we can obtain the asymptotic stability of the the system (32). To this end, we have to show zero is the simple eigenvalue of the system operator A. This result is given in the following theorem.
Theorem 5
Zero is the simple eigenvalue of the system operator A.
Proof
By the theory of the functional analysis, we know that the dual space of the state space X is X* = ℝ × (L∞(ℝ+))3 ×(L∞(ℝ+ × ℝ+))3 ×(L∞(ℝ+))10. For
If let
From the definition of the strictly dominant eigenvalue in [15], corollary 3 and the above theorem, we can get the following result.
Corollary 5
Zero is a strictly dominant eigenvalue of the system operator A.
Now, we show another main result of this note, that is the steady-state solution to the equations (1)∼(24) is existed and the dynamic solution converges to the steady-state solution in the norm topology of state space X. We elaborate on this property in following theorem. Indeed, the following theorem is a direct consequence of the theorem 2, theorem 5 and lemma 4.1 in [13], which is an extension theorem of Lyubich and Phong in [20].
Theorem 6
If
in which
On the other hand, we are able to obtain the asymptotic stability of the system by another way. For this purpose, we define the spectral project E(0, A) corresponding to the strictly dominant eigenvalue zero by
Thus, we can deduce the following theorem from the irreducibility of the semigroup T(t), which has been obtained in theorem 4.
Theorem 7
The semigroupT(t) is Cesáro mean ergodic and the state spaceXcan be decomposed into direct sum ofX = X1 ⊕ X2, in whichX1 = ker A = R(E(0, A)) andX2 = R(A) = ker E(0, A). Moreover, for any
Proof
By the Definition II 8.1 in [24], we know that the state space X is a AL space. Thus, the statements of the theorem are direct consequences of the lemma 5.1.1 and theorem 5.1.2 in [26].
In particular, if the vector
6 Conclusions and further researches
We obtained the well-posedness and asymptotic stability of the repairable system with three units and vacation by the theory of positive operator semigroups. That is to say the system equations have a unique non-negative dynamic solution and positive steady-state solution and dynamic solution strongly converges to steady-state solution. However, it is pity to enclose the manuscript that we did not give the reliability and exponential stability of the repairable system in this note. We shall show these properties and estimation of the availability of the system in another paper.
Appendix: Proofs of lemmas 1 to 5 and corollaries 1 and 2
Proof of lemma 1
The identities of the lemma 1 hold since
Proof of lemma 2
By simple computation we know
Thus, b1 + b2 = 1 holds. It is easy to see that
With the help of lemma 1 we have
Adding a1, a3 and a4 to a2 + a5 + a6 + a7 + a8, we would obtain
Proof of lemma 3
When Rez > 0, it is easy to obtain
When z = ai, we have
If the equality hold, that is
then
It follows from the formula of the module of complex number that
However, lemma 1 tells us
Thus
Since
Therefore, if a ≠ 0, then
This means that the first inequality holds for z = ai and a ≠ 0. The second inequality will be similarly proven.
Proof of corollary 1
If z ∈ ℂ, Rez > 0 or z = ai, a ≠ 0, since λi are positive, it follows that |λ1 + λ2 + λ3| < |z + λ1 + λ2 + λ3| and |gk| < 1 (k = 7, 8, 9, 12) hold. Moreover, we have
by lemma 3 and the notations and results of lemma 2. Thus, the statement of this corollary is true by the definition of the diagonally dominant matrix about column(see P. 184 of [21]).
Proof of corollary 2
The function gi given by corollary 2 is analytic in the right half-plane {z ∈ C|Rez > −c}. Moreover, we have
f(z) is then an analytic function in {z ∈ C|Rez > −c. Thus, the conclusion of this corollary results from
Proof of lemma 4
The absolute continuity of the functions fk(k = 1, 2, ⋯, 43) results from their special expressions. Concerning the absolute integrability, we just give an example. The other function is absolutely integrable is similarly derived. Indeed, we are able to obtain the following estimate
by the general hypotheses and integration by parts. Thus, f29(x) is an absolutely integrable function.
Proof of lemma 5
el(l = 0, 1, ⋯, 7) ∈ ℂ is equivalent to the module |el| of el is finite.
Only one example is given here. For instance, the module |e0| is finite due to
in which the same techniques of the above lemma are used.
Acknowledgements
The authors would like to thank the referees for their helpful suggestions and comments. This paper is supported by the National Natural Science Foundation of China (Grant No.11201037).
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Artikel in diesem Heft
- A Study on the Chinese Enterprise Annuity Replacement Rate Problem
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- Price Increasing Timing of Competitive Perishable Products
- Modeling and Forecasting Morbidity and Disease-mortality of Chinese Impaired Lives with the General Chronic Diseases
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- Well-posedness and Stability of the Repairable System with Three Units and Vacation
- Bayesian Subset Selection for Reproductive Dispersion Linear Models
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