Stability Analysis of a Repairable System with Warning Device and Repairman Vacation

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 𝐶₀ 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 𝐶₀ 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 𝑡=0, 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 𝐹(𝑡)=1−𝑒^−𝜆𝑡, 𝑡≥0, 𝜆 is a positive constant, and the distribution function of its repair time is 𝐺(𝑡)=∫^𝑡₀𝑔(𝑥)d𝑥=1−𝑒^{−∫𝑡0𝜇(𝑥)d𝑥} and ∫^∞₀𝑡d𝐺(𝑡)=1/𝑏.(2)The distribution function of the delayed-vacation time of the repairman is 𝐷(𝑡)=1−𝑒^−𝜀𝑡, 𝑡≥0, 𝜀 is a positive constant, and the distribution function of his vacation time is 𝑉(𝑡)=1−𝑒^−𝜇0𝑡, 𝜇₀ 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 𝑈(𝑡)=1−𝑒^−𝛼0𝑡, 𝑡≥0; 𝛼₀ is a positive constant.(4)The above stochastic variables are independent of each other.

Set 𝑁(𝑡) to be the state in which the system is at time 𝑡, 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: dd𝑡+𝜀+𝛼₀𝑃₀(𝑡)=𝜇₀𝑃₁(𝑡)+^∞₀𝜇(𝑥)𝑃₅()d𝑡,𝑥d𝑥,d𝑡+𝜇₀+𝛼₀𝑃₁(𝑡)=𝜀𝑃₀(𝑡),d𝑃d𝑡+𝜆₂(𝑡)=𝛼₀𝑃₀(𝑡)+𝜇₀𝑃₃(𝑡),dd𝑡+𝜇₀+𝜆𝑃₃(𝑡)=𝛼₀𝑃₁(𝑡),dd𝑡+𝜇₀𝑃₄(𝑡)=𝜆𝑃₃(𝑡),𝜕+𝜕𝜕𝑡(𝑥𝑃𝜕𝑥+𝜇)₅(𝑡,𝑥)=0.(1) The boundary condition is 𝑃₅(𝑡,0)=𝜆𝑃₂(𝑡)+𝜇₀𝑃₄(𝑡).(2) The initial conditions are 𝑃₀(0)=1,theothersequalto0.(3) Here, 𝑃_𝑖(𝑡) represents the probability that the system is in state 𝑖 at time 𝑡, 𝑖=0,1,…,4, and 𝑃₅(𝑡,𝑥)d𝑥 represents the probability that the system is in state 5 with elapsed repair time lying in [𝑥,𝑥+d𝑥) at time 𝑡.

Concerning the practical background, we can assume that 𝜇(𝑥)≥0,𝜇=sup_{[)𝑥∈0,∞}𝜇(𝑥)<∞.(4)

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 (P) by the method of characteristics. Secondly, the existence and uniqueness of the local solution of problem (P) are discussed by using the fixed point theory. Then, the existence and uniqueness of the global solution of problem (P) 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 𝐶¹ continuity of the system solution is also discussed.

3.1. Unique Existence of System Local Solution

For convenience, we will give some notations. Let 𝐿₁=𝐿¹[0,∞),𝑉₀=𝐶[0,𝑇],𝑉₁=𝐶[0,𝑇],𝐿₁(5) with norm 𝜑_𝐿1=^∞₀𝜑(𝑥𝑓)d𝑥,_𝑉0=max_{[]𝑡∈0,𝑇}𝑓(𝑡,𝑔_𝑉1)=max_{𝑡∈[0,𝑇]}^∞₀𝑔(𝑡,𝑥)d𝑥.(6) Choose 𝑀=𝑞∈𝑉₀∣𝑞(0)=1,𝑞≥0,𝑞≤2.(7) Clearly, 𝑀 is a closed subspace of 𝑉₀.

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

Theorem 1. For a given constant 𝑇>0, 𝑃_𝑖(𝑡)∈𝑉₀, 𝑖=0,1,…,4, 𝑃₅(𝑡,𝑥)∈𝑉₁ are the solution to (1)–(3) if and only if they are the solution to the following integral problem (𝑃): 𝑃₀(𝑡)=𝑒^{−(𝜀+𝛼0)𝑡}+^𝑡₀𝜇₀𝑃₁(𝑠)+^∞₀𝜇(𝑥)𝑃₅(𝑠,𝑥)d𝑥×𝑒^{−(𝜀+𝛼0)(𝑡−𝑠)}𝑃d𝑠,₁(𝑡)=^𝑡₀𝜀𝑃₀(𝑠)𝑒^{−(𝜇0+𝛼0)(𝑡−𝑠)}𝑃d𝑠,₂(𝑡)=^𝑡₀𝛼₀𝑃₀(𝑠)+𝜇₀𝑃₃(𝑠)𝑒^{−𝜆(𝑡−𝑠)}𝑃d𝑠,₃(𝑡)=^𝑡₀𝛼₀𝑃₁(𝑠)𝑒^{−(𝜇0+𝜆)(𝑡−𝑠)}𝑃d𝑠,₄(𝑡)=^𝑡₀𝜆𝑃₃(𝑠)𝑒^{−𝜇0(𝑡−𝑠)}𝑃d𝑠,₅(𝑡,𝑥)=0,𝑥≥𝑡𝜆𝑃₂(𝑡−𝑥)+𝜇₀𝑃₄(𝑡−𝑥)𝑒^{−∫𝑥0𝜇(𝜏)d𝜏}(P),𝑥<𝑡.

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 (P). To this end, for any 𝑞∈𝑉₀, we define six operators as follows: 𝐾₀𝑞(𝑡)=𝑒^{−(𝜀+𝛼0)𝑡}+^𝑡₀𝜇₀𝐾₁𝑞(𝑠)+^∞₀𝜇(𝑥)𝐾₅𝑞(𝑠,𝑥)d𝑥×𝑒^{−(𝜀+𝛼0)(𝑡−𝑠)}𝐾d𝑠,(8)₁𝑞(𝑡)=^𝑡₀𝜀𝑞(𝑠)𝑒^{−(𝜇0+𝛼0)(𝑡−𝑠)}𝐾d𝑠,(9)₂𝑞(𝑡)=^𝑡₀𝛼₀𝑞(𝑠)+𝜇₀𝐾₃𝑞(𝑠)𝑒^{−𝜆(𝑡−𝑠)}𝐾d𝑠,(10)₃𝑞(𝑡)=^𝑡₀𝛼₀𝐾₁𝑞(𝑠)𝑒^{−(𝜇0+𝜆)(𝑡−𝑠)}𝐾d𝑠,(11)₄𝑞(𝑡)=^𝑡₀𝜆𝐾₃𝑞(𝑠)𝑒^{−𝜇0(𝑡−𝑠)}𝐾d𝑠,(12)₅𝑞(𝑡,𝑥)=𝜆𝐾₂𝑞(𝑡−𝑥)+𝜇₀𝐾₄𝑞(𝑡−𝑥)×𝑒^{−∫𝑥0𝜇(𝜏)d𝜏}.(13) It can be seen that for 𝑞∈𝑉₀, if the operators 𝐾_𝑖, 𝑖=1,2,…,5 are determined, it needs only to get the fixed point of the operator 𝐾₀ in order to get the existence and uniqueness of the solution of the integral problem (P).

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

Lemma 2. For a given constant 𝑇>0 such that 𝑡∈[0,𝑇], then for any 𝑞∈𝑀, there exist unique and nonnegative 𝐾_𝑖(𝑞)∈𝑉₀, 𝑖=1,…,4 and 𝐾₅(𝑞)∈𝑉₁ satisfying (9)–(13).

Lemma 3. For a given constant 𝑇>0 such that 𝑡∈[0,𝑇], then for any 𝑞,𝑞∈𝑀, the following estimations hold: 𝐾₁𝑞𝐾≤2𝜀𝑇,₁𝑞−𝐾₁𝑞𝑞,𝐾≤𝜀𝑇𝑞−₃𝑞≤2𝛼₀𝜀𝑇²,𝐾₃𝑞−𝐾₃𝑞≤𝛼₀𝜀𝑇²𝑞,𝐾𝑞−₂𝑞≤2𝛼₀𝑇1+𝜇₀𝜀𝑇²,𝐾₂𝑞−𝐾₂𝑞≤𝛼₀𝑇1+𝜇₀𝜀𝑇²𝑞,𝐾𝑞−₄𝑞≤2𝜆𝛼₀𝜀𝑇³,𝐾₄𝑞−𝐾₄𝑞≤𝜆𝛼₀𝜀𝑇³𝑞,𝐾𝑞−₅𝑞≤2𝜆𝛼₀𝑇²1+2𝜇₀𝜀𝑇²,𝐾₅𝑞−𝐾₅𝑞≤𝜆𝛼₀𝑇²1+2𝜇₀𝜀𝑇²𝑞−𝑞.(14)

Theorem 4. There exists a 𝑇=𝑇₀>0, such that 𝐾₀ has a unique fixed point on 𝑀.

Proof. We prove the theorem in two steps. Firstly, we prove that the operator 𝐾₀ is a mapping from 𝑀 to 𝑀. From the definition of 𝐾₀, we can know that if 𝑞∈𝑀, then 𝐾₀(𝑞)∈𝑉₀ and 𝐾₀(𝑞)(0)=1. Choose 0<𝑇₀<1 satisfying 𝑇²₀𝜇₀𝜀+𝜇𝜆𝛼₀1+2𝜇₀𝜀<12.(15) Then, from (8) and Lemmas 2 and 3, it can be derived that 𝐾₀𝑞=max_{𝑡∈[0,𝑇]}𝐾₀𝑞(𝑡)≤1+2𝑇²×𝜇₀𝜀+𝜇𝜆𝛼₀𝑇1+2𝜇₀𝜀𝑇²<1+2𝑇²𝜇₀𝜀+𝜇𝜆𝛼₀1+2𝜇₀𝜀<2.(16) This implies that 𝐾₀(𝑞)∈𝑀.
Secondly, we prove that the operator 𝐾₀ is a strictly compressed mapping on 𝑀. For any 𝑞,𝑞∈𝑀, from (8) and Lemma 3, we have 𝐾₀𝑞−𝐾₀𝑞≤𝑇²𝜇₀𝜀+𝜇𝜆𝛼₀𝑇1+2𝜇₀𝜀𝑇²𝑞−𝑞<𝑇²𝜇₀𝜀+𝜇𝜆𝛼₀1+2𝜇₀𝜀𝑞<1𝑞−2𝑞−𝑞.(17) This means that 𝐾₀ is strictly compressed. According to the Banach contraction mapping principle combining the above two steps, it can be deduced readily that 𝐾₀ has a unique fixed point on 𝑀. 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 𝑇=𝑇₀>0 such that the system (1)–(3) has a unique nonnegative local solution (𝑃₀,𝑃₁,…,𝑃₄,𝑃₅)∈𝑉⁵₀×𝑉₁.

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 𝑇>0, if (𝑃₀,𝑃₁,…,𝑃₄,𝑃₅)∈𝑉⁵₀×𝑉₁ is the nonnegative solution of system (1)–(3), then one has: 𝐸(𝑡)≤𝑒^𝑄𝑇,∀𝑡∈[0,𝑇],(18) where 𝐸(𝑡)=∑⁴_𝑖=0𝑃_𝑖(𝑡)+∫^∞₀𝑃₅(𝑡,𝑥)d𝑥, 𝑄=max{𝜀+𝛼₀,𝜇₀+𝛼₀,𝜇₀+𝜆,𝜇} and 𝜇 is defined in (4).

Proof. Because the solution of system (1)–(3) is the solution of problem (𝑃), the estimation of the system solution can be obtained easily as follows: 𝑃₀(𝑡)≤𝑒^{−(𝜀+𝛼0)𝑡}+^𝑡₀𝜇₀𝑃₁(𝑠)+d𝑠_𝑡0∞0𝜇(𝑥)𝑃₅(𝑠,𝑥)d𝑥d𝑠,≤1+𝜇₀^𝑡₀𝑃₁(𝑠)d𝑠+𝜇^𝑡₀^∞₀𝑃₅()𝑃𝑠,𝑥d𝑥d𝑠,₁(𝑡)≤𝜀^𝑡₀𝑃₀(𝑠)𝑃d𝑠,₂(𝑡)≤𝛼₀^𝑡₀𝑃₀(𝑠)d𝑠+𝜇₀^𝑡₀𝑃₃(𝑠)𝑃d𝑠,₃(𝑡)≤𝛼₀^𝑡₀𝑃₁(𝑠)d𝑠,𝑃₄(𝑡)≤𝜆^𝑡₀𝑃₃(𝑠)d𝑠,_∞0𝑃₅(𝑡,𝑥)d𝑥≤𝜆^∞₀𝑃₂(𝑡−𝑥)d𝑥+𝜇₀^∞₀𝑃₄(𝑡−𝑥)d𝑥=𝜆^𝑡₀𝑃₂(𝑠)d𝑠+𝜇₀^𝑡₀𝑃₄(𝑠)d𝑠.(19) Thus, 𝐸(𝑡)=⁴_𝑖=0𝑃_𝑖(𝑡)+^∞₀𝑃₅(𝑡,𝑥)d𝑥≤1+𝜀+𝛼₀^𝑡₀𝑃₀(𝑠)+𝜇d𝑠₀+𝛼₀^𝑡₀𝑃₁(𝑠)d𝑠+𝜆^𝑡₀𝑃₂(𝑠)+𝜇d𝑠₀+𝜆^𝑡₀𝑃₃(𝑠)d𝑠+𝜇₀^𝑡₀𝑃₄(𝑠)d𝑠+𝜇^𝑡₀^∞₀𝑃₅(𝑠,𝑥)d𝑥d𝑠.(20) Let 𝑄=max{𝜀+𝛼₀,𝜇₀+𝛼₀,𝜇₀+𝜆,𝜇}, then 𝐸(𝑡)≤1+𝑄∫^𝑡₀𝐸(𝑠)d𝑠. The Gronwall Inequality follows the estimation immediately: 𝐸(𝑡)≤𝑒^𝑄𝑇, for all 𝑡∈[0,𝑇]. 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 𝑇>0, the system (1)–(3) has a unique nonnegative solution (𝑃₀,𝑃₁,…,𝑃₄,𝑃₅)∈𝑉⁵₀×𝑉₁.

3.3. Regularity of System Solution

In this section, we discuss the regularity or the 𝐶¹ continuity of the solution of system (1)–(3).

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

Theorem 8. For any 𝑇>0, if (𝑃₀,𝑃₁,…,𝑃₄,𝑃₅)∈𝑉⁵₀×𝑉₁ is the nonnegative solution of system (1)–(3), then 𝑃_𝑖∈𝐶[0,𝑇], 𝑖=0,1,…,4 and 𝑃₅∈𝐶([0,𝑇]×[0,∞)).

Theorem 9. For any 𝑇>0, assume 𝜇(𝑥) is continuous on [0,𝑇]. If (𝑃₀,𝑃₁,…,𝑃₄,𝑃₅)∈𝑉⁵₀×𝑉₁ is the nonnegative solution of system (1)–(3), then 𝑃_𝑖∈𝐶¹[0,𝑇], 𝑖=0,1,…,4, 𝑃₅∈𝐶¹(𝐷), where 𝐷={(𝑡,𝑥)∣0<𝑡≤𝑇,0≤𝑥<𝑡}.

Proof. From Theorem 1 and the expressions in problem (P) combing the assumption (4), it is not difficult to know that for any 𝑇>0, if (𝑃₀,𝑃₁,…,𝑃₄,𝑃₅)∈𝑉⁵₀×𝑉₁ is the nonnegative solution of system (1)–(3), then 𝑃_𝑖(𝑡) is differentiable on [0,𝑇] and 𝑃^′_𝑖(𝑡)∈𝐶[0,𝑇] by Theorem 8, 𝑖=0,1,…,4. That is, 𝑃_𝑖(𝑡)∈𝐶¹[0,𝑇], 𝑖=0,1,…,4. And with the expression of 𝑃₅(𝑡,𝑥) in problem (P), we have 𝜕𝑃₅(𝑡,𝑥)=𝜕𝑡𝜆𝑃^′₂(𝑡−𝑥)+𝜇₀𝑃^′₄(𝑡−𝑥)𝑒^{−∫𝑥0𝜇(𝜏)d𝜏},𝑥<𝑡0,𝑥>𝑡,𝜕𝑃₅(𝑡,𝑥)𝜕𝑥&=−𝜆𝑃^′₂(𝑡−𝑥)−𝜇₀𝑃^′₄(𝑡−𝑥)𝑒^{−∫𝑥0𝜇(𝜏)d𝜏}−𝜇(𝑥)𝜆𝑃₂(𝑡−𝑥)+𝜇₀𝑃₄(𝑡−𝑥)𝑒^{−∫𝑥0𝜇(𝜏)d𝜏},𝑥<𝑡0,𝑥>𝑡.(21) Then, 𝜕𝑃₅(𝑡,𝑥)𝜕𝑡^𝑥<𝑡_𝑥→𝑡=𝜆𝛼₀𝑒^{−∫𝑥0𝜇(𝜏)d𝜏},𝜕𝑃₅(𝑡,𝑥)𝜕𝑡^𝑥>𝑡_𝑥→𝑡=0,𝜕𝑃₅(𝑡,𝑥)𝜕𝑡^𝑥<𝑡_𝑥→𝑡=−𝜆𝛼₀𝑒^{−∫𝑥0𝜇(𝜏)d𝜏},𝜕𝑃₅(𝑡,𝑥)𝜕𝑡^𝑥>𝑡_𝑥→𝑡=0.(22) Therefore, by the continuity of 𝑃^′₂(𝑡), 𝑃^′₄(𝑡), and 𝜇(𝑥) on [0,𝑇], it can be yielded that 𝑃₅(𝑡,𝑥)∈𝐶¹(𝐷), where 𝐷={(𝑡,𝑥)∣0<𝑡≤𝑇,0≤𝑥<𝑡}. 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 𝑋 to be 𝑋=𝑃=𝑃₀,𝑃₁,…,𝑃₄,𝑃₅(𝑥)^T∣𝑃_𝑖𝑃∈ℝ,𝑖=0,1,…,4,₅(𝑥)∈𝐿¹ℝ₊,‖𝑃‖=⁴_𝑖=0𝑃_𝑖+𝑃_5𝐿1(ℝ+)<∞.(23) Here, ℝ₊ denotes the set of nonnegative real numbers. Obviously, 𝑋 is a Banach space.

Next, define operator 𝐴 as follows: −𝐴𝑃=𝜀+𝛼₀𝑃₀+𝜇₀𝑃₁+^∞₀𝜇(𝑥)𝑃₅(𝑥)−𝛼d𝑥₀+𝜇₀𝑃₁+𝜀𝑃₀−𝜆𝑃₂+𝛼₀𝑃₀+𝜇₀𝑃₃−𝜆+𝜇₀𝑃₃+𝛼₀𝑃₁−𝜇₀𝑃₄+𝜆𝑃₃−𝑃^′₅(𝑥)−𝜇(𝑥)𝑃₅(𝑥),𝐷(𝐴)=𝑃=𝑃₀,𝑃₁,…,𝑃₄,𝑃₅(𝑥)^T∈𝑋∣𝑃^′₅(𝑥)∈𝐿¹ℝ₊;𝑃₅(𝑥)𝑃isanabsolutelycontinuousfunctionsatisfying₅(0)=𝜆𝑃₂+𝜇₀𝑃₄.(24) Then, the system (1)–(3) can be rewritten as an abstract Cauchy problem in the Banach space 𝑋: d𝑃(𝑡,⋅)()𝑃()=𝑃d𝑡=𝐴𝑃𝑡,⋅,𝑡≥0𝑡,⋅₀(𝑡),…,𝑃₄(𝑡),𝑃₅(𝑡,𝑥)^T𝑃(0,⋅)≜𝑃₀=(1,0,…,0)^T.(25)

4.2. Properties of System Operator 𝐴

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

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

Proof. Firstly, we prove that 𝐴 is a closed operator. Choose 𝑃_𝑛=(𝑃_𝑛0,𝑃_𝑛1,…,𝑃_𝑛4,𝑃_𝑛5(𝑥))^T∈𝐷(𝐴), 𝑃_𝑛→𝑃=(𝑃₀,𝑃₁,…,𝑃₄,𝑃₅(𝑥))^T, 𝐴𝑃_𝑛→𝑄=(𝑄₀,𝑄₁,…,𝑄₄,𝑄₅(𝑥))^T, 𝑛→∞. By Proposition 1 ([29, II.2.10]), we know that the differential operator 𝒟 is the infinitesimal generator of a left translation semigroup {𝑇_𝑙(𝑡)}_𝑡≥0 with domain 𝐷(𝒟)=𝑓∈𝐿¹ℝ₊𝑓∣𝑓isabsolutelycontinuoussatisfying_′∈𝐿¹ℝ₊.(26) Because 𝐷(𝒟) is closed and 𝑃_𝑛5∈𝐷(𝒟), then 𝑃₅∈𝐷(𝒟), that is, 𝑃^′₅(𝑥)∈𝐿¹(ℝ₊), and 𝑃₅(𝑥) is absolutely continuous. Moreover, 𝑃_𝑛5(0)=𝜆𝑃_2𝑛+𝜇₀𝑃_4𝑛→𝜆𝑃₂+𝜇₀𝑃₄, 𝑛→∞. Thus, 𝑃∈𝐷(𝐴). Therefore, it is not difficult to get that 𝐴𝑃=𝑄 by noting the bounded measure of 𝜇(𝑥). This implies that 𝐴 is a closed operator.
Next, we prove that 𝐷(𝐴); the domain of 𝐴 is dense in 𝑋. For any 𝐹=(𝐹₀,𝐹₁,…,𝐹₄,𝐹₅(𝑥))^T∈𝑋, let 𝑃_𝑖=𝐹_𝑖, 𝑖=0,1,…,4. Because 𝐹₅(𝑥)∈𝐿¹(ℝ₊), then for any 𝜀>0, there exist 𝛿₁>0 and 𝐺>0 such that ^𝛿1₀𝐹₅(𝑥𝜀)d𝑥<6,^∞_𝐺𝐹₅(𝑥𝜀)d𝑥<3.(27) Set 𝛿=min{𝛿₁,1/6(1+|𝜆𝑃₂+𝜇₀𝑃₄|)}, and define 𝑃₅(𝑥)=𝜆𝑃₂+𝜇₀𝑃₄𝑔(𝑥),0≤𝑥<𝛿,𝛿≤𝑥≤𝐺0,𝑥>𝐺.(28) Here, 𝑔(𝑥) is continuously differentiable function on [𝛿,𝐺] satisfying 𝑔(𝛿)=𝜆𝑃₅+𝜇𝑃₄, 𝑔(𝐺)=0 and ^𝐺_𝛿𝑃₅(𝑥)(𝑥𝜀−𝑔)d𝑥<3.(29) Choose 𝑃=(𝑃₀,𝑃₁,…,𝑃₄,𝑃₅(𝑥))^T, then 𝑃∈𝐷(𝐴), and ‖𝑃−𝐹‖=^𝛿₀𝑃₅(𝑥)−𝐹₅(𝑥+)d𝑥_𝐺𝛿𝑃₅(𝑥)−𝐹₅(𝑥+)d𝑥_∞𝐺𝑃₅(𝑥)−𝐹₅(𝑥<)d𝑥_𝛿0𝑃₅(𝑥)d𝑥+^𝛿₀𝐹₅(𝑥+𝜀)d𝑥3+𝜀3<𝜆𝑃₂+𝜇₀𝑃₄𝜀𝛿+6+2𝜀3<𝜀.(30) This implies that 𝐷(𝐴) is dense in 𝑋.
Thirdly, we prove that 𝐴 is a dissipative operator. In fact, for any 𝑃=(𝑃₀,𝑃₁,…,𝑃₄,𝑃₅(𝑥))^T∈𝐷(𝐴), choose 𝑄=(𝑄₀,𝑄₁,…,𝑄₄,𝑄₅(𝑥))^T, where 𝑄_𝑖=‖𝑃‖sgn(𝑃_𝑖), 𝑖=0,1,…,4, 𝑄₅(𝑥)=‖𝑃‖sgn(𝑃₅(𝑥)). Clearly, 𝑄∈𝑋^∗=ℝ⁵×𝐿^∞(ℝ₊), the dual space of 𝑋, and ⟨𝑃,𝑄⟩=‖𝑃‖²=‖𝑄‖². Moreover, it is not difficult to know that ⟨𝐴𝑃,𝑄⟩≤0. This manifests that 𝐴 is a dissipative operator. The proof of Lemma 10 is completed.

Lemma 11. {𝛾∈ℂ∣Re𝛾>0 or 𝛾=𝑖𝑎, 𝑎∈ℝ⧵{0}}⊂𝜌(𝐴), the resolvent set of the system operator 𝐴.

Proof. For any 𝐺=(𝐺₀,𝐺₁,…,𝐺₄,𝐺₅(𝑥))^T∈𝑋, consider the operator equation (𝛾𝐼−𝐴)𝑃=𝐺. That is, 𝛾+𝜀+𝛼₀𝑃₀=𝐺₀+𝜇₀𝑃₁+^∞₀𝜇(𝑥)𝑃₅(𝑥)d𝑥,(31)𝛾+𝛼₀+𝜇₀𝑃₁=𝐺₁+𝜀𝑃₀,𝑃(32)𝛾+𝜆₂=𝐺₂+𝛼₀𝑃₀+𝜇₀𝑃₃,(33)𝛾+𝜆+𝜇₀𝑃₃=𝐺₃+𝛼₀𝑃₁,(34)𝛾+𝜇₀𝑃₄=𝐺₄+𝜆𝑃₃,𝑃(35)_′5(𝑥)+𝛾+𝜇(𝑥)𝑃₅(𝑥)=𝐺₅(𝑥),𝑃(36)₅(0)=𝜆𝑃₂+𝜇₀𝑃₄.(37) Solving (36) with the help of (37) yields 𝑃₅(𝑥)=𝑃₅(0)𝑒^{−∫𝑥0(𝛾+𝜇(𝑠))d𝑠}+^𝑥₀𝐺₅(𝜏)𝑒^{−∫𝑥𝜏(𝛾+𝜇(𝑠))d𝑠}=d𝜏𝜆𝑃₂+𝜇₀𝑃₄𝑒^{−∫𝑥0(𝛾+𝜇(𝑠))d𝑠}+𝑌₅(𝑥),(38) where 𝑌₅(𝑥)=∫^𝑥₀𝐺₅(𝜏)𝑒^{−∫𝑥𝜏[𝛾+𝜇(𝑠)]d𝑠}d𝜏. By [30], there exists a constant 𝑁, such that ^∞_𝑡𝑒^{−∫𝑥0𝜇(𝑠)d𝑠}d𝑥≤𝑁,∀𝑡≥0.(39) Thus, 𝑃₅(𝑥)∈𝐿¹(ℝ₊).
Substituting (38) into (31) derives 𝛾+𝜀+𝛼₀𝑃₀=𝐺₀+𝜇₀𝑃₁+^∞₀𝜇(𝑥)×𝑃₅(0)𝑒^{−∫𝑥0(𝛾+𝜇(𝑠))d𝑠}+𝑌₅(𝑥)d𝑥=𝜇₀𝑃₁+𝜆𝑃₂+𝜇₀𝑃₄×^∞₀𝜇(𝑥)𝑒^{−∫𝑥0(𝛾+𝜇(𝑠))d𝑠}d𝑥+𝑌₀,(40) where 𝑌₀=𝐺₀+∫^∞₀𝜇(𝑥)𝑌₅(𝑥)d𝑥.
Combing (40) and (32)–(35) follows the following matrix equation: 𝛾+𝜀+𝛼₀−𝜇₀−𝜆^∞₀𝜇(𝑥)𝑒^{∫𝑥0(𝛾+𝜇(𝑠))d𝑠}d𝑥0−𝜇₀^∞₀𝜇(𝑥)𝑒^{−∫𝑥0(𝛾+𝜇(𝑠))d𝑠}d𝑥−𝜀𝛾+𝛼₀+𝜇₀000−𝛼₀0𝛾+𝜆−𝜇₀00−𝛼₀0𝛾+𝜆+𝜇₀0000−𝜆𝛾+𝜇₀𝑃₀𝑃₁𝑃₂𝑃₃𝑃₄𝑌₀𝐺₁𝐺₂𝐺₃𝐺₄.(41)For Re𝛾>0 or 𝛾=𝑖𝑎, 𝑎∈ℝ⧵{0}, it is not difficult to get the following estimation from the definition of modulus of complex number: ^∞₀𝜇(𝑥)𝑒^{−∫𝑥0(𝛾+𝜇(𝑠))d𝑠}d𝑥<1.(42) 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 (𝑃₀,𝑃₁,𝑃₂,𝑃₃,𝑃₄)^T. Combing (38), it can be seen that (31)–(37) have a unique solution 𝑃=(𝑃₀,𝑃₁,𝑃₂,𝑃₃,𝑃₄,𝑃₅(𝑥))^T∈𝐷(𝐴). This means that 𝛾𝐼−𝐴 is surjective. Because 𝛾𝐼−𝐴 is closed and 𝐷(𝐴) is dense in 𝑋, then (𝛾𝐼−𝐴)⁻¹ exists and is bounded by Inverse Operator Theorem, for any Re𝛾>0 or 𝛾=𝑖𝑎, 𝑎∈ℝ⧵{0}. The proof of Lemma 11 is completed.

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

Proof. Consider the operator equation (𝛾𝐼−𝐴)𝑃=0. Let 𝐷(𝛾) be the determinant of coefficient of the matrix equation (41), then we have𝐷𝛾=𝛾+𝜀+𝛼₀−𝜇₀−𝜆1−𝛾𝑔𝛾0−𝜇₀1−𝛾𝑔𝛾−𝜀𝛾+𝛼₀+𝜇₀000−𝛼₀0𝛾+𝜆−𝜇₀00−𝛼₀0𝛾+𝜆+𝜇₀0000−𝜆𝛾+𝜇₀=𝛾𝐹𝛾.(43)Here, 𝑔(𝛾)=∫^∞₀𝑒^{−∫𝑥0(𝛾+𝜇(𝑠))d𝑠} and 𝐹𝛾=𝛼₀𝜆𝜇₀𝑔𝛾𝜆+𝜇₀𝜀+𝛼₀+𝜇₀+𝛼₀𝜀𝜇²₀+𝜇³+𝜇²₀𝛼₀+𝜆𝛼₀+𝜇₀𝜆𝜇₀+𝜀+𝜇₀𝛼₀+𝜀+𝜇₀+𝛼₀𝜆𝛾𝑔𝛾2𝜀𝜇₀+2𝛼₀𝜇₀+3𝜇²₀+2𝜆𝜇₀+𝛼₀𝜆+𝛾𝜆𝜇₀+𝜆𝛼₀+𝛼₀𝜇₀4𝜇₀+𝜆+2𝜀+𝜆₂+𝜇²₀𝜇₀+𝜀+2𝜇₀𝛼²₀+𝜀𝜆𝜇₀+𝜆𝛼²₀+𝛼₀𝜆𝛾²𝑔𝛾3𝜇₀+𝛼₀+𝜆+𝛾²5𝜆𝜇₀+2𝜀𝜆+𝜆²+𝛼²₀+3𝜇²₀+𝜀𝛼₀+5𝜇₀𝛼₀+3𝜆𝛼₀+2𝜀𝜇₀+𝛾³2𝜆+𝜀+2𝛼₀+3𝜇₀+𝛼₀𝜆𝑔𝛾+𝛾⁴.(44) Because 𝐹(0)=𝛼₀𝜆𝜇₀𝑔(0)𝜆+𝜇₀𝜀+𝛼₀+𝜇₀+𝛼₀𝜀𝜇²₀+𝜇³+𝜇²₀𝛼₀+𝜆𝛼₀+𝜇₀×𝜆𝜇₀+𝜀+𝜇₀𝛼₀+𝜀+𝜇₀>0.(45) This means 𝛾=0 is an eigenvalue of the system operator 𝐴 with algebraic multiplicity one. The proof of Lemma 12 is completed.

4.3. Properties of Adjoint Operator 𝐴^∗

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

The dual space of 𝑋 is 𝑋^∗=ℝ⁵×𝐿^∞ℝ₊(46) with norm ‖𝑄‖=sup{|𝑄_𝑖|,‖𝑄₅‖_{𝐿∞(ℝ+)},𝑖=0,1,…,4} for 𝑄=(𝑄₀,𝑄₁,…,𝑄₄,𝑄₅(𝑥))^T∈𝑋^∗.

Lemma 13. 𝐴^∗, the adjoint operator of the system operator 𝐴 is as follows: 𝐴^∗−𝑄=𝜀+𝛼₀𝑄₀+𝜀𝑄₁+𝛼₀𝑄₂−𝛼₀+𝜇₀𝑄₁+𝜇₀𝑄₀+𝛼₀𝑄₃−𝜆𝑄₂+𝜆𝑄₅(0)−𝜆+𝜇₀𝑄₃+𝜇₀𝑄₂+𝜆𝑄₄−𝜇₀𝑄₄+𝜇₀𝑄₅(0)𝑄^′₅(𝑥)+𝜇(𝑥)𝑄₀−𝑄₅(𝑥≜()𝑄)𝐶+𝐷(47) with domain 𝐷𝐴^∗=𝑄=𝑄₀,𝑄₁,…,𝑄₄,𝑄₅(𝑥)^T∈𝑋^∗∣𝑄^′₅(𝑥)∈𝐿^∞ℝ₊,𝑄₅(𝑥)𝑄𝑖𝑠𝑎𝑏𝑠𝑜𝑙𝑢𝑡𝑒𝑙𝑦𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠𝑠𝑎𝑡𝑖𝑠𝑓𝑦𝑖𝑛𝑔₅(∞)<∞.(48) Here, 𝐶𝑄=diag−𝜀+𝛼₀,−𝛼₀+𝜇₀,−𝜆,−𝜆+𝜇₀,−𝜇₀,dd𝑥−𝜇(𝑥)𝑄,𝐷𝑄=0𝜀𝛼₀𝜇000₀00𝛼₀(⋅)0000000𝜆𝜃00𝜇₀0𝜆000000𝜇₀𝜃(⋅)𝜇(𝑥)𝑄00000(49) and 𝜃(⋅)∶𝐿^∞(ℝ₊)→ℂ, satisfying 𝜃(𝑓)=𝑓(0).

Proof. For any 𝑃∈𝐷(𝐴) and 𝑄∈𝑋^∗, 𝐴^∗ and its domain 𝐷(𝐴^∗) can be readily derived by the equality ⟨𝐴𝑃,𝑄⟩=⟨𝑃,𝐴^∗𝑄⟩. The proof of Lemma 13 is completed.

Lemma 14. 𝑆={𝛾∈ℂ∣sup{(𝜀+𝛼₀)/|𝛾+𝜀+𝛼₀|, (𝜇₀+𝛼₀)/|𝛾+𝜇₀+𝛼₀|, 𝜆/|𝛾+𝜆|, (𝜆+𝜇₀)/|𝛾+𝜆+𝜇₀|, 𝜇₀/|𝛾+𝜇₀|, 𝑀/(Re𝛾+𝑀)}<1}⊂𝜌(𝐴^∗), the resolvent set of 𝐴^∗, where 𝑀=sup_𝑥≥0𝜇(𝑥).

Proof. For any 𝑊=(𝑊₀,𝑊₁,…,𝑊₄,𝑊₅(𝑥))^T∈𝑋^∗, consider the equation (𝛾𝐼−𝐶)𝑄=𝐷𝑊. That is, 𝛾+𝜀+𝛼₀𝑄₀=𝜀𝑊₁+𝛼₀𝑊₂,(50)𝛾+𝛼₀+𝜇₀𝑄₁=𝜇₀𝑊₀+𝛼₀𝑊₃,(51)𝛾+𝜆𝑄₂=𝜆𝑊₅(0),(52)𝛾+𝜆+𝜇₀𝑄₃=𝜇₀𝑊₂+𝜆𝑊₄,(53)𝛾+𝜇₀𝑄₄=𝜇₀𝑊₅(0),d(54)𝑄₅(𝑥)d𝑥=𝛾+𝜇(𝑥)𝑄₅(𝑥)−𝜇(𝑥)𝑊₀.(55) Solving (50)–(54) yields 𝑄₀=𝜀𝑊₁+𝛼₀𝑊₂𝛾+𝜀+𝛼₀,𝑄₁=𝜇₀𝑊₀+𝛼₀𝑊₃𝛾+𝛼₀+𝜇₀,𝑄₂=𝜆𝑊₅(0),𝑄𝛾+𝜆₃=𝜇₀𝑊₂+𝜆𝑊₄𝛾+𝜆+𝜇₀,𝑄₄=𝜇₀𝑊₅(0)𝛾+𝜇₀.(56) Solving (55) derives 𝑄₅(𝑥)=𝑒^{∫𝑥0(𝛾+𝜇(𝑠))d𝑠}𝑄₅(0)−^𝑥₀𝑊₀𝜇(𝜏)𝑒^{−∫𝜏0(𝛾+𝜇(𝑠))d𝑠}d𝜏(57) multiplying 𝑒^{−∫𝑥0(𝛾+𝜇(𝑠))d𝑠} to the two sides of (57) and letting 𝑥→∞ by noting 𝑄₅(∞)<∞ follows 𝑄₅(0)=^∞₀𝑊₀𝜇(𝜏)𝑒^{−∫𝜏0(𝛾+𝜇(𝑠))d𝑠}d𝜏.(58) Substituting (58) into (57) yields 𝑄₅(𝑥)=𝑒^{∫𝑥0(𝛾+𝜇(𝑠))d𝑠∞}_𝑥𝑊₀𝜇(𝜏)𝑒^{−∫𝜏0(𝛾+𝜇(𝑠))d𝑠}d𝜏.(59) Thus, we can get the following estimation: 𝑄_{5𝐿∞[0,∞)}=sup_{[)𝑥∈0,∞}𝑄₅(𝑥)=sup_{𝑥∈[0,∞)}𝑒^{∫𝑥0(𝛾+𝜇(𝑠))d𝑠}×^∞_𝑥𝑊₀𝜇(𝜏)𝑒^{−∫𝜏0(𝛾+𝜇(𝑠))d𝑠}d𝜏≤‖𝑊‖sup_{𝑥∈[0,∞)}𝑒^{∫𝑥0(Re𝛾+𝜇(𝑠))d𝑠}×^∞_𝑥−𝑒^{−Re𝛾𝜏}d𝑒^{−∫𝜏0𝜇(𝑠)d𝑠}=‖𝑊‖sup_{𝑥∈[0,∞)}𝑒^{∫𝑥0(Re𝛾+𝜇(𝑠))d𝑠}×𝑒^{−∫𝑥0(Re𝛾+𝜇(𝑠))d𝑠}−Re𝛾^∞_𝑥𝑒^{−∫𝜏0(Re𝛾+𝜇(𝑠))d𝑠}d𝜏=‖𝑊‖sup_{[)𝑥∈0,∞}1−Re𝛾𝑒^Re𝛾𝑥×^∞_𝑥𝑒^{−∫𝜏𝑥(Re𝛾+𝜇(𝑠))d𝑠}d𝜏≤‖𝑊‖sup_{𝑥∈[0,∞)}1−Re𝛾𝑒^Re𝛾𝑥×^∞_𝑥𝑒^{−∫𝜏𝑥(Re𝛾+𝑀)d𝑠}d𝜏=‖𝑊‖sup_{𝑥∈[0,∞)}1−Re𝛾𝑒^{(Re𝛾+𝑀)𝑥}×^∞_𝑥𝑒^{−(Re𝛾+𝑀)𝜏}d𝜏=‖𝑊‖sup_{𝑥∈[0,∞)}1−Re𝛾𝑒^{(Re𝛾+𝑀)𝑥}×𝑒^{−(Re𝛾+𝑀)𝑥}=𝑀Re𝛾+𝑀‖𝑊‖,Re𝛾+𝑀(60) where 𝑀=sup_𝑥≥0𝜇(𝑥).
Equation (56) derive the following estimations: 𝑄₀≤𝜀+𝛼₀𝛾+𝜀+𝛼₀‖𝑊‖,𝑄₁≤𝜇₀+𝛼₀𝛾+𝛼₀+𝜇₀‖𝑊‖,𝑄₂≤𝜆‖𝑊‖,𝑄𝛾+𝜆³≤𝜆+𝜇₀𝛾+𝜆+𝜇₀‖𝑊‖,𝑄₄≤𝜇₀𝛾+𝜇₀‖𝑊‖,𝑄₅≤𝑀‖𝑊‖.Re𝛾+𝑀(61)
Then, for 𝛾∈𝑆, we have ‖𝑄‖=⁴_𝑖=0𝑄_𝑖+𝑄₅≤sup𝜀+𝛼₀𝛾+𝜀+𝛼₀,𝜇₀+𝛼₀𝛾+𝛼₀+𝜇₀,𝜆,𝛾+𝜆𝜆+𝜇₀𝛾+𝜆+𝜇₀,𝜇₀𝛾+𝜇₀,𝑀𝑊‖<‖𝑊‖.Re𝛾+𝑀‖(62) This implies that ‖(𝛾𝐼−𝐶)⁻¹𝐷‖<1. Thus, [𝐼−(𝛾𝐼−𝐶)⁻¹𝐷] is invertible. Therefore, (𝛾𝐼−𝐴^∗) is invertible and 𝛾𝐼−𝐴^∗−1=𝛾𝐼−(𝐶+𝐷)⁻¹=𝐼−𝛾𝐼−𝐶⁻¹𝐷⁻¹𝛾𝐼−𝐶⁻¹,𝛾∈𝑆.(63) The proof of Lemma 14 is completed.

Lemma 15. 0 is an eigenvalue of operator 𝐴^∗ with algebraic multiplicity one.

Proof. We prove Lemma 15 in two steps. Firstly, we prove that 0 is an eigenvalue of operator 𝐴^∗. For any 𝑄∈𝐷(𝐴^∗), consider the operator equation 𝐴^∗𝑄=0, that is, 𝜀+𝛼₀𝑄₀=𝜀𝑄₁+𝛼₀𝑄₂,(64)𝛼₀+𝜇₀𝑄₁=𝜇₀𝑄₀+𝛼₀𝑄₃,(65)𝜆𝑄₂=𝜆𝑄₅(0),(66)𝜆+𝜇₀𝑄₃=𝜇₀𝑄₂+𝜆𝑄₄,(67)𝜇₀𝑄₄=𝜇₀𝑄₅(0),(68)𝑄^′₅(𝑥)=𝜇(𝑥)𝑄₅(𝑥)−𝑄₀𝜇(𝑥).(69) Solving (69) yields 𝑄₅(𝑥)=𝑒^{∫𝑥0𝜇(𝑠)d𝑠}𝑄₅(0)−𝑄₀^𝑥₀𝜇(𝜏)𝑒^{−∫𝜏0𝜇(𝑠)d𝑠}d𝜏(70) multiplying 𝑒^{−∫𝑥0𝜇(𝑠)d𝑠} to the two sides of (70) and letting 𝑥→∞ by noting 𝑄₅(∞)<∞ derive 𝑄₅(0)=𝑄₀^∞₀𝜇(𝜏)𝑒^{−∫𝜏0𝜇(𝑠)d𝑠}d𝜏=𝑄₀.(71) Combing (64)–(68) with (71) follows 𝑄₀=𝑄₁=𝑄₂=𝑄₃=𝑄₄=𝑄₅(0).(72) Substituting (71) into (70) yields 𝑄₅(𝑥)=𝑄₀𝑒^{∫𝑥0𝜇(𝑠)d𝑠}1+^𝑥₀d𝑒^{−∫𝜏0𝜇(𝑠)d𝑠}=𝑄₀.(73) This implies that 𝜂𝑄^∗(𝜂≠0) is the eigenfunction corresponding to eigenvalue 0 of operator 𝐴^∗, where 𝑄^∗=(1,1,…,1)^T.
Next, we prove that the algebraic multiplicity of 0 in 𝑋^∗ is one. From the above step, we can see that the geometric multiplicity of 0 in 𝑋^∗ is one. Then, we only need to verify the algebraic index of 0 in 𝑋^∗ which is also one according to [31]. We use the reduction to absurdity. Suppose that the algebraic index of 0 in 𝑋^∗ is 2 without loss of generality, then there exists 𝑌∈𝑋^∗, such that 𝐴^∗𝑌=𝑄^∗. It is obvious that 0=⟨𝐴𝑃,𝑌⟩=𝑃,𝐴^∗𝑌=𝑃,𝑄^∗,(74) where 𝑃 is the positive eigenfunction corresponding to eigenvalue 0 of 𝐴. However, ⟨𝑃,𝑄^∗⟩=⁴_𝑖=0𝑃_𝑖+^∞₀𝑃₅(𝑥)d𝑥>0(75) which contradicts (74). Thus, the algebraic index of 0 in 𝑋^∗ is one. Therefore, the algebraic multiplicity of 0 in 𝑋^∗ 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 𝐶₀ 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 𝐴 generates a positive 𝐶₀ semigroup of contraction 𝑇(𝑡).

Theorem 17. The system (25) has a unique nonnegative time-dependent solution 𝑃(𝑡,⋅) which satisfies ‖𝑃(𝑡,⋅)‖=1,∀𝑡∈[0,∞).(76)

Proof. From Theorem 16 and [32], it can be derived that the system (25) has a unique nonnegative solution 𝑃(𝑡,⋅) which can be expressed as 𝑃(𝑡,⋅)=𝑇(𝑡)𝑃₀,∀𝑡∈[0,∞).(77) Because 𝑃(𝑡,⋅) satisfies (1)–(3), it is not difficult to know that d‖𝑃(𝑡,⋅)‖d𝑡=0.(78) Therefore, ‖𝑃(𝑡,⋅)‖=𝑇(𝑡)𝑃₀=𝑃₀=1,∀𝑡∈[0,∞).(79) This just reflects the physical meaning of 𝑃(𝑡,⋅). The proof of Theorem 17 is completed.

Remark 18. Because the initial value 𝑃₀ of the system (25) belongs to the domain 𝐷(𝐴) of the system operator 𝐴, then the nonnegative time-dependent solution of the system expressed in (77) is the strong solution of the system (25).
Noting that the 𝐶₀ semigroup 𝑇(𝑡) generated by 𝐴 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 𝑃 be the eigenfunction corresponding to eigenvalue 0 of the system operator 𝐴 satisfying ‖𝑃‖=1, and let 𝑄^∗ be defined in Lemma 15, then the time-dependent solution 𝑃(𝑡,⋅) of the system (25) converges to the nonnegative steady-state solution 𝑃. That is, lim_𝑡→∞𝑃(𝑡,⋅)=𝑃₀,𝑄^∗𝑃=𝑃,(80) where 𝑃₀ 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 𝐴 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 𝐶₀ semigroup. Therefore, the system solution is exponentially stable.

For convenience, we will introduce three operators first: 𝐵𝑃=^∞₀𝜇(𝑥)𝑃₅(𝑥)d𝑥+𝜇₀𝑃₁,𝜀𝑃₀,𝛼₀𝑃₀+𝜇₀𝑃₃,𝛼₀𝑃₁,𝜆𝑃₃,0^Twith𝐷(𝐵)=𝑋,𝐴=𝐴−𝐵with𝐷𝐴=𝑃=𝐷𝑃=₀,𝑃₁,…,𝑃₄,𝑃₅(𝑥)^T∈𝑋∣𝑃^′₅(𝑥)∈𝐿¹ℝ₊,𝑃₅(𝑥)isanabsolutelycontinuousfunction,𝐴₀=𝐴with𝐷𝐴₀=𝑃∈𝐷∣𝑃₅(0)=0.(81) It is easy to know that 𝐴 and 𝐴₀ are both closed operators with dense domains in 𝑋. And with the perturbation of 𝐶₀ semigroup, it is clear that 𝐴 also generates a 𝐶₀ semigroup 𝑆(𝑡).

Lemma 20. Assume that the mean of the repair rate exists and greater than zero, that is, 0<𝜇=lim_𝑥→∞1𝑥^𝑥₀𝜇(𝑠)d𝑠.(82) Then, 𝐴₀ generates a quasicompact semigroup 𝑇₀(𝑡).

Proof. Firstly, we will prove that 𝐴₀ generates a 𝐶₀ semigroup 𝑇₀(𝑡). Consider the following abstract Cauchy problem: d𝑃(𝑡,⋅)d𝑡=𝐴₀𝑃()𝑃()𝑡,⋅,𝑡≥0,0,⋅=Φ,(83) where Φ=(𝜑₀,𝜑₁,…,𝜑₄,𝜑₅(𝑥))^T∈𝑋. That is, dd𝑡+𝜀+𝛼₀𝑃₀(𝑡)=0,(84)dd𝑡+𝛼₀+𝜇₀𝑃₁(𝑡)=0,(85)dd𝑡+𝜆𝑃₂(𝑡)=0,(86)dd𝑡+𝜆+𝜇₀𝑃₃(𝑡)=0,(87)dd𝑡+𝜇₀𝑃₄(𝑡)=0,(88)𝜕+𝜕𝜕𝑡(𝑥𝑃𝜕𝑥+𝜇)₅(𝑡,𝑥)=0,(89)𝑃₅(𝑡,0)=0,(90)𝑃_𝑖(0)=𝜑_𝑖,𝑖=0,1,…,4,(91)𝑃₅(0,𝑥)=𝜑₅(𝑥).(92) Solving (84)–(88) with the help of (91) yields 𝑃₀(𝑡)=𝜑₀𝑒^{−(𝜀+𝛼0)𝑡},𝑃₁(𝑡)=𝜑₁𝑒^{−(𝛼0+𝜇0)𝑡},𝑃₂(𝑡)=𝜑₂𝑒^−𝜆𝑡,𝑃₃(𝑡)=𝜑₃𝑒^{−(𝜆+𝜇0)𝑡},𝑃₄(𝑡)=𝜑₄𝑒^−𝜇0𝑡.(93) Solving (89) with the help of (90) and (92) by the method of characteristics yields 𝑃₅()=𝑃𝑡,𝑥⁵(𝑡−𝑥,0)𝑒^{−∫𝑥0𝜇(𝜏)d𝜏}𝑃=0,𝑡>𝑥₅(0,𝑥−𝑡)𝑒^{−∫𝑡0𝜇(𝑥−𝑡+𝜏)d𝜏}=𝜑₅(𝑥−𝑡)𝑒^{−∫𝑥𝑥−𝑡𝜇(𝜏)d𝜏},𝑡≤𝑥.(94) Therefore, it is easy to prove that 𝐴₀ generates a 𝐶₀ semigroup 𝑇₀(𝑡) satisfying 𝑇₀(𝑡)Φ(𝑥)=Ψ₁,0^TΨ,𝑥<𝑡₁,𝜑₅(𝑥−𝑡)𝑒^{−∫𝑥𝑥−𝑡𝜇(𝜏)d𝜏T},𝑥≥𝑡.(95) Here, Ψ₁=𝜑₀𝑒^{−(𝜀+𝛼0)𝑡},𝜑₁𝑒^{−(𝛼0+𝜇0)𝑡},𝜑₂𝑒^−𝜆𝑡,𝜑₃𝑒^{−(𝜆+𝜇0)𝑡},𝜑₄𝑒^−𝜇0𝑡.(96)
Next, we will prove that 𝑇₀(𝑡) is quasicompact. We only need to prove that the essential growth bound 𝑊_ess(𝐴₀) is less than zero.
The assumption condition (82) implies that for any 𝜀>0, there exists 𝑡₀>0 such that 1𝑡^𝑥_𝑥−𝑡𝜇(𝑠)d𝑠>𝜇−𝜀,𝑥≥𝑡≥𝑡₀.(97) With the help of (97), it is not difficult to deduce that 𝑇₀(𝑡)Φ=𝜑₀𝑒^{−(𝜀+𝛼0)𝑡}+𝜑₁𝑒^{−(𝛼0+𝜇0)𝑡}+𝜑₂𝑒^−𝜆𝑡+𝜑₃𝑒^{−(𝜆+𝜇0)𝑡}+𝜑₄𝑒^−𝜇0𝑡+^∞_𝑡𝜑₅(𝑥−𝑡)𝑒^{−∫𝑥𝑥−𝑡𝜇(𝜏)d𝜏}d𝑥<𝜑₀𝑒^{−(𝜀+𝛼0)𝑡}+𝜑₁𝑒^{−(𝛼0+𝜇0)𝑡}+𝜑₂𝑒^−𝜆𝑡+𝜑₃𝑒^{−(𝜆+𝜇0)𝑡}+𝜑₄𝑒^−𝜇0𝑡+^∞_𝑡𝜑₅(𝑥−𝑡)𝑒^{−(𝜇−𝜀)𝑡}d𝑥=𝜑₀𝑒^{−(𝜀+𝛼0)𝑡}+𝜑₁𝑒^{−(𝛼0+𝜇0)𝑡}+𝜑₂𝑒^−𝜆𝑡+𝜑₃𝑒^{−(𝜆+𝜇0)𝑡}+𝜑₄𝑒^−𝜇0𝑡+𝑒^{−(𝜇−𝜀)𝑡∞}₀𝜑₅(𝑥)d𝑥≤𝑒^{−min{𝜀+𝛼0,𝛼0+𝜇0,𝜆,𝜆+𝜇0,𝜇0,𝜇−𝜀}𝑡}‖Φ‖.(98) This manifests that 𝑇₀(𝑡)≤𝑒^{−min{𝜀+𝛼0,𝜆,𝜇0,𝜇−𝜀}𝑡}.(99) Then, 𝑊_ess𝐴₀≤𝑊𝐴₀=lim_𝑡→∞ln𝑇₀(𝑡)𝑡≤−min𝜀+𝛼₀,𝜆,𝜇₀,𝜇−𝜀<0.(100) Therefore, 𝐴₀ generates a quasicompact 𝐶₀ semigroup 𝑇₀(𝑡). The proof of Lemma 20 is completed.

For 𝛾>0, 𝑃∈𝑋, let Φ_𝛾(𝑃)(𝑥)=diag0,0,0,0,0,𝜆𝑃₂+𝜇₀𝑃₄⋅𝐸_𝛾(𝑥),(101) where 𝐸_𝛾(𝑥)=(0,0,0,0,0,𝑒^{−∫𝑥0[𝛾+𝜇(𝑠)]d𝑠})^T∈Ker(𝛾𝐼−𝐴) and Φ_𝛾 is a compact operator. Then, it is not difficult to obtain the following result.

Lemma 21. 𝐼+Φ_𝛾 is a bijection from 𝐷(𝐴₀) to 𝐷(𝐴) and 𝛾𝐼−(𝐴−𝐵)𝐼+Φ_𝛾=𝛾𝐼−𝐴₀.(102)

Lemma 22. 𝑆(𝑡)−𝑇₀(𝑡) is a compact operator, for any 𝑡≥0. Here 𝑆(𝑡) is the 𝐶₀ semigroup generated by 𝐴.

Proof. From Lemma 21, we can see that 𝑅(𝛾,𝐴−𝐵)≥𝑅(𝛾,𝐴₀), for any 𝛾>0. Therefore 𝑆(𝑡)≥𝑇₀(𝑡), for any 𝑡≥0.
For 𝑃∈𝐷(𝐴₀), set Ψ(𝑠)𝑃=𝑆(𝑡−𝑠)𝐼+Φ_𝛾𝑇₀(𝑠)𝑃,(103) where 0≤𝑠≤𝑡, 𝛾>0. Recalling the properties of 𝐶₀-semigroup and Lemma 21, we can obtain Ψ^′(𝑠)𝑃=−𝑆(𝑡−𝑠)(𝐴−𝐵)𝐼+Φ_𝛾𝑇₀(𝑠)𝑃+𝑆(𝑡−𝑠)𝐼+Φ_𝛾𝐴₀𝑇₀(𝑠)𝑃=𝑆(𝑡−𝑠)𝛾𝐼−(𝐴−𝐵)𝐼+Φ_𝛾𝑇₀(𝑠)𝑃+𝑆(𝑡−𝑠)𝐼+Φ_𝛾−𝛾𝐼+𝐴₀𝑇₀(𝑠)𝑃=𝑆(𝑡−𝑠)𝛾𝐼−𝐴₀𝑇₀(𝑠)𝑃+𝑆(𝑡−𝑠)𝐼+Φ_𝛾−𝛾𝐼+𝐴₀𝑇₀(𝑠)𝑃=𝑆(𝑡−𝑠)Φ_𝛾−𝛾𝐼+𝐴₀𝑇₀(𝑠)𝑃.(104) Since [Ψ(𝑡)−Ψ(0)]𝑃=∫^𝑡₀Ψ^′(𝑠)𝑃d𝑠, then [Ψ(𝑡)−Ψ(0)]𝑃=^𝑡₀𝑆(𝑡−𝑠)Φ_𝛾−𝛾𝐼+𝐴₀𝑇₀(𝑠)𝑃d𝑠.(105) That is, 𝑆(𝑡)𝑃−𝑇₀(𝑡)𝑃=−^𝑡₀𝑆(𝑡−𝑠)Φ_𝛾−𝛾𝐼+𝐴₀×𝑇₀(𝑠)𝑃d𝑠+Φ_𝛾𝑇₀(𝑡)𝑃−𝑆(𝑡)Φ_𝛾𝑃.(106) Therefore, 𝑆(𝑡)−𝑇₀(𝑡)(𝑡≥0) is compact because the right-hand side of the above equation is the sum of three compact operators for the compactness of Φ_𝛾. The proof of Lemma 22 is completed.

From the above preparations, we can present the main results of this section.

Theorem 23. 𝐶₀ semigroup 𝑇(𝑡) generated by the system operator 𝐴 is quasicompact.

Proof. According to Proposition 9.20 (see [34]) combing Lemmas 22 and 20, we can deduce that 𝑊_ess𝐴≤𝑊𝐴₀<0.(107) This shows that 𝑆(𝑡), the 𝐶₀ semigroup generated by 𝐴 is quasicompact. Because 𝐵 is a compact operator, then according to [35], it is evident that 𝑊_ess(𝐴)=𝑊_ess(𝐴−𝐵)<0.(108) This implies that 𝑇(𝑡) is quasicompact. The proof of Theorem 23 is completed.

Theorem 24. The time-dependent solution of the system (1)–(3) strongly converges to its steady-state solution. That is, lim_𝑡→∞𝑃(𝑡,⋅)=𝑃.(109) Moreover, there exist 𝐶>0 and 𝜀>0 such that 𝑃(𝑡,⋅)−𝑃≤𝐶𝑒^−𝜀𝑡.(110) Here, 𝑃 is defined in Theorem 19.

Proof. Recalling Theorem 2.10 (see [35]) combing Theorem 23, we can derive that 𝐶₀ semigroup 𝑇(𝑡) generated by the system operator 𝐴 can be decomposed as 𝑇(𝑡)=𝑃₀+𝑅(𝑡), where 𝑃₀ is the residue corresponding to eigenvalue 0 and ‖𝑅(𝑡)‖≤𝐶𝑒^−𝜀𝑡 for suitable constants 𝜀>0 and 𝐶>0.
However, by Theorem 19, the nonnegative solution of the system (1)–(3) can be expressed as 𝑃(𝑡,⋅)=𝑇(𝑡)𝑃₀, 𝑡∈[0,∞). Then, combing Theorem 12.3 in [36], we can derive that 𝑃(𝑡,⋅)=𝑇(𝑡)𝑃₀=𝑃₀+𝑅(𝑡)𝑃₀=𝑃₀,𝑄^∗𝑃+𝑅(𝑡)𝑃₀=𝑃+𝑅(𝑡)𝑃₀,(111) where 𝑄^∗ is defined in Lemma 15. Hence, we can get 𝑃(𝑡,⋅)−𝑃≤𝐶𝑒^−𝜀𝑡.(112) The proof of Theorem 24 is completed.

5. Reliability Indices

In this section, we will discuss some reliability indices of the system. Noting that the eigenfunction corresponding to eigenvalue 0 of the system operator 𝐴 is just the steady-state solution of system (1)–(3), we are dedicated to studying some primary steady-state indices of the system from the point of eigenfunction.

We first analyze the eigenfunction corresponding to eigenvalue 0 of the system operator 𝐴. In the proof process of Lemma 11, solving (31)–(36) with the help of (37) by letting 𝛾=0 and 𝐺=0 derives 𝑃₁=𝜀𝛼₀+𝜇₀𝑃₀,(113)𝑃₂=𝛼₀𝜆𝑃₀+𝜇₀𝜆𝑃₃=𝛼₀𝜆+𝜀𝛼₀𝜇₀𝜆𝜆+𝜇₀𝛼₀+𝜇₀𝑃₀,(114)𝑃₃=𝛼₀𝜆+𝜇₀𝑃₁=𝜀𝛼₀𝜆+𝜇₀𝛼₀+𝜇₀𝑃₀,(115)𝑃₄=𝜆𝜇₀𝑃₃=𝜆𝜀𝛼₀𝜇₀𝜆+𝜇₀𝛼₀+𝜇₀𝑃₀,(116)𝑃₅(𝑥)=𝛼₀𝜀1+𝛼₀+𝜇₀𝑃₀𝑒^{−∫𝑥0𝜇(𝑠)d𝑠}.(117) Let 𝑃₅=^∞₀𝑃₅(𝑥)d𝑥=𝛼₀𝜀1+𝛼₀+𝜇₀𝑃₀𝑔(0),(118) where 𝑔(0)=∫^∞₀𝑒^{−∫𝑥0𝜇(𝑠)d𝑠}d𝑥 and 𝑆=⁵_𝑖=0𝑃_𝑖=𝜇₀𝜆+𝜇₀𝛼₀+𝜇₀+𝜀𝜆+𝛼₀+𝜆𝛼₀𝑔(0)+𝜆²𝜀𝛼₀𝜆𝜇₀𝛼₀+𝜇₀𝜆+𝜇₀𝑃₀.(119)

Theorem 25. The steady-state availability of the system is 𝐴_𝑣=𝜇₀𝜆+𝛼₀𝜆+𝜇₀𝛼₀+𝜇₀+𝜀𝜇₀𝜆+𝜇₀𝛼₀+𝜇₀+𝜀𝜆+𝛼₀+𝜆𝛼₀𝑔(0)+𝜆²𝜀𝛼₀.(120)

Proof. The instantaneous availability of the system at time 𝑡 is 𝐴_𝑣(𝑡)=³_𝑖=0𝑃_𝑖(𝑡).(121) Let 𝑡→∞, then the steady-state availability of the system is obtained as follows: 𝐴_𝑣=∑³_𝑖=0𝑃_𝑖𝑆=𝜇₀𝜆+𝛼₀𝜆+𝜇₀𝛼₀+𝜇₀+𝜀𝜇₀𝜆+𝜇₀𝛼₀+𝜇₀+𝜀𝜆+𝛼₀+𝜆𝛼₀𝑔(0)+𝜆²𝜀𝛼₀.(122) The proof of Theorem 25 is completed.

Theorem 26. The steady-state probability of the repairman vacation is 𝑃_𝑣=𝜆𝜀𝛼₀+𝜇₀𝜆+𝜇₀𝜇₀𝜆+𝜇₀𝛼₀+𝜇₀+𝜀𝜆+𝛼₀+𝜆𝛼₀𝑔(0)+𝜆²𝜀𝛼₀.(123)

Proof. The instantaneous probability of the repairman vacation at time 𝑡 is 𝑃_𝑣(𝑡)=𝑃₁(𝑡)+𝑃₃(𝑡)+𝑃₄(𝑡).(124) Letting 𝑡→∞ derives the steady-state probability of the repairman vacation: 𝑃_𝑣=𝑃₁+𝑃₃+𝑃₄𝑆=𝜆𝜀𝛼₀+𝜇₀𝜆+𝜇₀𝜇₀𝜆+𝜇₀𝛼₀+𝜇₀+𝜀𝜆+𝛼₀+𝜆𝛼₀𝑔(0)+𝜆²𝜀𝛼₀.(125) The proof of Theorem 26 is completed.

Theorem 27. The steady-state probability of the system in warning state is 𝑃_𝑤=𝛼₀𝜇₀𝜆+𝜇₀𝛼₀+𝜇₀+𝜀𝜇₀𝜆+𝜇₀𝛼₀+𝜇₀+𝜀𝜆+𝛼₀+𝜆𝛼₀𝑔(0)+𝜆²𝜀𝛼₀.(126)

Proof. The instantaneous probability of the system in warning state at time 𝑡 is 𝑃_𝑤(𝑡)=𝑃₂(𝑡)+𝑃₃(𝑡).(127) Letting 𝑡→∞ yields the steady-state probability of the system in warning state: 𝑃_𝑤=𝑃₂+𝑃₃𝑆=𝛼₀𝜇₀𝜆+𝜇₀𝛼₀+𝜇₀+𝜀𝜇₀𝜆+𝜇₀𝛼₀+𝜇₀+𝜀𝜆+𝛼₀+𝜆𝛼₀𝑔(0)+𝜆²𝜀𝛼₀.(128) The proof of Theorem 27 is completed.

Theorem 28. The steady-state failure frequency of the system is 𝑊_𝑓=𝜆𝑃_𝑤.(129)

Proof. Let 𝑃₅(𝑡)=∫^∞₀𝑃₅(𝑡,𝑥)d𝑥 and 𝜇(𝑡)=∫^∞₀𝜇(𝑥)𝑃₅(𝑡,𝑥)d𝑥/𝑃₅(𝑡). Then, the matrix of the transition probability of the system (1)–(3) can be obtained by (1)-(2) as follows: 𝑇=−𝜀−𝛼₀𝜇₀000𝜇(𝑡)𝜀−𝛼₀−𝜇₀𝛼0000₀0−𝜆𝜇₀000𝛼₀0−𝜆−𝜇₀00000𝜆−𝜇₀000𝜆0𝜇₀(𝑡).−𝜇(130) Thus, by [37] the instantaneous failure frequency of the system at time 𝑡 can be derived as 𝑊_𝑓(𝑡)=𝜆𝑃₂(𝑡)+𝑃₃(𝑡).(131) Let 𝑡→∞, then the steady-state failure frequency is immediate 𝑊_𝑓=𝜆𝑃₂+𝑃₃𝑆=𝜆𝑃_𝑤.(132) The proof of Theorem 28 is completed.

6. Applications and Numerical Examples

Reference [38] discussed the effects of the delayed vacation and vacation policies on a system. That is, the shorter the delayed vacation time, the larger the reliability and failure frequency of a system; and the reliability of a system with multiple vacations is smaller than that of a system with single vacation, while the profit of a system with multiple vacations is larger than that of a system with single vacation. Therefore, in this section, we only concentrate on that how the warning device will affect the system. Specifically, we will compare the reliability, availability, and profit of the system with warning device and those of the system without warning device and present some numerical examples.

6.1. System without Warning Device

The simple repairable system without warning device and with a repairman following delayed-multiple vacations policy is as follows: dd𝑡+𝜀+𝜆𝑃₀(𝑡)=𝜇₀𝑃₁(𝑡)+^∞₀𝜇(𝑥)𝑃₅(𝑡,𝑥)d𝑥,(133)dd𝑡+𝜇₀+𝜆𝑃₁(𝑡)=𝜀𝑃₀(𝑡),(134)dd𝑡+𝜇₀𝑃₄(𝑡)=𝜆𝑃₁(𝑡),(135)𝜕+𝜕𝜕𝑡(𝑥𝑃𝜕𝑥+𝜇)₅(𝑡,𝑥)=0(136) with boundary condition 𝑃₅(𝑡,0)=𝜆𝑃₀(𝑡)+𝜇₀𝑃₄(𝑡)(137) and initial conditions 𝑃₀(0)=1,theothersequalto0.(138)

Then, by the same method of Theorems 25, 26, and 28, the corresponding reliability indices of the system (133)–(138) are as follows.

Theorem 29. The steady-state availability of the system (133)–(138) is 𝐴_𝑣=𝜇₀𝜇₀+𝜆+𝜀𝜇₀𝜇₀(0),+𝜆+𝜀1+𝜆𝑔+𝜆𝜀(139) where 𝑔(0) is defined in Section 5.

Theorem 30. The steady-state probability of the repairman vacation of the system (133)–(138) is 𝑃_𝑣=𝜀𝜇₀+𝜆𝜇₀𝜇₀(0).+𝜆+𝜀1+𝜆𝑔+𝜆𝜀(140)

Theorem 31. The steady-state failure frequency of the system (133)–(138) is 𝑊_𝑓=𝜆𝐴_𝑣.(141)

6.2. Numerical Examples

By comparing the two groups of (120) and (139) and (129) and (141), it is not difficult to deduce the following results.(i)The steady-state availability of the system with warning device (i.e., system (1)–(3)) is larger than that of the system without warning device (i.e., system (133)–(138)). That is, 𝐴_𝑣>𝐴_𝑣.(ii)If 𝛼₀≤𝜆, then the steady-state failure frequency of the system with warning device is less than that of the system without warning device. That is, 𝑊_𝑓<𝑊_𝑓. While if 𝛼₀>𝜆, the magnitude of 𝑊_𝑓 and 𝑊_𝑓 cannot be determined.

Let 𝐼 and 𝐼̃ be the total profit of the system with and without warning device, respectively. That is, 𝐼=𝑐₁𝐴_𝑣−𝑐₂𝑊_𝑓+𝑐₃𝑃_𝑣,𝐼̃=𝑐₁𝐴_𝑣−𝑐₂𝑊_𝑓+𝑐₃𝑃_𝑣.(142) Here, 𝑐₁, 𝑐₂, and 𝑐₃ represent the income of the system for working unit per unit time, the loss of the system for failed unit per unit time, and the income of the system for the repairman vacation per unit time, respectively. Given 𝜀=1, 𝜇₀=0.5, 𝛼₀=0.2, 𝑐₁=50, 𝑐₂=15, 𝑐₃=30, let 𝐷=𝐼−𝐼̃. Then, 𝐷 is a function of 𝜆 and 𝜇. From Figure 10, we can know that the profit of the system with warning device is larger than that of the system without warning device.

From the above discussions, we can deduce that because both the availability and profit of a system with warning device are larger than those of a system without warning device. Then, a system with warning device is better than a system without warning device in practice.

In the following, we will present some numerical examples to illustrate the conclusions.

(1) Let 𝜀=1, 𝜇₀=𝜇=0.5, 𝛼₀=0.2, 𝜆=0.01,0.05,0.1, and 0.5, respectively. Figures 1 and 2 present the instantaneous probabilities of the systems with and without warning device in good state with different values of 𝜆. Let 𝜀=1, 𝜆=0.1, 𝜇₀=0.5, 𝛼₀=0.2, 𝜇=0.1,0.3,0.5, and 1, respectively. Figures 3 and 4 present the instantaneous probabilities of the systems with and without warning device in good state with different values of 𝜇. From the four figures, we can see that the instantaneous probabilities of the system without warning device in good state decrease with the increasing of 𝜆, while the instantaneous probabilities of the system with warning device in good state increase with the increasing of 𝜆. But both the instantaneous probabilities of the systems with and without warning device in good state increase with the increasing of 𝜇 in general.

Figure 1 

Instantaneous probabilities of the system without warning device in good state with different 𝜆.

Figure 2 

Instantaneous probabilities of the system with warning device in good state with different 𝜆.

Figure 3 

Instantaneous probabilities of the system without warning device in good state with different 𝜇.

Figure 4 

Instantaneous probabilities of the system with warning device in good state with different 𝜇.

(2) Let 𝜀=1, 𝜆=0.1, 𝜇₀=0.5, 𝜇=0.2, 𝛼₀=0.2. Figure 5 presents the instantaneous availabilities of the systems with and without warning device. From the figure, we can see that the availability of the system with warning device is larger than that of the system without warning device.

Figure 5 

Instantaneous availabilities of the systems with and without warning device.

(3) Let 𝜀=1, 𝜆=0.1, 𝜇₀=0.5, 𝜇=0.2 and 𝛼₀=0.01,0.2,1, and 5, respectively. Figures 6, 7, 8, and 9 present the instantaneous failure frequencies of the systems with and without warning device. It can be derived from the four figures that when 𝛼₀<𝜆, the instantaneous failure frequency of the system with warning device is less than that of the system without warning device. But when 𝛼₀≥1≫𝜆, the instantaneous failure frequency of the system with warning device is larger than that of the system without warning device.

Figure 6 

Instantaneous failure frequencies of the systems with and without warning device with 𝛼=0.01.

Figure 7 

Instantaneous failure frequencies of the systems with and without warning device with 𝛼=0.2.

Figure 8 

Instantaneous failure frequencies of the systems with and without warning device with 𝛼=1.

Figure 9 

Instantaneous failure frequencies of the systems with and without warning device with 𝛼=5.

Figure 10 

Profit difference between the systems with and without warning device.

(4) Let 𝜀=1, 𝜇₀=0.5, 𝛼₀=0.2, 𝑐₁=50, 𝑐₂=15, 𝑐₃=30. Figure 10 presents the profit difference between the systems with and without warning device. From the figure, it can be derived easily that the profit the system with warning device is more than that of the system without warning device.

7. Conclusion

In this paper, we proposed a simple repairable system with a warning device and a repairman who can have delayed-multiple vacations. Because the two hypotheses used for Laplace transform in order to obtain the steady-state solution of a repairable system in traditional reliability research that needs to be verified, and the substitution of steady-state solution for the dynamic one that should be based on some conditions, the study of well-posedness of the time-dependent solution of a system is in demand in terms of theory and practice. In this paper, we first transformed the system model into a group of operator equations and obtained the existence and uniqueness as well as 𝐶¹ continuity of the system solution by functional analysis method. Then to study the stability of the system, we translated the system model into an abstract Cauchy problem in a suitable Banach space. The asymptotic stability and further the exponential stability of the system solution were derived by using 𝐶₀ semigroup theory and compact operator disturbance theorem. Because the stable solution of the system is just the eigenfunction corresponding to eigenvalue 0 of the system operator, we also presented some reliability indices, such as reliability, failure frequency, probabilities of repairman vacation, and system in warning state of the system in the viewpoint of eigenfunction. At the end of the paper, by the theoretical and numerical analyses, we give the conclusion that the system with warning device is better than the system without warning device in practice.

Acknowledgments

This work is supported by the TianYuan Special Funds of the National Natural Science Foundation of China under Grant no. 11226249, the Foundation Research Project of Shanxi Province (The Youth) under Grant no. 2012021015-5, the Youth Foundation of Taiyuan University of Technology under Grant no. 2012L031, the Special Funds of the National Natural Science Foundation of China under Grant no. 61250011, and the Foundation Research Project of Shanxi Province (The Natural Science) under Grant no. 2012011004-4.