The Well-Posedness and Stability Analysis of a Computer Series System

Abstract

A repairable computer system model which consists of hardware and software in series is established in this paper. This study is devoted to discussing the unique existence of the solution and the stability of the studied system. In view of 𝑐₀ semigroup theory, we prove the existence of a unique nonnegative solution of the system. Then by analyzing the spectra distribution of the system operator, we deduce that the transient solution of the system strongly converges to the nonnegative steady-state solution which is the eigenvector corresponding to eigenvalue 0 of the system operator. Finally, some reliability indices of the system are provided at the end of the paper with a new method.

1. Introduction

With the development of the modern technology and the extensive use of the electronic products, the reliability problem of the repairable systems has become a hot topic. It is well known that the reliability of a system is an important concept in engineering. The high degree of reliability is usually achieved by introducing redundancy or repairman (e.g., [1–4]) or applying preventive maintenance (e.g., [5, 6]), optimal inspection plans (e.g., [7–9]), or optimal replacement policy (e.g., [10]). The aim is to increase the performance of the system by reducing the downtime or the maintenance and inspection cost of the system.

In the general reliability analysis of the computer system, however, because of different characteristics of the hardware and software, we cannot simply take the hardware and software as a unit or two different types of units [9]. Then, it is rare to analyze synthetically [11]. With the passage of the using time and the number of failures increasing, the reliability of the hardware would descend, and the repair time would be longer [12]. During the software debugging and testing stages, as the failures occur, potential software error is discovered and corrected constantly which make the software reliability grow [13]. Since the hardware failure or software failure leads to the whole computer system failure, the computer system can be formulated as a series system with hardware and software (namely, hardware and software in series). There are some obstacles to overcome to obtain the main result since our model is more complicated than that of [11–13].

In this paper, we study a repairable computer system which is composed of hardware and software in series. The unique existence of the system solution is obtained by using 𝑐₀ semigroup theory. The exponential stability of the system is further achieved by analyzing the spectrum distribution of the system operator given by (2)–(4), which shows that the solution to the system (2)–(4) is exponentially stable. Thus we not only provide strict theoretical foundation for reliability study but also make it more valuable in practice.

The remainder of the paper is organized as follows. In Section 2 we formulate the mathematical model of the system with concerned notations; in Section 3.1 we show the unique existence of the dynamic solution of the system. In Section 3.2 we study the unique existence of the solution of the abstract Cauchy problem corresponding to the system and present a detailed spectral analysis of the system operator; some steady-state reliability indices of the system are presented in Section 4, and Section 5 concludes the paper.

2. Mathematical Model Formulation

In the reliability analysis of repair system, it is usually assumed that the repaired units which compose system are as good as new, and the failed units are repaired immediately. However, in reality it is usually not the case. In real life, it is possible that the reliability reduces after the software failure each time. That is, the condition 𝐹_𝑆^(𝑛)(𝑡)=𝐹_𝑆(𝑎^𝑛−1𝑡)=1−𝑒^{−𝑎𝑛−1𝜆𝑠𝑡}, 𝑡≥0, 𝜆_𝑠>0, and the coefficient 𝑎>1. With the number of repair times increasing, the failure rate is increasing gradually. In view of the aging and accumulative wear, the repair time will become longer and longer and tend towards infinity; that is, the system is nonrepairable. Therefore, we first suppose that the software is overhauled (replaced) to be as good as new after the (𝑁−1)th minimal repair, and studying the number of minimal repair before overhaul repair is more appropriate. And we will also discuss how its reliability will be affected by the number of minimal repair and overhaul. In [14], the author supposed that the software cannot be repaired as good as new and utilized the geometric process and supplementary variable technique to analyze the system reliability. However, the life and repair times of the hardware and software are supposed to follow exponential distribution. In this paper, under assumption that the life time of the hardware and software follows exponential distribution and repair time is subject to the general distribution, we set up a mathematical model of the repairable computer system by the supplementary variable method, which is composed of the hardware and software in series. The hardware is repaired to be as good as new, the software is repaired periodically, and restored software life decreases. After a period of time, an overhaul makes it as a new one.

The system model is formulated specifically as follows. (i)The computer system is composed of hardware 𝐻 and software 𝑆 in series.(ii)The distribution function of the life time 𝑋_𝐻 of hardware 𝐻 is 𝐹_𝐻(𝑡)=1−𝑒^{−𝜆ℎ𝑡}, 𝑡≥0, 𝜆_ℎ>0.(iii)The distribution function of the life time 𝑋_𝑆^(𝑛) of software 𝑆 during its 𝑛th period (e.g., the time between the completion of its (𝑛−1)th repair and that of the 𝑛th repair) is 𝐹^𝑛_𝑆(𝑡)=𝐹_𝑆(𝑎^𝑛−1𝑡)=1−𝑒^{−𝑎𝑛−1𝜆𝑠𝑡}, 𝑡≥0, 𝜆_𝑠>0, 𝑎>1, 𝑛=1,2,…,𝑁.(iv)Let 𝑌₁ be the repair time of hardware 𝐻, 𝑌₂ the repair time of software 𝑆 after its 𝑛th failure (𝑛=1,2,…,𝑁−1), and 𝑌₃ the repair time of software 𝑆 after its 𝑛th failure, respectively. Their distribution functions are 𝐺_𝑗(𝑡)=∫^𝑡₀𝑔_𝑗(𝑥)𝑑𝑥=1−𝑒^{−∫𝑡0𝜇𝑗(𝑥)𝑑𝑥} and 𝐸[𝑌_𝑗]=1/𝜇_𝑗, 𝑗=1,2,3, where 𝜇₃(𝑥)>𝜇₂(𝑥), for all 𝑥≥0.(v)The hardware 𝐻 is repaired as good as new. The software 𝑆 is performed by a minimal repair (e.g., a maintenance action performed on a failed system by which its survival time is decreasing) during its 𝑛th (𝑛=1,2,…,𝑁−1) period and an overhaul repair (e.g., a maintenance action performed on a failed system by which it is repaired as good as new) during its 𝑁th period. The above stochastic variables are independent of each other.

Let 𝑁(𝑡) be the system state at time 𝑡, and assume all the possible states as below.

0𝑖 (𝑖=1,2,…,𝑁) the system is working.

1𝑖 (𝑖=1,2,…,𝑁) the system is failed because the failure hardware 𝐻 is being repaired in the 𝑖th time.

2𝑖 (𝑖=1,2,…,𝑁) the system has in failure state because the failure software 𝑆 is being repaired minimally in the 𝑖th time (𝑖=1,2,…,𝑁−1) and the software 𝑆 is being overhauled in the 𝑁th time.

Then the system state space is 𝐸={0𝑖,1𝑖,2𝑖} (𝑖=1,2,…,𝑁), in which the working state space is 𝑊={0𝑖} (𝑖=1,2,…,𝑁) and the failure state space is 𝐹={1𝑖,2𝑖} (𝑖=1,2,…,𝑁).

When 𝑁(𝑡)=𝑗𝑖  (𝑗=0,1,2; 𝑖=1,2,…,𝑁) supplement variable 𝑌_𝑗(𝑡) (𝑗=1,2,3) which denotes the elapsed repair time of hardware 𝐻, the elapsed minimal repair time of software 𝑆, and its elapsed overhaul time at time 𝑡, respectively, then {𝑁(𝑡),𝑌_𝑗(𝑡)} constitutes a matrix Markov process whose state probabilities are defined as follows: 𝑃_0𝑖(𝑡){𝑁(𝑡)}𝑃=𝑃=0𝑖,𝑖=1,2,…,𝑁,_1𝑖(𝑥,𝑡)=𝑃𝑥<𝑌₁(𝑡)(𝑡),𝑃≤𝑥+𝑑𝑥,𝑁=1𝑖𝑖=1,2,…,𝑁,_2𝑖(𝑥,𝑡)=𝑃𝑥<𝑌₂(𝑡)(𝑡),𝑃≤𝑥+𝑑𝑥,𝑁=2𝑖𝑖=1,2,…,𝑁−1,_2𝑁(𝑥,𝑡)=𝑃𝑥<𝑌₃(𝑡)≤𝑥+𝑑𝑥,𝑁(𝑡)=2𝑁. (1) Then by using the method of probability analysis, the system under consideration can be formulated as the following equations: 𝑑𝑃₀₁(𝑡)𝜆𝑑𝑡=−_ℎ+𝜆_𝑠𝑃₀₁(𝑡)+^∞₀𝑃₁₁(𝑥,𝑡)𝜇₁(𝑥)+𝑑𝑥_∞0𝑃_2𝑁(𝑥,𝑡)𝜇₃(𝑥)𝑑𝑥,𝑑𝑃_0𝑖(𝑡)𝜆𝑑𝑡=−_ℎ+𝑎^𝑖−1𝜆_𝑠𝑃_0𝑖(𝑡)+^∞₀𝑃_1𝑖(𝑥,𝑡)𝜇₁(𝑥)𝑑𝑥+^∞₀𝑃_2(𝑖−1)(𝑥,𝑡)𝜇₂(𝑥)𝑑𝑥,𝑖=2,3,…,𝑁,𝜕𝑃_1𝑖(𝑥,𝑡)+𝜕𝑥𝜕𝑃_1𝑖(𝑥,𝑡)𝜕𝑡=−𝜇₁(𝑥)𝑃_1𝑖(𝑥,𝑡),𝑖=1,2,…,𝑁,𝜕𝑃_2𝑖(𝑥,𝑡)+𝜕𝑥𝜕𝑃_2𝑖(𝑥,𝑡)𝜕𝑡=−𝜇₂(𝑥)𝑃_2𝑖(𝑥,𝑡),𝑖=1,2,…,𝑁−1,𝜕𝑃_2𝑁(𝑥,𝑡)+𝜕𝑥𝜕𝑃_2𝑁(𝑥,𝑡)𝜕𝑡=−𝜇₃(𝑥)𝑃_2𝑁(𝑥,𝑡). (2) The boundary conditions are 𝑃_1𝑖(0,𝑡)=𝜆_ℎ𝑃_0𝑖(𝑡)𝑃,𝑖=1,2,…,𝑁,_2𝑖(0,𝑡)=𝑎^𝑖−1𝜆_𝑠𝑃_0𝑖(𝑡),𝑖=1,2,…,𝑁. (3) The initial conditions are 𝑃₀₁(0)=1,andtheothersareequalto0. (4) Taking into account the practical background, we assume that 0<𝐾=sup_{[)𝑥∈0,∞}𝜇_𝑗(𝑥)<∞,_𝑇0𝜇_𝑗(𝑥)𝑑𝑥<∞,0<𝑇<∞,_∞0𝜇_𝑗(𝑥)𝑑𝑥=∞,𝑗=1,2,3. (5)

And then, we may know that many repairs/services are done periodically in practice. So we can suppose that the mean of repair/service rate exists and does not equal to 0 ([15, 16]): 0<𝜇_𝑗=lim_𝑥→∞1𝑥^𝑥₀𝜇_𝑗(𝜏)𝑑𝜏<∞,𝑗=1,2,3. (6)

3. Stability Analysis

In this section, we firstly study the unique existence of the classical solution of the system by pure analysis method in Section 3.1. In Section 3.2, we will formulate the problem into a suitable Banach space. Then we explain that the system has a unique generalized solution, and it is just the classical one when 𝑡>0. We also carry out a detailed spectral analysis of the system operator 𝐴+𝐸. Finally, the exponential stability of the system can be readily achieved.

3.1. Unique Existence of the Classical Solution

Theorem 1. The system (2)–(4) has a unique nonnegative solution in 𝐶[0,𝑇], for any 𝑇>0.

Proof. Solving (2)–(4) with the method in [17] one gets 𝑃₀₁(𝑡)=𝑒^−𝜋1𝑡+^∞₀𝑘₁₁𝑡−𝜂𝑃₁₁+0,𝜂𝑑𝜂_∞0𝑘_2𝑁𝑡−𝜂𝑃_2𝑁𝑃0,𝜂𝑑𝜂,_0𝑖(𝑡)=𝑒^{−𝜋𝑖𝑡}+^∞₀𝑘_1𝑖𝑡−𝜂𝑃_1𝑖+0,𝜂𝑑𝜂_∞0𝑘_2(𝑖−1)𝑡−𝜂𝑃_2(𝑖−1)𝑃0,𝜂𝑑𝜂,𝑖=2,3…,𝑁,₁₁(0,𝑡)=𝜆_ℎ𝑒^−𝜋1𝑡+^∞₀𝑘₁₁𝑡−𝜂𝑃₁₁+0,𝜂𝑑𝜂_∞0𝑘_2𝑁𝑡−𝜂𝑃_2𝑁,𝑃0,𝜂𝑑𝜂_1𝑖(0,𝑡)=𝜆_ℎ𝑒^{−𝜋𝑖𝑡}+^∞₀𝑘_1𝑖𝑡−𝜂𝑃_1𝑖+0,𝜂𝑑𝜂_∞0𝑘_2(𝑖−1)𝑡−𝜂𝑃_2(𝑖−1),𝑃0,𝜂𝑑𝜂𝑖=2,3,…,𝑁,₂₁(0,𝑡)=𝜆_𝑠𝑒^−𝜋1𝑡+^∞₀𝑘₁₁𝑡−𝜂𝑃₁₁+0,𝜂𝑑𝜂_∞0𝑘_2𝑁𝑡−𝜂𝑃_2𝑁,𝑃0,𝜂𝑑𝜂_2𝑖(0,𝑡)=𝑎^𝑖−1𝜆_𝑠𝑒^{−𝜋𝑖𝑡}+^∞₀𝑘_1𝑖𝑡−𝜂𝑃_1𝑖0,𝜂𝑑𝜂+^∞₀𝑘_2(𝑖−1)𝑡−𝜂𝑃_2(𝑖−1)0,𝜂𝑑𝜂,𝑖=2,3,…,𝑁, (7) where 𝜋_𝑖=𝜆_ℎ+𝑎^𝑖−1𝜆_𝑠,𝑘_1𝑖𝑡−𝜂=₀^𝑡−𝜂𝑒^{−𝜋𝑖(𝑡−𝜂)𝜋𝑖𝑣−∫𝑣0𝜇1(𝜏)𝑑𝜏}𝜇₁(𝑣)𝑘𝑑𝑣,𝑖=1,2,…,𝑁,_2𝑖𝑡−𝜂=₀^𝑡−𝜂𝑒^{−𝜋𝑖+1(𝑡−𝜂)𝜋𝑖+1𝑣−∫𝑣0𝜇2(𝜏)𝑑𝜏}𝜇₂(𝑣)𝑘𝑑𝑣,𝑖=1,2,…,𝑁−1,_2𝑁𝑡−𝜂=₀^𝑡−𝜂𝑒^{−𝜋1(𝑡−𝜂)𝜋1𝑣−∫𝑣0𝜇3(𝜏)𝑑𝜏}𝜇₃(𝑣)𝑑𝑣. (8)
With the help of (7), we can get the following convolution-type integral equation 𝑃(𝑡)=𝐹(𝑡)+^𝑡₀𝐾𝑡−𝜂𝑃𝜂𝑑𝜂, (9) where 𝑃(𝑡)=𝑃₀₁(𝑡),…,𝑃_0𝑁(𝑡),𝑃₁₁(0,𝑡),…,𝑃_1𝑁(),𝑃0,𝑡₂₁(0,𝑡),…,𝑃_2𝑁(0,𝑡)^𝑇,𝐹(𝑡)=(𝑓(𝑡),𝜆_ℎ𝑓(𝑡),𝜆_𝑠𝑒^−𝜋1𝑡,𝑎𝑒^−𝜋2𝑡,…,𝑎^𝑁−1𝑒^{−𝜋𝑁𝑡𝑇},𝑓(𝑡)=𝑒^−𝜋1𝑡,𝑒^−𝜋2𝑡,…,𝑒^{−𝜋𝑁𝑡},𝐾=𝑂𝑡−𝜂^𝑛×𝑛𝐾¹¹𝑡−𝜂𝐾¹²𝑂𝑡−𝜂_𝑛×𝑛𝐾²¹𝑡−𝜂𝐾²²𝑂𝑡−𝜂_𝑛×𝑛𝐾³¹𝑡−𝜂𝐾³²,𝐾𝑡−𝜂₁₂=𝑡−𝜂0⋯0𝑘_2𝑁𝑘𝑡−𝜂₂₁𝑡−𝜂⋯00⋮⋱⋮⋮0⋯𝑘_2(𝑁−1)𝑡−𝜂0_𝑛×𝑛,𝐾²²𝑡−𝜂=𝜆_ℎ𝐾¹²,𝐾𝑡−𝜂₃₂=𝑡−𝜂_{0⋯0𝜆𝑠𝑘2𝑁(𝑡−𝜂)𝑎𝜆𝑠𝑘21(𝑡−𝜂)⋯00⋮⋱⋮⋮0⋯𝑎𝑁−1𝜆𝑠𝑘2(𝑁−1)(𝑡−𝜂)0𝑛×𝑛},𝐾¹¹𝑡−𝜂=diag𝑘₁₁𝑡−𝜂,𝑘₁₁𝑘𝑡−𝜂,…,₁₁𝑡−𝜂_𝑛×𝑛,𝐾²¹𝑡−𝜂=diag𝜆_ℎ𝑘₁₁𝑡−𝜂,𝜆_ℎ𝑘₁₁𝜆𝑡−𝜂,…,_ℎ𝑘₁₁𝑡−𝜂_𝑛×𝑛,𝐾³¹𝑡−𝜂=diag𝜆_𝑠𝑘₁₁𝑡−𝜂,𝑎𝜆_𝑠𝑘₁₁𝑎𝑡−𝜂,…,_𝑁−1𝜆_𝑠𝑘₁₁𝑡−𝜂_𝑛×𝑛. (10)
It is clear that the unique existence of the nonnegative solution of the system (2)–(4) is equal to that of the convolution-type integral equation (9). Since, for any 𝑇>0, each coordinate of 𝐹(𝑡) and 𝐾(𝑡−𝜂) is nonnegatively bounded function and is absolutely integrable in 𝐶[0,𝑇], we can obtain that the convolution-type integral equation (9) has a unique nonnegative solution in 𝐶[0,𝑇] by using the related theory in integral equation. For this reason, the system (2)–(4) has a unique nonnegative solution in 𝐶[0,𝑇], for any 𝑇>0. The proof is complete.

3.2. Exponential Stability

In this subsection, in order to further study the properties of the studied system, we will formulate the problem into a suitable Banach space. Then we study the unique existence of its solution and explain the exponential stability of the system by analyzing the spectrum distribution of the system operator in detail.

Firstly, let the state space 𝑋 be 𝑋=𝑃∈𝑅^𝑛×𝐿¹𝑅⁺×𝐿¹𝑅^+𝑛∣‖𝑃‖=𝑃⁰+²_𝑖=1𝑃^𝑖<∞, (11) where 𝑃=𝑃⁰,𝑃¹(𝑥),𝑃²(𝑥,𝑃₀)=𝑃₀₁,𝑃₀₂,…,𝑃_0𝑁^𝑇,𝑃¹(𝑥)=𝑃₁₁(𝑥),𝑃₁₂(𝑥),…,𝑃_1𝑁(𝑥)^𝑇,𝑃²(𝑥)=𝑃₂₁(𝑥),𝑃₂₂(𝑥),…,𝑃_2𝑁(𝑥)^𝑇,𝑃⁰=^𝑁_𝑖=1𝑃_0𝑖,𝑃^𝑗=^𝑁_𝑖=1𝑃_{𝑗𝑖𝐿1(𝑅+)},𝑗=1,2. (12) It is obvious that 𝑋 is a Banach space.

Secondly, we will introduce some operators in 𝑋.

Consider 𝐴𝑃=(diag(−(𝜆_ℎ+𝜆_𝑠),−(𝜆_ℎ+𝑎𝜆_𝑠),…,−(𝜆_ℎ+𝑎^𝑁−1𝜆_𝑠))𝑃⁰,diag(−(𝑑/𝑑𝑥)−𝜇₁𝜇(𝑥),…,−(𝑑/𝑑𝑥)−₁(𝑥))𝑃¹(𝑥),diag(−(𝑑/𝑑𝑥)−𝜇₂(𝑥),…,−(𝑑/𝑑𝑥)−𝜇₂(𝑥),−(𝑑/𝑑𝑥)−𝜇₃(𝑥))𝑃²(𝑥)).

Taking 𝐷(𝐴)={𝑃=(𝑃⁰,𝑃¹(𝑥),𝑃²(𝑥))∈𝑋∣(𝑑𝑃_𝑗𝑖(𝑥)/𝑑𝑥)∈𝐿¹(𝑅⁺), 𝑗=1,2, 𝑖=1,2,…,𝑁, 𝑃_𝑗𝑖(𝑥)  (𝑗=1,2, 𝑖=1,2,…,𝑁) are absolutely continuous functions satisfying 𝑃(0)=(𝑃⁰,𝑃¹(0),𝑃²(0))=(𝑃⁰,𝜆_ℎ𝑃⁰,𝜆_𝑠(𝑃₀₁,𝑎𝑃₀₂,…,𝑎^𝑁−1𝑃_0𝑁)^𝑇)}, 𝑂𝐸𝑃=^𝑛×𝑛𝐸₁𝐸₂𝑂_2𝑛×𝑛𝑂_2𝑛×𝑛𝑂_2𝑛×𝑛𝑃⁰𝑃¹(𝑥)𝑃²(𝑥),𝐷(𝐸)=𝑋, (13) where 𝑂_𝑛×𝑛, 𝑂_2𝑛×𝑛 are zero vectors and 𝐸₁=diag(𝜇₁(𝑥),𝜇₁(𝑥),…,𝜇₁(𝑥))_𝑛×𝑛, 𝐸₂=00⋯0∫^∞₀𝜇₃(𝑥)𝑑𝑥0∫^∞₀𝜇₂(𝑥)𝑑𝑥⋯00⋮⋮⋱⋮⋮00⋯∫^∞₀𝜇₂(𝑥)𝑑𝑥0_𝑛×𝑛. (14)

Then the former equations (2)–(4) can be formulated as an abstract Cauchy problem into the suitable Banach space 𝑋, that is, 𝑑𝑃()=()𝑃()𝑃()=𝑃𝑑𝑡⋅,𝑡𝐴+𝐸⋅,𝑡,𝑡≥0,⋅,𝑡⁰(𝑡),𝑃¹(⋅,𝑡),𝑃²(),𝑃()⋅,𝑡⋅,0=𝑃₀=(1,0,…,0)^𝑇_1×𝑛,𝑂_𝑛×1,𝑂_𝑛×1. (15)

Since the system (2)–(4) is rewritten as an abstract Cauchy problem, it is necessary to prove the well-posedness of the system (15). Next, we will prove that the system (15) has a unique nonnegative solution by using 𝑐₀ semigroup theory. We present the expression of the dynamic solution of the system equation.

For convenience, we will present four useful lemmas.

Lemma 2. There exists constant 𝐿>0 such that for any 𝑡>0 (see [18]) ^∞_𝑡𝑒^{−∫𝑥𝑡𝜇𝑗(𝜏)𝑑𝜏}𝑑𝑥≤𝐿,𝑗=1,2,3. (16)

Lemma 3. For any 𝑟∈{𝑟∈ℂ∣Re𝑟>0 or 𝑟=𝑖𝑎, 𝑎∈𝑅,𝑎≠0} (see [18]), ^∞₀𝜇_𝑗(𝑥)𝑒^{−∫𝑥0(𝑟+𝜇𝑗(𝜏))𝑑𝜏}𝑑𝑥<1,𝑗=1,2,3. (17)

Lemma 4. The system operator 𝐴+𝐸 is a dispersive operator (see [19]) with dense domain.

The proof of Lemma 4 can be seen in [20–22].

Lemma 5. {𝑟∈ℂ∣Re𝑟>0 or 𝑟=𝑖𝑎, 𝑎∈𝑅⧵{0}}⊆𝜌(𝐴+𝐸).

Proof. Firstly, for any 𝐺=(𝑔⁰,𝑔¹(𝑥),𝑔²(𝑥))∈𝑋, here 𝑔^𝑗=(𝑔_𝑗1,𝑔_𝑗2,…,𝑔_𝑗𝑁)^𝑇, 𝑗=0,1,2, considering [𝑟𝐼−(𝐴+𝐸)]𝑃=𝐺. That is, 𝑟+𝜆_ℎ+𝜆_𝑠𝑃₀₁−^∞₀𝜇₁(𝑥)𝑃₁₁(𝑥)−𝑑𝑥_∞0𝜇₃(𝑥)𝑃_2𝑁(𝑥)𝑑𝑥=𝑔₀₁,𝑟+𝜆_ℎ+𝑎^𝑗−1𝜆_𝑠𝑃_0𝑗−^∞₀𝜇₁(𝑥)𝑃_1𝑗(𝑥)−𝑑𝑥_∞0𝜇₂(𝑥)𝑃_2(𝑗−1)(𝑥)𝑑𝑥=𝑔_0𝑗,𝑑𝑗=2,…,𝑁,𝑃𝑑𝑥^1𝑗(𝑥)+𝑟+𝜇₁(𝑥)𝑃_1𝑗(𝑥)=𝑔_1𝑗(𝑥),𝑑𝑗=1,2,…,𝑁,𝑃𝑑𝑥^2𝑗(𝑥)+𝑟+𝜇₂(𝑥)𝑃_2𝑗(𝑥)=𝑔_2𝑗(𝑥),𝑑𝑗=2,3,…,𝑁−1,𝑃𝑑𝑥^2𝑁(𝑥)+𝑟+𝜇₃(𝑥)𝑃_2𝑁(𝑥)=𝑔_2𝑁(𝑥). (18)(19)(20)(21)(22) And we can suppose 𝑃_1𝑗(0)=𝜆_ℎ𝑃_0𝑗𝑃,𝑗=1,2,…,𝑁,_2𝑗(0)=𝑎^𝑗−1𝜆_𝑠𝑃_0𝑗,𝑗=1,2,…,𝑁. (23) Solving (20)–(22) with the help of (23) one gets 𝑃_1𝑗(𝑥)=𝑃_1𝑗(0)𝑒^{−∫𝑥0[𝑟+𝜇1(𝜂)]𝑑𝜂}+^𝑥₀𝑒^{−∫𝑥𝜏[𝑟+𝜇1(𝜂)]𝑑𝜂}𝑔_1𝑗(𝜏)𝑃𝑑𝜏,𝑗=1,2,…,𝑁,_2𝑗(𝑥)=𝑃_2𝑗(0)𝑒^{−∫𝑥0[𝑟+𝜇2(𝜂)]𝑑𝜂}+^𝑥₀𝑒^{−∫𝑥𝜏[𝑟+𝜇2(𝜂)]𝑑𝜂}𝑔_2𝑗(𝜏)𝑃𝑑𝜏,𝑗=1,2,…,𝑁−1,_2𝑁(𝑥)=𝑃_2𝑁(0)𝑒^{−∫𝑥0[𝑟+𝜇3(𝜂)]𝑑𝜂}+^𝑥₀𝑒^{−∫𝑥𝜏[𝑟+𝜇3(𝜂)]𝑑𝜂}𝑔_2𝑁(𝜏)𝑑𝜏. (24) Noticing that 𝑔_𝑗𝑖(𝑥)∈𝐿¹[0,∞), 𝑗=1,2, 𝑖=1,2,…,𝑁, together with Lemma 2, we know 𝑃_𝑗𝑖(𝑥)∈𝐿¹[0,∞), 𝑗=1,2, 𝑖=1,2,…,𝑁. This implies that [𝑟𝐼−(𝐴+𝐸)] is an onto mapping.
Secondly, we will prove that this operator is also an injective mapping. That is, the operator equation [𝑟𝐼−(𝐴+𝐸)]𝑃=0 has a unique solution 0. Set 𝐺=0 in the former discussion. Then we can obtain the following matrix equation by combing (18)-(19) with (24): 𝑟+𝑏₁0⋯0−𝑎^𝑁−1𝜆_𝑠𝑊₃−𝜆_𝑠𝑊₂𝑟+𝑏₂⋯000−𝑎𝜆_𝑠𝑊₂⋱00⋮⋮⋱⋮⋮00⋯𝑟+𝑏_𝑁−1000⋯−𝑎^𝑁−2𝜆_𝑠𝑊₂𝑟+𝑏_𝑁×𝑃₀₁𝑃₀₂𝑃₀₃⋮𝑃_0(𝑁−1)𝑃_0𝑁=000⋮00, (25) where 𝑏_𝑗=𝜆_ℎ1−𝑊₁+𝑎^𝑗−1𝜆_𝑠,𝑊𝑗=1,2,…,𝑁_𝑗=^∞₀𝜇_𝑗(𝑥)𝑒^{−∫𝑥0[𝑟+𝜇𝑗(𝜏)]𝑑𝜏}𝑑𝑥,𝑗=1,2,3. (26)
It is clear that 𝑟+𝑏_𝑗=𝑟+𝜆_ℎ1−𝑊₁+𝑎^𝑗−1𝜆_𝑠>𝑟+𝑎^𝑗−1𝜆_𝑠>𝑎^𝑗−1𝜆_𝑠>−𝑎^𝑗−1𝜆_𝑠𝑊₂,𝑗=1,2,…,𝑁−1,𝑟+𝑏_𝑁=𝑟+𝜆_ℎ1−𝑊₁+𝑎^𝑁−1𝜆_𝑠>𝑟+𝑎^𝑁−1𝜆_𝑠>𝑎^𝑁−1𝜆_𝑠>−𝑎^𝑁−1𝜆_𝑠𝑊₃>0, (27) for 𝑟>0 or 𝑟=𝑖𝑎, 𝑎∈𝑅⧵{0}. From Lemma 3, we can obtain |𝑊_𝑗|<1, 𝑗=1,2,3. Thus the matrix of coefficients of the linear equations (25) is a strictly diagonal-dominant matrix about column. Therefore, this matrix is invertible, which manifests that operator [𝑟𝐼−(𝐴+𝐸)] is a one-to-one mapping.
Because [𝑟𝐼−(𝐴+𝐸)] is densely defined closed in 𝑋, we can derive that [𝑟𝐼−(𝐴+𝐸)]⁻¹ exists and is bounded by recalling inverse operator theorem and closed graph theorem. That is, set {𝑟∈ℂ∣Re𝑟>0 or 𝑟=𝑖𝑎, 𝑎∈𝑅⧵{0}} belongs to the resolvent set of the system operator 𝐴+𝐸. Thus we complete the proof of Lemma 5.

Theorem 6. The simple eigenvalue of system operator 𝐴+𝐸 is 0.

Proof. Firstly, we will explain that 0 is the eigenvalue of 𝐴+𝐸 with positive eigenvector.
Consider (𝐴+𝐸)𝑃=0 and assume that 𝑃 satisfies the boundary conditions (23). Then repeating the proof process of the injective mapping in Lemma 5 with 𝑟=0, we can get the following solutions: 𝑃_0𝑗=𝑎^𝑁−𝑗𝑃_0𝑁,𝑃𝑗=1,2,…,𝑁_1𝑗(𝑥)=𝑎^𝑁−𝑗𝜆_ℎ𝑃_0𝑁𝑒^{−∫𝑥0𝜇1(𝜂)𝑑𝜂},𝑃𝑗=1,2,…,𝑁_2𝑗(𝑥)=𝑎^𝑁−𝑗𝜆_𝑠𝑃_0𝑁𝑒^{−∫𝑥0𝜇2(𝜂)𝑑𝜂},𝑃𝑗=1,2,…,𝑁−1_2𝑁(𝑥)=𝑎^𝑁−1𝜆_𝑠𝑃_0𝑁𝑒^{−∫𝑥0𝜇3(𝜂)𝑑𝜂}, (28) where 𝑃_0𝑁 is an arbitrary real number. Then it can be derived that 𝑃_𝑗𝑖(𝑥), 𝑗=1,2, 𝑖=1,2,…,𝑁, for all 𝑥∈[0,∞) by taking 𝑃_0𝑁>0 without loss of generality. Since the vector 𝑃^∗=𝑃^∗₀₁,…,𝑃^∗_0𝑁^𝑇,𝑃^∗₁₁(𝑥),…,𝑃^∗_1𝑁(𝑥)^𝑇,𝑃^∗₂₁(𝑥),…,𝑃^∗_2𝑁(𝑥)^𝑇 (29) is the positive eigenvector corresponding to eigenvalue 0 of the system operator 𝐴+𝐸 and it is also the positive steady-state solution of the system, here 𝑃^∗_0𝑖 and 𝑃^∗_𝑗𝑖(𝑥), respectively, signify 𝑃_0𝑖 and 𝑃_𝑗𝑖(𝑥) showed in (28), 𝑗=1,2, 𝑖=1,2,…,𝑁. In addition, it is easy to see that the geometric multiplicity of eigenvalue 0 in 𝑋 is one.
Secondly, we will explain that the algebraic multiplicity of eigenvalue 0 is one.
Taking vector 𝑄=(𝐼⁰_𝑛×1,𝐼¹_𝑛×1,𝐼²_𝑛×1), here 𝐼^𝑗=(1,1,…,1)^𝑇, 𝑗=0,1,2. Then 𝑄∈𝑋^∗. For any 𝑃∈𝐷(𝐴+𝐸), it is not difficult to show that ⟨(𝐴+𝐸)𝑃,𝑄⟩=0 by noticing the boundary conditions. Therefore, we can deduce that ⟨𝑃,(𝐴+𝐸)^∗𝑄⟩=0, for all 𝑃∈𝑋, for 𝐷(𝐴) is dense in 𝑋. This manifests that 𝑄 is the eigenvector of (𝐴+𝐸)^∗, the adjoint operator of 𝐴+𝐸, corresponding to eigenvalue 0.
In the light of [23, 24], we only need to explain that the algebraic index of 0 in 𝑋 is one. We use the reduction to absurdity. Assume that the algebraic index of 0 is 2 without loss of generality. Thus there exists 𝑈∈𝑋 such that (𝐴+𝐸)𝑈=𝑃^∗. Therefore, 𝑃^∗,𝑄=⟨(𝐴+𝐸)𝑈,𝑄⟩=𝑈,(𝐴+𝐸)^∗𝑄=0. (30) However, 𝑃^∗,𝑄=^𝑁_𝑖=1𝑃_0𝑖+^2𝑁_𝑗=1𝑖=1^∞₀𝑃_𝑗𝑖(𝑥)𝑑𝑥>0. (31) Equation (30) contradicts (31). Then the algebraic index of 0 in 𝑋 is one. Then the algebraic multiplicity of 0 in 𝑋 is one. The proof of Theorem 6 is complete.

Lemma 5 and Theorem 6 can imply that several important results hold. Firstly, they imply that the spectral bound 𝑠(𝐴+𝐸) of 𝐴+𝐸 is zero. Secondly, Lemma 5 and Theorem 6 illustrate 0 is a strictly dominant eigenvalue of the operator 𝐴+𝐸.

The following task is to verify the operator 𝐴+𝐸 generates some 𝑐₀ semigroups 𝑇(𝑡).

Theorem 7. The system operator 𝐴+𝐸 generates a positive contraction 𝑐₀ semigroup 𝑇(𝑡).

Proof. We can get the proof of Theorem 7 by the Phillips theorem (see [25]), Lemma 4, and Lemma 5.

Theorem 8. The system (15) has a unique nonnegative time-dependent solution 𝑃(⋅,𝑡), which satisfies ‖𝑃(⋅,𝑡)‖=1, for all 𝑡∈[0,∞).

Proof. In view of Theorem 7 and [25], we have that the system (15) has a unique nonnegative solution 𝑃(⋅,𝑡) and it can be expressed as 𝑃(⋅,𝑡)=𝑇(𝑡)𝑃₀,∀𝑡∈[0,∞). (32) Because 𝑃(⋅,𝑡) satisfies (2)–(4), it is easy to receive that 𝑑‖𝑃(⋅,𝑡)‖/𝑑𝑡=0. Here ‖𝑃(⋅,𝑡)‖=‖𝑇(𝑡)𝑃₀‖=‖𝑃₀‖=1, for all 𝑡∈[0,∞).
Because 𝑃₀∉𝐷(𝐴), (32) is the mild solution of the system. However, Theorem 1 implies that the classical solution of the system uniquely exists for 𝑡>0. Hence, the mild solution 𝑃(⋅,𝑡)=𝑇(𝑡)𝑃₀ is just the classical one for 𝑡>0. Thus the abstract Cauchy problem (15) is well posed.

Theorem 9. The time-dependent solution of the system (2)–(4) strongly converges to its steady-state solution. That is, lim_𝑡→∞𝑃(⋅,𝑡)=𝑃^∗, where 𝑃^∗ is the eigenvector corresponding to 0 in 𝑋 satisfying ‖𝑃^∗‖=1.

Proof. In the light of Theorem 8, the nonnegative solution of the system (2)–(4) can be expressed as 𝑃(⋅,𝑡)=𝑇(𝑡)𝑃₀, for all 𝑡∈[0,∞). Thus combing Theorem 2.10 (see [19]) together with Theorem 12.3 in [26], we can deduce that 𝑃(⋅,𝑡)=𝑇(𝑡)𝑃₀=𝑃₀,𝑄𝑃^∗+𝑅(𝑡)𝑃₀=𝑃^∗+𝑅(𝑡)𝑃₀, (33) where 𝑄 is the same as defined in Theorem 6 and 𝑅(𝑡)=𝐶𝑒^−𝜀𝑡 for suitable constants 𝜀>0 and 𝐶>0. Hence we have lim_𝑡→∞𝑃(⋅,𝑡)=𝑃₀,𝑄𝑃^∗=𝑃^∗. (34)
As a result, the exponential stability of the solution of the studied system was obtained.

Thus we show that the studied system has exponential stability. Exponential stability is a very important property in reliability study. We can overcome some problems readily and deduce some better conclusions by using it. For example, by using the property, the governors can make up their mind how to arrange the repairman to do minimal repair or overhaul in his work time to increase the profit of the system benefit.

As far as such a problem is concerned, previous literatures such as [27] only pointed out to when the profit of the system benefit with repairman vacation in steady state is larger than that of the classical system benefit. But this is a less practical condition because they cannot solve the following problems. Firstly, how long time the system will take to get the stability state. Secondly, whether the steady-state indices such as steady-state availability can substitute for the transient ones. Thirdly, what is the probability that the repairman can carry out minimal repair.

However, by studying the exponential stability of the system, all these problems can be solved easily. Actually, for a given fault, the system can get to the steady state at a very fast speed and its steady-state availability can substitute for the dynamic one by considering a safety factor. Moreover, 𝑃_0𝑖(𝑡) (𝑖=1,2,…,𝑁) means the probability that the system is operating normally after every minimal repair, and the repairman is on vacation at time 𝑡≥0 and 𝑃_0𝑖(𝑡)→𝑃_0𝑖>0; here 𝑃_0𝑖 (𝑖=1,2,…,𝑁) is the first 𝑁 coordinate of the eigenvector 𝑃^∗ in Theorem 9. Then Theorem 9 indicates that, after a certain time 𝑡>0, the repairman can always be urged to overhaul with a fixed probability to increase the total profit of the system benefit.

4. Reliability Indices

In this section, we first present the steady-state probability and frequency of failure of the system with traditional method. Second, we propose the steady-state availability and the failure frequency of the system with one new method different from the traditional one (see [28]). And the two methods were compared; we have the second method is more practical and simple.

Firstly, the above equations (2) are valid for any 𝑡≥0. Since we are interested in the steady-state behavior of our system, we will seek the long-run probabilities which are the solution of the following equations obtained from (2) taking the limits as 𝑡→∞: 𝜆_ℎ+𝜆_𝑠𝑃₀₁=^∞₀𝑃₁₁(𝑥)𝜇₁(𝑥)𝑑𝑥+^∞₀𝑃_2𝑁(𝑥)𝜇₃(𝑥)𝜆𝑑𝑥,_ℎ+𝑎^𝑖−1𝜆_𝑠𝑃_0𝑖=^∞₀𝑃_1𝑖(𝑥)𝜇₁(𝑥)𝑑𝑥+^∞₀𝑃_2(𝑖−1)(𝑥)𝜇₂(𝑥)𝑑𝑑𝑥,𝑖=2,3,…,𝑁,𝑑𝑥+𝜇₁(𝑥)𝑃_1𝑖(𝑥)𝑑=0,𝑖=1,2,…,𝑁,𝑑𝑥++𝜇₂(𝑥)𝑃_2𝑖(𝑥)𝑑=0,𝑖=1,2,…,𝑁−1,𝑑𝑥+𝜇₃(𝑥)𝑃_2𝑁(𝑥)=0. (35)

Equations (35) are to be solved under the following conditions: 𝑃_1𝑖(0)=𝜆_ℎ𝑃_0𝑖𝑃,𝑖=1,2,…,𝑁,_2𝑖(0)=𝑎^𝑖−1𝜆_𝑠𝑃_0𝑖,𝑖=1,2,…,𝑁. (36)

The steady-state probabilities are 𝑃_0𝑖, 𝑃_1𝑖=∫₀^+∞𝑃_1𝑖(𝑥)𝑑𝑥, 𝑃_2𝑖=∫₀^+∞𝑃_2𝑖(𝑥)𝑑𝑥, where 𝑖=1,2,…,𝑁, respectively. The steady-state probabilities must satisfy the total probability equation ^𝑁_𝑖=1𝑃_0𝑖+𝑃_1𝑖+𝑃_2𝑖=1. (37) Probability and frequency of failure are given by 𝑃_𝑓=∑^𝑁_𝑖=1(𝑃_1𝑖+𝑃_2𝑖) and 𝐹_𝑓=∑^𝑁_𝑖=1(𝜆_ℎ+𝑎^𝑖−1𝜆_𝑠)𝑃_0𝑖.

Next, we obtain the steady-state availability and the failure frequency of the system by using the proposed method in this paper.

Theorem 10. The steady-state availability after the 𝑁th time repair is 𝐴_𝑣𝑁=∑^𝑁_𝑖=1𝑎^𝑖−1∑^𝑁_𝑖=1𝑎^𝑖−11+𝜆_ℎ𝐸₁+𝑎^𝑁−1𝜆_𝑠(𝑁−1)𝐸₂+𝐸₃, (38) where 𝐸_𝑖=∫^∞₀𝑒^{−∫𝑥0𝜇𝑖(𝑠)𝑑𝑠}𝑑𝑥(𝑖=1,2,3).

Proof. Let 𝑀=^2𝑁_𝑗=0𝑖=1𝑃^∗_𝑗𝑖≜^𝑁_𝑖=1𝑃^∗_0𝑖+^𝑁_𝑖=1^∞₀𝑃^∗_1𝑖(𝑥)𝑑𝑥+^𝑁_𝑖=1^∞₀𝑃^∗_2𝑖(𝑥)=𝑑𝑥_𝑁𝑖=1𝑎^𝑖−11+𝜆_ℎ𝐸₁+𝑎^𝑁−1𝜆_𝑠(𝑁−1)𝐸₂+𝐸₃𝑃^∗_0𝑁, (39) where 𝑃^∗_0𝑖,𝑃^∗_𝑗𝑖(𝑥) (𝑗=1,2, 𝑖=1,2,…,𝑁) are the corresponding coordinates of 𝑃^∗ presented in (29). Then the steady-state availability of the system can be expressed as 𝐴_𝑣𝑁=∑^𝑁_𝑖=1𝑃^∗_0𝑖∑^𝑁_𝑖=1𝑃^∗_0𝑖+𝑃^∗_1𝑖(𝑥)+𝑃^∗_2𝑖(𝑥)=∑^𝑁_𝑖=1𝑎^𝑖−1∑^𝑁_𝑖=1𝑎^𝑖−11+𝜆_ℎ𝐸₁+𝑎^𝑁−1𝜆_𝑠(𝑁−1)𝐸₂+𝐸₃, (40)(41) where 𝐸_𝑖=∫^∞₀𝑒^{−∫𝑥0𝜇𝑖(𝑠)𝑑𝑠}𝑑𝑥 (𝑖=1,2,3).

From the above availability expression, the system steady-state availability decreases with the number of the minimal repair times increasing. So, the system steady-state availability is gradually decreasing (for 𝑎>1).

Theorem 11. The steady-state failure frequency after the 𝑁th time repair is 𝑊_𝑓𝑁=𝜆_ℎ∑^𝑁_𝑖=1𝑎^𝑖−1+𝜆_𝑠𝑁𝑎^𝑁−1∑^𝑁_𝑖=1𝑎^𝑖−11+𝜆_ℎ𝐸₁+𝑎^𝑁−1𝜆_𝑠(𝑁−1)𝐸₂+𝐸₃, (42) where 𝐸_𝑖=∫^∞₀𝑒^{−∫𝑥0𝜇𝑖(𝑠)𝑑𝑠}𝑑𝑥 (𝑖=1,2,3).

The proof of Theorem 11 is the same as Theorem  4.1 in [28].

Of course, we can receive the formulations of the instantaneous reliability indices and their corresponding steady-state values as well, such as the reliability of the system and the probability of the repairman being busy.

As we all know, the reliability indices are ordinarily obtained by the Tauberian theorem and Laplacian transformation. However, the proposed method in this paper is probably more simply and more valuable in some practice applications, because the only thing needs to be considered is the eigenvector corresponding to eigenvalue 0 of the system operator.

In the light of these two methods, the first method is more idealistic and does not exist in real life. Compared with the first method, the second method is more practical and simple.

5. Conclusion

In this paper, we dealt with a repairable computer system which composed of hardware and software in series. We were dedicated to studying the unique existence and the exponential stability of the solution of the system. The exponential stability of the system guaranteed that the stability of the system was not easy to be affected by some factors such as failure rate and repair rate. And we presented a new method to receive the steady-state indices of the system by using the eigenvector corresponding to eigenvalue 0 of the system operator. It was more simple and practical than the traditional one.

However, it was well known that it was difficult or even impossible to obtain the time-dependent solution and the dynamic indices of the system. This paper presented a new method to overcome these problems from the view point of theory.

Acknowledgments

The authors thank the referees for their useful comments and suggestions. The research is supported by NSFC (no. 11201037) and Natural Science Foundation of Heilongjiang Province, China (no. QC2010024).