#### Abstract

This paper considers a simple repairable system with a warning device and a repairman who can have delayed-multiple vacations. By Markov renewal process theory and the probability analysis method, the system is first transformed into a group of integrodifferential equations. Then, the existence and uniqueness as well as regularity of the system dynamic solution are discussed with the functional analysis method. Further, the asymptotic stability, especially the exponential stability of the system dynamic solution, is studied by using the strongly continuous semigroup theory or semigroup theory. The reliability indices and some applications (such as the comparisons of indices and profit of systems with and without warning device), as well as numerical examples, are presented at the end of the paper.

#### 1. Introduction

A repairable system is a system which, after failing to perform one or more of its functions satisfactorily, can be restored to fully satisfactory performance by any method, rather than the replacement of the entire system. With different repair levels, repair can be broken down into three categories (see [1]): perfect repair, normal repair, and minimal repair. A perfect repair can restore a system to an “as good as new” state, a normal repair is assumed to bring the system to any condition, and a minimal repair, or imperfect repair, can restore the system to the exact state it was before failure.

Repairable system is not only a kind of important system discussed in reliability theory but also one of the main objects studied in reliability mathematics. Since the 1960s, various repairable system models have been established and researched.

However, in traditional repairable systems, it is assumed that the repairman or server remains idle until a failed component presents. But as Mobley [2] pointed out, one-third of all maintenance costs were wasted as the result of unnecessary or improper maintenance activities. Today, the role of maintenance tends to be a “profit contributor.” Therefore, much more profit can be produced when the repairman in a system might take a sequence of vacations in the idle time. Repairman’s vacation may literally mean a lack of work or repairman taking another assigned job. From the perspective of rational use of human resources, the introduction of repairman’s vacation makes modeling of the repairable system more realistic and flexible. This is due to the fact that in practice, the vast majority of small-and medium-sized enterprises (SMEs) cannot afford to hire a full-time repairman. So, the repairman in SMEs usually plays two roles: one for looking after the equipment and one for other duties. Under normal circumstances, the repairman has to periodically check the status of the system. If he finds that the system failed, he repairs it immediately after the end of vacation; otherwise, he will leave the system for other duties or for a vacation.

Vacation model originally arised in queueing theory and has been well studied in the past three decades and successfully applied in many areas such as manufacturing/service and computer/communication network systems. Excellent surveys on the earlier works of vacation models have been reported by Doshi [3], Takagi [4], and Tian and Zhang [5]. A number of works (e.g., please see [6–10] and references therein) have recently appeared in the queueing literature in which concepts of different control operating politics along with vacations have been discussed. And Ke et al. [11] provided a summary of the most recent research works on vacation queueing systems in the past 10 years, in which a wide class of vacation policies for governing the vacation mechanism is presented.

In the past decade, inspired by the vacation queueing theory, some researchers introduced vacation model into repairable systems. The available references concerning repairman vacation in repairable systems can be classified into two categories: one is focused on the system indices and the other is the optimization problems.

For the first category, Jain and Rakhee [12] considered the bilevel control policy for a machining system having two repairmen. One turns on when queue size of failed units reaches a preassigned level. The other’s provision in case of long queue of failed units may be helpful in reducing the backlog. The steady state queue size distribution is obtained by applying the recursive method. Hu et al. [13] studied the steady-state availability and the mean up-time of a series-parallel repairable system consisting of one master control unit, two slave units, and a single repairman who operates single vacation by using the supplementary variable method and the vector Markov process theory. Q. T. Wu and S. M. Wu [14] analyzed some reliability indices of a cold standby system consisting of two repairable units, a switch and a repairman who may not always be at the job site or take vacation. Yuan [15] and Yuan and Cui [16] studied a k-out-of-n:G system and a consecutive-k-out-of-n:F system, respectively, with R repairmen who can take multiple vacations and by using Markov model; the analytical solution of some reliability indices was discussed. Yuan and Xu [17] studied a deteriorating system with a repairman who can have multiple vacations. By means of the geometric process and the supplementary variable techniques, a group of partial differential equations of the system was presented, and some reliability indices were derived. Ke and Wu [18] studied a multiserver machine repair model with standbys and synchronous multiple vacations, and the stationary probability vectors were obtained by using the matrix-analytical approach and the technique of matrix recursive.

For the second category, Ke and Wang [19] studied a machine repair problem consisting of M operating machines with two types of spare machines and R servers (repairmen) who can take different vacation policies. The steady-state probabilities of the number of failed machines in the system as well as the performance measures were derived by using the matrix geometric theory, and a direct search algorithm was used to determine the optimal values of the number of two types of spares and the number of servers while maintaining a minimum specified level of system availability. Jia and Wu [20] considered a replacement policy for a repairable system that can not be repaired “as good as new” with a repairman who can have multiple vacations. By using geometric processes, the explicit expression of the expected cost rate was derived, and the corresponding optimal policy was determined analytically and numerically. Yuan and Xu [21, 22] considered, respectively, a deteriorating repairable system and a cold standby repairable system with two different components of different priority in use, both with one repairman who can take multiple vacations. The explicit expression of the expected cost rate was given, and an optimal replacement policy was discussed. Yu et al. [23] analyzed a phase-type geometric process repair model with spare device procurement lead time and repairman’s multiple vacations. Employing the theory of renewal reward process, the explicit expression of the long-run average profit rate for the system was derived, and the optimal maintenance policy was also numerically determined.

However, to the best knowledge of the authors, whichever the catalogue, the references above only concentrated on the steady state (the steady-state indices or the steady-state optimization problems) of the systems. It is because that the transient behavior of a system is difficult to be studied. Therefore, in reliability study researchers usually substitute the steady-state solution for the instantaneous one of a system, for the steady-state solution can be easily obtained by Laplace transform and a limit theorem. Whereas, Laplace transform should be based on the two hypotheses: (1) the instantaneous solution of the interested system existed and (2) the instantaneous solution of the system is stable. Whether the hypotheses hold or not is still an open question and should be justified. Moreover, the substitution of the steady-state solution for the instantaneous one is not always rational. For detailed information or explanations, please see [24, 25].

Warning systems emerge in the background of repairable systems which are stepping into the times of requiring of both advanced warning and real-time fault detection. The so-called warning system is able to send emergency signals and report dangerous situations prior to disasters, catastrophes and/or other dangers need to watch out based on previous experiences and/or observed possible omens. Real-time warning systems play an important role in fault management in banking, telecommunications, securities, electric power, and other industries. If the warning prompts during system operation, operating staff can choose shut down the system, operate carefully, or repair the system. Warning systems can help users to achieve the 24-hour uninterrupted real-time monitoring and alerting during running of various types of network infrastructure sand application services. Therefore, there is a need to study the repairable systems with warning device.

This paper considers a simple repairable system with a warning device and a repairman who can have delayed-multiple vacations. The delayed-multiple vacations mean that the repairman will not leave for a vacation immediately if there is no component failed. However, there is a stochastic vacation-preparing period in which if a failed component appears he will stop the vacation preparing and serve it immediately; otherwise, he will take a rest on the end of the vacation-preparing period. When he returns from a vacation, he will either deal with the failed components waiting in the system or prepare for another vacation. In this paper, we are devoted to studying the asymptotic behavior of the system by strongly continuous semigroup theory and make comparisons of indices (such as reliability, availability, and the probability of the repairman’s vacation) and profit of the two systems with and without warning device.

The paper is structured as follows. The coming section introduces the system model specifically and expresses it into a group of integrodifferential equations by Markov renewal process theory and the probability analysis method. Section 3 discusses the existence and uniqueness as well as the regularity of the system dynamic solution by the functional analysis method. Section 4 studies the asymptotic behavior of the system by strongly continuous semigroup theory or semigroup theory. Section 5 presents some reliability indices of the system, and the steady-state indices are discussed from the viewpoint of eigenfunction of the system operator. In Section 6, comparisons of indices and profit of systems with and without warning device are made. And a brief conclusion is offered in the last section.

#### 2. System Formulation

The system model of interest is a simple repairable system (i.e., a repairable system with a unit and a repairman) with repairman vacation and a warning device. It is described specifically as follows: at the initial time , the unit is new, the system begins to work, and the repairman starts to prepare for the vacation. If the unit fails in the delayed-vacation period, the repairman deals with it immediately, and the delayed vacation is terminated. Otherwise, he leaves for a vacation after the delayed-vacation period ends. If the warning device sends alerts in the delayed-vacation period, the repairman will stay in the system until the unit fails. Whenever the repairman returns from a vacation, he either prepares for the next vacation if the unit is working or deals with the failed unit immediately or stays in the system if the warning device has sent alerts. The repair facility neither failed nor deteriorated. The unit is repaired as good as new. Further, we assume the following.(1)The distribution function of the working time of the unit is , , is a positive constant, and the distribution function of its repair time is and .(2)The distribution function of the delayed-vacation time of the repairman is , , is a positive constant, and the distribution function of his vacation time is , is a positive constant.(3)The distribution function of the time of the warning device from its beginning to work to its first sending alerts is , ; is a positive constant.(4)The above stochastic variables are independent of each other.

Set to be the state in which the system is at time , and assume all the possible states as follows:0:the system is working, and the repairman is preparing for the vacation;1:the system is working, and the repairman is on vacation;2:the system is warning, and the repairman is in the system;3:the system is warning, and the repairman is on vacation;4:the unit failed, and the repairman is on vacation;5:the repairman is dealing with the failed unit.

Then, by using probability analysis method, the system model can be described as the following group of integrodifferential equations: The boundary condition is The initial conditions are Here, represents the probability that the system is in state at time , , and represents the probability that the system is in state 5 with elapsed repair time lying in at time .

Concerning the practical background, we can assume that

#### 3. Existence and Uniqueness of System Solution

In this section, we will study the existence and uniqueness as well as the regularity of the system solution. Firstly, we will transform the system (1)–(3) into an equivalent integral problem by the method of characteristics. Secondly, the existence and uniqueness of the local solution of problem are discussed by using the fixed point theory. Then, the existence and uniqueness of the global solution of problem is further studied by a uniform priori estimate. Thus, the existence and uniqueness of the solution of system (1)–(3) are obtained. Moreover, the regularity or the continuity of the system solution is also discussed.

##### 3.1. Unique Existence of System Local Solution

For convenience, we will give some notations. Let with norm Choose Clearly, is a closed subspace of .

By the method of characteristics [26], the following equivalent proposition can be easily obtained [27, 28].

Theorem 1. *For a given constant , , , are the solution to (1)–(3) if and only if they are the solution to the following integral problem :
*

Clearly, to get the existence and uniqueness of the solution of system (1)–(3), it is necessary to study the existence and uniqueness of the solution of the above integral problem . To this end, for any , we define six operators as follows: It can be seen that for , if the operators , are determined, it needs only to get the fixed point of the operator in order to get the existence and uniqueness of the solution of the integral problem .

From (9)–(13), the following two lemmas can be easily obtained.

Lemma 2. *For a given constant such that , then for any , there exist unique and nonnegative , and satisfying (9)–(13).*

Lemma 3. *For a given constant such that , then for any , the following estimations hold:
*

Theorem 4. *There exists a , such that has a unique fixed point on .*

*Proof. *We prove the theorem in two steps. Firstly, we prove that the operator is a mapping from to . From the definition of , we can know that if , then and . Choose satisfying
Then, from (8) and Lemmas 2 and 3, it can be derived that
This implies that .

Secondly, we prove that the operator is a strictly compressed mapping on . For any , from (8) and Lemma 3, we have
This means that is strictly compressed. According to the Banach contraction mapping principle combining the above two steps, it can be deduced readily that has a unique fixed point on . The proof of Theorem 4 is completed.

Theorems 1 and 4 combing Lemma 2 follows the existence and uniqueness of the local solution of system (1)–(3).

Theorem 5 (existence and uniqueness of local solution). *There exists a such that the system (1)–(3) has a unique nonnegative local solution .*

##### 3.2. Unique Existence of System Global Solution

In this section, we will prove the existence and uniqueness of the global solution of system (1)–(3) by a uniform priori estimate and extension theorem.

Lemma 6. *For a given constant , if is the nonnegative solution of system (1)–(3), then one has:
**
where , and is defined in (4).*

*Proof. *Because the solution of system (1)–(3) is the solution of problem , the estimation of the system solution can be obtained easily as follows:
Thus,
Let , then . The Gronwall Inequality follows the estimation immediately: , for all . The proof of Lemma 6 is completed.

From Theorem 5, Lemma 6, and extension theorem, the existence and uniqueness of the system solution can be derived readily as below.

Theorem 7 (existence and uniqueness of global solution). *For any , the system (1)–(3) has a unique nonnegative solution .*

##### 3.3. Regularity of System Solution

In this section, we discuss the regularity or the continuity of the solution of system (1)–(3).

From Theorem 1 and the expressions in problem and noting the assumption (4), the following result is obvious.

Theorem 8. *For any , if is the nonnegative solution of system (1)–(3), then , and .*

Theorem 9. *For any , assume is continuous on . If is the nonnegative solution of system (1)–(3), then , , , where .*

*Proof. *From Theorem 1 and the expressions in problem combing the assumption (4), it is not difficult to know that for any , if is the nonnegative solution of system (1)–(3), then is differentiable on and by Theorem 8, . That is, , . And with the expression of in problem , we have
Then,
Therefore, by the continuity of , , and on , it can be yielded that , where . The proof of Theorem 9 is completed.

#### 4. Stability of System Solution

In this section, we will study the asymptotic stability and exponential stability of the solution of system (1)–(3). For convenience, we will first translate the system equations into an abstract Cauchy problem in a Banach space. Then, the asymptotic stability of the system solution is discussed by analyzing the spectral distributions of the system operator and that of its adjoint operator. Further, the exponential stability of the system solution is studied by analyzing the essential spectrum bound of the system operator.

##### 4.1. System Transformation

In this section, we will translate the system equations into an abstract Cauchy problem in a suitable Banach space.

First, choose the state space to be Here, denotes the set of nonnegative real numbers. Obviously, is a Banach space.

Next, define operator as follows: Then, the system (1)–(3) can be rewritten as an abstract Cauchy problem in the Banach space :

##### 4.2. Properties of System Operator

In this section, we will study some properties of the system operator .

Lemma 10. *The system operator is a densely closed dissipative operator.*

*Proof. *Firstly, we prove that is a closed operator. Choose , , , . By Proposition 1 ([29, II.2.10]), we know that the differential operator is the infinitesimal generator of a left translation semigroup with domain
Because is closed and , then , that is, , and is absolutely continuous. Moreover, , . Thus, . Therefore, it is not difficult to get that by noting the bounded measure of . This implies that is a closed operator.

Next, we prove that ; the domain of is dense in . For any , let , . Because , then for any , there exist and such that
Set , and define
Here, is continuously differentiable function on satisfying , and
Choose , then , and
This implies that is dense in .

Thirdly, we prove that is a dissipative operator. In fact, for any , choose , where , , . Clearly, , the dual space of , and . Moreover, it is not difficult to know that . This manifests that is a dissipative operator. The proof of Lemma 10 is completed.

Lemma 11. * or , , the resolvent set of the system operator .*

*Proof. *For any , consider the operator equation . That is,
Solving (36) with the help of (37) yields
where . By [30], there exists a constant , such that
Thus, .

Substituting (38) into (31) derives
where .

Combing (40) and (32)–(35) follows the following matrix equation: For or , , it is not difficult to get the following estimation from the definition of modulus of complex number:
Thus, the coefficient matrix of the matrix equation (41) is a strictly diagonally dominant matrix for column. So, it is inverse, and the matric equation (41) has a unique solution . Combing (38), it can be seen that (31)–(37) have a unique solution . This means that is surjective. Because is closed and is dense in , then exists and is bounded by Inverse Operator Theorem, for any or , . The proof of Lemma 11 is completed.

Lemma 12. *0 is an eigenvalue of the system operator with algebraic multiplicity one.*

*Proof. *Consider the operator equation . Let be the determinant of coefficient of the matrix equation (41), then we haveHere, and
Because
This means is an eigenvalue of the system operator with algebraic multiplicity one. The proof of Lemma 12 is completed.

##### 4.3. Properties of Adjoint Operator

In this section, we will study some properties of , the adjoint operator of system operator .

The dual space of is with norm for .

Lemma 13. *, the adjoint operator of the system operator is as follows:
**
with domain
**
Here,
**
and , satisfying .*

*Proof. *For any and , and its domain can be readily derived by the equality . The proof of Lemma 13 is completed.

Lemma 14. *, , , , , , the resolvent set of , where .*

*Proof. *For any , consider the equation . That is,
Solving (50)–(54) yields
Solving (55) derives
multiplying to the two sides of (57) and letting by noting follows
Substituting (58) into (57) yields
Thus, we can get the following estimation:
where .

Equation (56) derive the following estimations:

Then, for , we have
This implies that . Thus, is invertible. Therefore, is invertible and
The proof of Lemma 14 is completed.

Lemma 15. *0 is an eigenvalue of operator with algebraic multiplicity one.*

*Proof. *We prove Lemma 15 in two steps. Firstly, we prove that 0 is an eigenvalue of operator . For any , consider the operator equation , that is,
Solving (69) yields
multiplying to the two sides of (70) and letting by noting derive
Combing (64)–(68) with (71) follows
Substituting (71) into (70) yields
This implies that is the eigenfunction corresponding to eigenvalue 0 of operator , where .

Next, we prove that the algebraic multiplicity of 0 in is one. From the above step, we can see that the geometric multiplicity of 0 in is one. Then, we only need to verify the algebraic index of 0 in which is also one according to [31]. We use the reduction to absurdity. Suppose that the algebraic index of 0 in is 2 without loss of generality, then there exists , such that . It is obvious that
where is the positive eigenfunction corresponding to eigenvalue 0 of . However,
which contradicts (74). Thus, the algebraic index of 0 in is one. Therefore, the algebraic multiplicity of 0 in is one. The proof of Lemma 15 is completed.

##### 4.4. Asymptotic Stability of System Solution

In this section, we will present the asymptotic stability of the system solution by using semigroup theory.

Recalling Phillips Theorem (see [32]) together with Lemmas 10, 11, and 3, we can obtain the following results.

Theorem 16. *The system operator generates a positive semigroup of contraction .*

Theorem 17. *The system (25) has a unique nonnegative time-dependent solution which satisfies
*

*Proof. *From Theorem 16 and [32], it can be derived that the system (25) has a unique nonnegative solution which can be expressed as
Because satisfies (1)–(3), it is not difficult to know that
Therefore,
This just reflects the physical meaning of . The proof of Theorem 17 is completed.

*Remark 18. *Because the initial value of the system (25) belongs to the domain of the system operator , then the nonnegative time-dependent solution of the system expressed in (77) is the strong solution of the system (25).

Noting that the semigroup generated by is contractive according to Theorem 16, it is uniformly bounded certainly. Thus, recalling [33] combining Lemmas 11, 12, 14, and 15, we can know that the time-dependent solution of the system strongly converges to its steady-state solution. That is the following result.

Theorem 19. *Let be the eigenfunction corresponding to eigenvalue 0 of the system operator satisfying , and let be defined in Lemma 15, then the time-dependent solution of the system (25) converges to the nonnegative steady-state solution . That is,
**
where is the initial value of the system.*

##### 4.5. Exponential Stability of System Solution

In Section 4.4, we have obtained the asymptotic stability of the system. In other words, the dynamic solution of the system asymptotically converges to its steady-state solution. However, there are still two problems: first, the convergence rate is unknown; second, the convergence is subject to some factors such as failure rate and repair rate. Both can be well settled if the system is exponentially stable. For this purpose, in this section, we will discuss the exponential stability of the system.

For simplicity, we will divide the system operator into two operators. The one is a compact operator, and the other generates a quasicompact semigroup. Then, by the perturbation of compact operator, it is derived readily that the system operator also generates a quasicompact semigroup. Therefore, the system solution is exponentially stable.

For convenience, we will introduce three operators first: It is easy to know that and are both closed operators with dense domains in . And with the perturbation of semigroup, it is clear that also generates a semigroup .

Lemma 20. *Assume that the mean of the repair rate exists and greater than zero, that is,
**
Then, generates a quasicompact semigroup .*

*Proof. *Firstly, we will prove that generates a semigroup . Consider the following abstract Cauchy problem:
where . That is,
Solving (84)–(88) with the help of (91) yields
Solving (89) with the help of (90) and (92) by the method of characteristics yields
Therefore, it is easy to prove that generates a semigroup satisfying
Here,

Next, we will prove that is quasicompact. We only need to prove that the essential growth bound is less than zero.

The assumption condition (82) implies that for any , there exists such that
With the help of (97), it is not difficult to deduce that
This manifests that
Then,
Therefore, generates a quasicompact semigroup . The proof of Lemma 20 is completed.

For , , let where and is a compact operator. Then, it is not difficult to obtain the following result.

Lemma 21. * is a bijection from to and
*

Lemma 22. * is a compact operator, for any . Here is the semigroup generated by .*

*Proof. *From Lemma 21, we can see that , for any . Therefore , for any .

For , set
where , . Recalling the properties of -semigroup and Lemma 21, we can obtain