A New Approach for Analyzing the Reliability of the Repair Facility in a Series System with Vacations

Abstract

Based on the renewal process theory we develop a decomposition method to analyze the reliability of the repair facility in an n-unit series system with vacations. Using this approach, we study the unavailability and the mean replacement number during (0,𝑡] of the repair facility. The method proposed in this work is novel and concise, which can make us see clearly the structures of the facility indices of a series system with an unreliable repair facility, two convolution relations. Special cases and numerical examples are given to show the validity of our method.

1. Introduction

It is well known that the repairable systems with an unreliable repair facility are frequently studied in the literature of reliability theory. Theoretically, they are in a class of more general repairable systems, and the usual repairable systems are their special cases [1].

Common approaches in the reliability analysis of repairable systems with a repair facility include the Markov renewal process method, the geometric process method, the supplementary variable method, and the strong stability method. The Markov renewal process method has important applications in reliability theory. By means of this method, the reliability indices of various systems were obtained in the previous literatures (see, e.g., [2, 3]). The geometric process proposed by Lam [4] is a special monotone process, which can model the working times and the repair times of a deteriorating system. Much research has been conducted by Jia and Wu [5, 6], Wu and Clements-Croome [7], Zhang and Wang [8], and others under the monotone process model. Moreover, with the help of the geometric process method, a deteriorating system with its repairman having multiple vacations was studied by Yuan and Xu [9]. They obtained the optimal replacement policy which could make the long-run expected cost per unit time minimize. The strong stability method studied by Rahmoune and Aissani [10] and other researchers is a more powerful tool. For example, Rahmoune and Aissani [10] applied this method to analyze the characteristics of the 𝑀/𝐺/1/𝑁 queue with multiple vacations where the rate of the vacations is sufficiently small. This method is also useful in analyzing complex repairable systems, such as, redundant repairable systems with unreliable facilities, with vacations, or with priority in use. The supplementary variable technique was first introduced by Cox [11], and it had been widely applied to complex repairable systems by Li et al. [12], Liu et al. [13, 14], and many others. Also, recently, Guo et al. [15] have derived the steady-state reliability indices of a series system by using the eigenfunction corresponding to eigenvalue 0 of the system operator.

However, these methods mentioned above usually become too complicated to be solved especially when dealing with some complex systems with many random variables following general distributions. In this paper, a new approach, called decomposition method here, is developed based on the renewal process theory. With this novel method, we analyze the unavailability and the mean replacement number during (0,𝑡] of the repair facility in an 𝑛-unit series system with an unreliable repair facility and vacations. To be precise, we obtain two convolution relations for the unavailability and the mean replacement number. The two convolution relations turn studying the unavailability and the mean replacement number during (0,𝑡] into studying the corresponding reliability indices in the classical one-unit system and the “generalized busy period of the repair facility” (see Definition 2.3). In addition, some new reliability results are also obtained.

To introduce our method, we consider an 𝑛-unit series system with a nonreliable repair facility and multiple adaptive vacations. We aim at obtaining two key reliability quantities of the repair facility, that are, the replacement probability (the unavailability of the repair facility) and the average number of replacements during (0,𝑡]. Note that the two indices have been investigated by Liu et al. [13, 14] with the help of the supplementary variable method. In this paper we study again the above two indices by means of a decomposition method, which is completely different from the methods used in [2–15]. Our main idea is as follows: (1) with the definition of “generalized busy period of the repair facility,” we obtain the probability that the repair facility is during its generalized busy period at time 𝑡; (2) based on this, using our decomposition method, we derive the unavailability and average replacement number of the repair facility, which lead to two convolution relations; (3) finally, we present some special cases and numerical examples to show the validity of our method.

The rest of the paper is organized as follows. The next section gives the assumptions of the considered model and some preliminaries. In Section 3 we use a decomposition method to discuss two key reliability indices of the repair facility. In Sections 4 and 5 special cases and numerical examples are presented to validate the derived results. Conclusions are finally drawn in Section 6.

2. Assumptions and Preliminaries

We consider a series repairable system consisting of 𝑛 dissimilar units, a repairman, and an unreliable repair facility by making the following assumptions.(1)Whenever one unit fails, the system breaks down. It is assumed that 𝑛 units shut each other off, and each unit after repair is as good as new. As soon as the repair of the failed unit is completed, the system starts to operate immediately.(2)The repairman will take a random maximum number, denoted by 𝐻, of vacations whenever there is no failed unit waiting for repair in the system. The probability mass function of 𝐻 is 𝑃(𝐻=𝑗)=𝑐_𝑗, 𝑗=1,2,…. The random variable 𝐻 may represent the maximum number tasks or jobs available for repairman to work on if the system is operating. Denote the repairman’s 𝑖th vacation time by 𝑉_𝑖, 𝑖≥1. Assume that 𝑉_𝑖, 𝑖≥1 are independent and identically distributed random variables each with the probability distribution function 𝑉(𝑡) and a mean vacation time 𝐸(𝑉). At each vacation completion instant, the repairman checks the system to see if there is a failed unit waiting and decides the action to take according to the state of the system. There are three cases: (a) if there is a failed unit waiting, the repairman will repair it immediately; (b) if there is no failed unit waiting and the total number of vacations is still less than 𝐻, the repairman will take another vacation; (c) if there is no failed unit waiting and the total number of vacations is equal to 𝐻, the repairman will stay idle and wait for the next failure.(3)The lifetime 𝑋_𝑖 of unit 𝑖 follows an exponential distribution 𝐹_𝑖(𝑡)=1−𝑒^{−𝜆𝑖𝑡}, 𝑡≥0, 0≤𝜆_𝑖<∞. The repair time 𝑌_𝑖 of the failed unit 𝑖 obeys a general distribution function 𝐺_𝑖(𝑡), 𝑡≥0 with a mean repair time 𝐸(𝑌_𝑖), 𝑖=1,2,…,𝑛.(4)It is assumed that the repair facility neither fails nor deteriorates in its idle periods and fails only during its “generalized busy periods” presented by this paper. The life span 𝑈 of the repair facility has an exponential distribution 𝑈(𝑡)=1−𝑒^𝛼𝑡, 𝑡≥0, 0≤𝛼<∞.(5)When the repair facility fails, it is replaced by a new and identical one immediately and the unit that is being repaired has to wait for repair. The replacement time 𝐵 of the repair facility has an arbitrary distribution 𝐵(𝑡) with a mean replacement time 𝐸(𝐵). After replacement, the new repair facility starts its repair to the failed unit whose repair was stopped. The repair time for the failed unit is cumulative, that is, the repaired time of this failed unit is still valid.(6)Initially, the system with 𝑛 new units begins to operate, the new repair facility is during its idle period, and the repairman begins to take multiple adaptive vacations. All random variables are mutually independent.

Remark 2.1. The multiple adaptive vacation policy presented by this paper is introduced by Tian and Zhang [16], which is a generalization of single vacation, multiple vacations and variant vacation, and so forth. The analysis in this work shows that our method is valid in studying the reliability indices of the repair facility of complex vacation systems with an unreliable repair facility.
For ease of reference, we list some notations and state two definitions as shown at the end of the paper.

Definition 2.2. The “generalized repair time of unit 𝑖” 𝑌_𝑖, 𝑖=1,2,…,𝑛 denotes the time interval from the time when the repair facility begins to repair the failed unit 𝑖 until the repair of the failed unit 𝑖 ends, which includes some possible replacement times due to the facility failures in the process of repairing the failed unit 𝑖.
For 𝑡≥0, let 𝐺_𝑖(𝑡)=𝑃(𝑌_𝑖≤𝑡), then (see [1]) 𝐺_𝑖(𝑡)=^∞_𝑘=0^𝑡₀𝐵^(𝑘)()()𝑡−𝑥𝛼𝑥^𝑘𝑒𝑘!^−𝛼𝑥𝑑𝐺_𝑖(𝑥). (2.1) The Laplace-Stieltjes transform of 𝐺_𝑖(𝑡) is given by 𝑔_𝑖(𝑠)=^∞₀𝑒^−st𝑑𝐺_𝑖(𝑡)=𝑔_𝑖(𝑠+𝛼−𝛼𝑏(𝑠)),ℜ(𝑠)>0, (2.2) where 𝑔_𝑖(𝑠)=∫^∞₀𝑒^−st𝑑𝐺_𝑖(𝑡) and 𝑏(𝑠)=∫^∞₀𝑒^−st𝑑𝐵(𝑡) are the Laplace-Stieltjes transforms of 𝐺_𝑖(𝑡) and 𝐵(𝑡), respectively. The mean of the generalized repaired time of the failed unit 𝑖 is given by 𝐸𝑌_𝑖=−𝑑𝑔_𝑖(𝑠)𝑑𝑠_𝑠=0=𝐸𝑌_𝑖(1+𝛼𝐸(𝐵)),𝑖=1,2,…,𝑛. (2.3)

Definition 2.3. The “generalized busy period of the repair facility” 𝐷 denotes the time interval from the time when the repair facility begins to repair a failed unit until the repair of the failed unit completes, where 𝐷 contains some possible replacement times due to the facility failures in the process of repairing the failed unit.

Remark 2.4. If the system’s operating time min_{1≤𝑖≤𝑛}𝑋_𝑖=𝑋_𝑙, then 𝐷=𝑌_𝑙, 𝑙=1,2,…,𝑛.

3. Reliability Analysis of the Repair Facility

In this section, by means of a decomposition method we will discuss the probability that the repair facility is during its generalized busy period at time 𝑡. On this basis, we study the unavailability and the replacement number during (0,𝑡] of the repair facility. Finally, we analyze the structures of the facility indices in our system.

3.1. The Probability That the Facility Is during Its Generalized Busy Period

For 𝑡≥0, let 𝐴(𝑡)=𝑃therepairfacilityisduringitsgeneralizedbusyperiodattime𝑡. (3.1)

Theorem 3.1. If ℜ(𝑠)>0, then the Laplace transform of 𝐴(𝑡) is 𝑎^∗(𝑠)=Θ(𝑠)𝑠1−Θ(𝑠)∑^𝑛_𝑙=1𝜆_𝑙/Λ𝑔_𝑙(𝑠)^𝑛_𝑙=1𝜆_𝑙Λ1−𝑔_𝑙(𝑠), (3.2) and, in steady state, the probability that the repair facility is during its generalized busy period is given by lim_𝑡→∞𝐴(𝑡)=(1/Λ)∑^𝑛_𝑖=1𝜆_𝑖𝐸𝑌_𝑖(1+𝛼𝐸(𝐵))(1/Λ)∑^𝑛_𝑖=1𝜆_𝑖𝐸𝑌_𝑖((𝐵(𝑉)/((Λ+[(𝑉)/((Λ𝐻(𝑣(Λ,1+𝛼𝐸))+𝐸1−𝑣))1/Λ−𝐸1−𝑣))])) (3.3) where Θ(𝑠)=Λ𝐻(𝑣(+𝑣(𝑠)()𝑠+Λ𝑠+Λ))−𝑣𝑠+Λ()[(𝑣(.1−𝑣𝑠+Λ1−𝐻𝑠+Λ))] (3.4)

Proof. Let Δ_𝑡 be the event that the repair facility is during its generalized busy period at time 𝑡. Denote 𝜉₁ and 𝜂₁ be the first operating time and the first repair time of the system, respectively. By applying the total probability decomposition technique and noting that the time points that the system begins to operate are renewal ones, we have 𝐴(𝑡)=𝑃Δ_𝑡,𝜉₁>𝑡+𝑃Δ_𝑡,𝜉₁Δ≤𝑡=𝑃_𝑡,𝜉₁≤𝑡,𝜉₁>^𝐻_𝑖=1𝑉_𝑖+𝑃Δ_𝑡,𝜉₁≤𝑡,𝜉₁≤^𝐻_𝑖=1𝑉_𝑖. (3.5) The first term of (3.5) can be decomposed as ^∞_𝑗=1𝑐_𝑗𝑃𝜉₁>^𝑗_𝑖=1𝑉_𝑖,𝜉₁≤𝑡<𝜉₁+𝜂₁+𝑃Δ_𝑡,𝜉₁>^𝑗_𝑖=1𝑉_𝑖,𝑡≥𝜉₁+𝜂₁=^∞_𝑗=1𝑐_𝑗^𝑛_𝑙=1𝑃𝜉₁=𝑋_𝑙,𝑋_𝑙>^𝑗_𝑖=1𝑉_𝑖,𝑋_𝑙≤𝑡<𝑋_𝑙+𝑌_𝑙+𝑃Δ_𝑡,𝜉₁=𝑋_𝑙,𝑋_𝑙>^𝑗_𝑖=1𝑉_𝑖,𝑡≥𝑋_𝑙+𝑌_𝑙=^∞_𝑗=1𝑐_𝑗^𝑛_𝑙=1^𝑡₀1−𝐺_𝑙(𝑡−𝑥)𝜆_𝑙𝑒^−Λ𝑥𝑉^(𝑗)(𝑥)+𝑑𝑥_{𝑡00𝑡−𝑥}𝐴𝑡−𝑥−𝑦𝑑𝐺_𝑙𝑦𝜆_𝑙𝑒^−Λ𝑥𝑉^(𝑗)(𝑥)𝑑𝑥. (3.6) Similarly, we can decompose the second term of (3.5) as ^∞_𝑗=1𝑐_𝑗^𝑗_𝑘=1𝑃^𝑘−1_𝑖=1𝑉_𝑖≤𝜉₁<^𝑘_𝑖=1𝑉_𝑖,^𝑘_𝑖=1𝑉_𝑖<𝑡≤^𝑘_𝑖=1𝑉_𝑖+𝜂₁+𝑃Δ_𝑡,^𝑘−1_𝑖=1𝑉_𝑖≤𝜉₁<^𝑘_𝑖=1𝑉_𝑖,𝑡≥^𝑘_𝑖=1𝑉_𝑖+𝜂₁=^∞_𝑗=1𝑐_𝑗^𝑗𝑛_𝑘=1𝑙=1𝑃𝜉₁=𝑋_𝑙,^𝑘−1_𝑖=1𝑉_𝑖≤𝑋_𝑙<^𝑘_𝑖=1𝑉_𝑖,^𝑘_𝑖=1𝑉_𝑖<𝑡≤^𝑘_𝑖=1𝑉_𝑖+𝑌_𝑙+𝑃Δ_𝑡,𝜉₁=𝑋_𝑙,^𝑘−1_𝑖=1𝑉_𝑖≤𝑋_𝑙<^𝑘_𝑖=1𝑉_𝑖,𝑡≥^𝑘_𝑖=1𝑉_𝑖+𝑌_𝑙=^∞_𝑗=1𝑐_𝑗^𝑗𝑛_𝑘=1𝑙=1𝜆_𝑙Λ^𝑡₀₀^𝑡−𝑥1−𝐺_𝑙(𝑡−𝑥−𝑢)1−𝑒^−Λ𝑢𝑑𝑉(𝑢)𝑒^−Λ𝑥𝑑𝑉^(𝑘−1)(𝑥)+^𝑡₀₀^𝑡−𝑥₀^{𝑡−𝑥−𝑢}𝐴𝑡−𝑥−𝑢−𝑦𝑑𝐺_𝑙𝑦×1−𝑒^−Λ𝑢𝑑𝑉(𝑢)𝑒^−Λ𝑥𝑑𝑉^(𝑘−1)(𝑥). (3.7) Taking the Laplace transforms of (3.5)–(3.7) and simplifying lead to 𝑎^∗(𝑠)=^∞_𝑗=1𝑐_𝑗^𝑛_𝑙=1𝜆_𝑙1−𝑔_𝑙(𝑠)𝑠()[𝑣(𝑠+Λ𝑠+Λ)]^𝑗+𝜆_𝑙𝑎^∗(𝑠)𝑔_𝑙(𝑠)[𝑣(𝑠+Λ𝑠+Λ)]^𝑗+^∞_𝑗=1𝑐_𝑗^𝑗𝑛_𝑘=1𝑙=1𝜆_𝑙Λ1−𝑔_𝑙(𝑠)𝑠[𝑣(𝑠+Λ)]^𝑘−1[𝑣(𝑠)−𝑣(𝑠+Λ)]+𝑎^∗(𝑠)𝑔_𝑙(𝑠)[𝑣(𝑠)−𝑣(𝑠+Λ)][𝑣(𝑠+Λ)]^𝑘−1. (3.8) Solving (3.8), we can get (3.2). Applying the well-known Tauberian theorem [17], we have lim_𝑡→∞𝐴(𝑡)=lim_𝑠→0𝑠𝑎^∗(𝑠). (3.9) Putting (3.2) into the above equation and using L' Hospital's rule complete the proof of Theorem 3.1.

To study the unavailability and the replacement number during (0, 𝑡] of the repair facility, we first consider a classical one-unit replaceable system [18, 19]. When the unit fails, it is replaced by a new and identical one immediately. The lifetime 𝑈 of the unit has an exponential distribution 𝑈(𝑡)=1−𝑒^−𝛼𝑡 (𝑡≥0,0≤𝛼<∞), and the replacement time 𝐵 of the failed unit obeys an arbitrary distribution 𝐵(𝑡) with a mean replacement time 𝐸(𝐵). 𝑈 and 𝐵 are mutually independent. After replacement, the new unit begins to operate immediately. For 𝑡≥0, let Φ₀(𝑡),𝑀=𝑃theunitisbeingreplacedattime𝑡₀(𝑡)=𝐸thereplacementnumberoftheunitduring(0,𝑡], (3.10) and 𝜑^∗₀(𝑠)=∫^∞₀𝑒^−stΦ₀(𝑡)𝑑𝑡, 𝑚₀(𝑠)=∫^∞₀𝑒^−st𝑑𝑀₀(𝑡).

Lemma 3.2 (see [18, 19]). If ℜ(𝑠)>0, then 𝜑^∗₀(𝑠)=𝛼[1−𝑏(𝑠)]𝑠[𝑠+𝛼−𝛼𝑏(𝑠)],𝑚₀(𝑠)=𝛼(𝑠),𝑠+𝛼−𝛼𝑏lim_𝑡→∞Φ₀(𝑡)=lim_𝑠→0𝑠𝜑^∗₀(𝑠)=𝛼𝐸(𝐵)1+𝛼𝐸(𝐵),lim_𝑡→∞𝑀₀(𝑡)𝑡=lim_𝑠→0𝑠𝑚₀(𝑠)=𝛼(𝐵).1+𝛼𝐸 (3.11)

3.2. The Unavailability of the Repair Facility

Now, we discuss the unavailability of the repair facility at time 𝑡, that is, the probability that the repair facility is being replaced at time 𝑡. For 𝑡≥0, let Φ(𝑡)=𝑃therepairfacilityisbeingreplacedattime𝑡. (3.12)

Theorem 3.3. Let 𝜑^∗(𝑠)=∫^∞₀𝑒^−𝑠𝑡Φ(𝑡)𝑑𝑡. If ℜ(𝑠)>0, then 𝜑^∗(𝑠)=𝜑^∗₀(𝑠)𝑠𝑎^∗(𝑠), (3.13) and the steady-state unavailability of the repair facility is given by lim_𝑡→∞Φ(𝑡)=(𝛼/Λ)∑^𝑛_𝑙=1𝜆_𝑙𝐸𝑌_𝑙𝐸(𝐵)(1/Λ)∑^𝑛_𝑙=1𝜆_𝑙𝐸𝑌_𝑙((𝐵(𝑉)/((Λ+[(𝑉)/((Λ𝐻(𝑣(Λ,1+𝛼𝐸))+𝐸1−𝑣))1/Λ−𝐸1−𝑣))])) (3.14) where 𝜑^∗₀(𝑠) and 𝑎^∗(𝑠) are given by Lemma 3.2 and (3.2), respectively.

Proof. (i) In the light of the assumptions, the repair facility neither fails nor deteriorates in its idle period and it is available at the beginning time as well as the ending time of each generalized busy period, then the repair facility is replaced at time 𝑡 if and only if the time 𝑡 is during one generalized busy period of the repair facility, and the repair facility is replaced at time 𝑡. Therefore, according to total probability decomposition and the renewal point technique, Φ(𝑡) is decomposed as Φ(𝑡)=𝑃therepairfacilityisbeingreplacedattime𝑡,𝜉₁>^𝐻_𝑖=1𝑉_𝑖,𝜉₁≤𝑡+𝑃therepairfacilityisbeingreplacedattime𝑡,𝜉₁≤^𝐻_𝑖=1𝑉_𝑖,𝜉₁=≤𝑡_∞𝑗=1𝑐_𝑗^𝑛_𝑙=1𝑃therepairfacilityisbeingreplacedattime𝑡,𝜉₁=𝑋_𝑙,𝑋_𝑙>^𝑗_𝑖=1𝑉_𝑖,𝑋_𝑙≤𝑡<𝑋_𝑙+𝑌_𝑙𝜉+𝑃thefacilityisbeingreplacedattime𝑡,₁=𝑋_𝑙,𝑋_𝑙>^𝑗_𝑖=1𝑉_𝑖,𝑋_𝑙+𝑌_𝑙+≤𝑡_∞𝑗=1𝑐_𝑗^𝑗𝑛_𝑘=1𝑙=1𝑃therepairfacilityisbeingreplacedattime𝑡,𝜉₁=𝑋_𝑙,_{𝑘−1𝑖=1}𝑉_𝑖≤𝑋_𝑙<^𝑘_𝑖=1𝑉_𝑖,^𝑘_𝑖=1𝑉_𝑖≤𝑡<^𝑘_𝑖=1𝑉_𝑖+𝑌_𝑙+𝑃therepairfacilityisbeingreplacedattime𝑡,𝜉₁=𝑋_𝑙,_{𝑘−1𝑖=1}𝑉_𝑖≤𝑋_𝑙<^𝑘_𝑖=1𝑉_𝑖,^𝑘_𝑖=1𝑉_𝑖+𝑌_𝑙=≤𝑡_∞𝑗=1𝑐_𝑗^𝑛_𝑙=1^𝑡₀𝑆_𝑙(𝑡−𝑥)𝜆_𝑙𝑒^−Λ𝑥𝑉^(𝑗)(𝑥)+𝑑𝑥_{𝑡00𝑡−𝑥}Φ𝑡−𝑥−𝑦𝑑𝐺_𝑙𝑦𝜆_𝑙𝑒^−Λ𝑥𝑉^(𝑗)(𝑥)+𝑑𝑥_∞𝑗=1𝑐_𝑗^𝑗𝑛_𝑘=1𝑙=1𝜆_𝑙Λ^𝑡₀₀^𝑡−𝑥𝑆_𝑙(𝑡−𝑥−𝑢)1−𝑒^−Λ𝑢𝑑𝑉(𝑢)𝑒^−Λ𝑥𝑑𝑉^(𝑘−1)(𝑥)+^𝑡₀₀^𝑡−𝑥₀^{𝑡−𝑥−𝑢}Φ𝑡−𝑥−𝑢−𝑦𝑑𝐺_𝑙𝑦×1−𝑒^−Λ𝑢𝑑𝑉(𝑢)𝑒^−Λ𝑥𝑑𝑉^(𝑘−1)(𝑥), (3.15) where 𝑆_𝑙(𝑡)=𝑃(𝐷=𝑌_𝑙>𝑡≥0; the repair facility is being replaced at time 𝑡).
(ii) For 𝑡≥0, 𝑆_𝑙(𝑡)=Φ₀(𝑡)−^𝑡₀Φ₀(𝑡−𝑥)𝑑𝐺_𝑙(𝑥), (3.16) where Φ₀(𝑡) is determined by Lemma 3.2.
In fact, in the generalized busy period the repair facility may be in one of two states: operation or replacement, and it is available at the beginning and ending times of each generalized busy period. According to the fact that the life span of the repair facility has an exponential distribution, and conditioning on the generalized busy period, we have the decomposition of Φ₀(𝑡) as follows: Φ₀(𝑡)=𝑃therepairfacilityisbeingreplacedattime𝑡;𝐷=𝑌_𝑙𝑌≤𝑡+𝑃therepairfacilityisbeingreplacedattime𝑡;𝐷=_𝑙=>𝑡_𝑡0𝑃therepairfacilityisbeingreplacedattime𝑡−𝑥∣𝐷=𝑥𝑑𝐺_𝑙(𝑥)+𝑆_𝑙(𝑡)=^𝑡₀Φ₀(𝑡−𝑥)𝑑𝐺_𝑙(𝑥)+𝑆_𝑙(𝑡). (3.17) So (3.16) holds.
(iii) Substituting (3.16) into (3.15) and making the Laplace transform we get (3.13). Equation (3.14) is obtained by the Tauberian theorem, Lemma 3.2 and (3.3).

3.3. The Replacement Number of the Repair Facility

For 𝑡≥0, let 𝑀(𝑡)=𝐸thereplacementnumberoftherepairfacilityduring(0,𝑡]. (3.18)

Theorem 3.4. Let 𝑚(𝑠)=∫^∞₀𝑒^−𝑠𝑡𝑑𝑀(𝑡). If ℜ(𝑠)>0, then 𝑚(𝑠)=𝑚₀(𝑠)𝑠𝑎^∗(𝑠), (3.19) and the steady-state replacement frequency of the repair facility is lim_𝑡→∞𝑀(𝑡)𝑡=(𝛼/Λ)∑^𝑛_𝑙=1𝜆_𝑙𝐸𝑌_𝑙(1/Λ)∑^𝑛_𝑙=1𝜆_𝑙𝐸𝑌_𝑙((𝐵(𝑉)/((Λ+)(𝑉)/((Λ𝐻(𝑣(Λ.1+𝛼𝐸))+𝐸1−𝑣))[(1/Λ−𝐸1−𝑣))])) (3.20)

Proof. (1) For 𝑡≥0, let 𝐼_𝑙(𝑡)=𝐸0≤𝑡<𝑌_𝑙(],𝐽;thereplacementnumberoftherepairfacilityduring0,𝑡_𝑙(𝑡)=𝐸𝑌_𝑙≤𝑡;thereplacementnumberoftherepairfacilityduring0,𝑌_𝑙, (3.21) then similar to (3.16) above, by conditioning on the generalized busy period we have 𝐼_𝑙(𝑡)+𝐽_𝑙(𝑡)=𝑀₀(𝑡)−^𝑡₀𝑀₀(𝑡−𝑥)𝑑𝐺_𝑙(𝑥), (3.22) where 𝑀₀(𝑡) is determined by Lemma 3.2.
(2) Using total probability decomposition, we have the decomposition of 𝑀(𝑡) as follows: 𝑀(𝑡)=^∞_𝑗=1𝑐_𝑗^𝑗𝑛_𝑘=1𝑙=1𝐸(],𝜉thereplacementnumberofthefacilityduring0,𝑡₁=𝑋_𝑙,^𝑘−1_𝑖=1𝑉_𝑖<𝑋_𝑙≤^𝑘_𝑖=1𝑉_𝑖,^𝑘_𝑖=1𝑉_𝑖≤𝑡<^𝑘_𝑖=1𝑉_𝑖+𝑌_𝑙+^∞_𝑗=1𝑐_𝑗^𝑛_𝑙=1𝐸(],𝜉thereplacementnumberofthefacilityduring0,𝑡₁=𝑋_𝑙,𝑋_𝑙>^𝑗_𝑖=1𝑉_𝑖,𝑋_𝑙≤𝑡<𝑋_𝑙+𝑌_𝑙+^∞_𝑗=1𝑐_𝑗^𝑗𝑛_𝑘=1𝑙=1𝐸(],𝜉thereplacementnumberofthefacilityduring0,𝑡₁=𝑋_𝑙,^𝑘−1_𝑖=1𝑉_𝑖<𝑋_𝑙≤^𝑘_𝑖=1𝑉_𝑖,𝑡≥^𝑘_𝑖=1𝑉_𝑖+𝑌_𝑙+^∞_𝑗=1𝑐_𝑗^𝑛_𝑙=1𝐸(],𝜉thereplacementnumberofthefacilityduring0,𝑡₁=𝑋_𝑙,𝑋_𝑙>^𝑗_𝑖=1𝑉_𝑖,𝑡≥𝑋_𝑙+𝑌_𝑙. (3.23) Based on the definition of 𝐼_𝑙(𝑡) and the renewal point technique, the sum of the first and second terms of (3.23) gives ^∞_𝑗=1𝑐_𝑗^𝑗𝑛_𝑘=1𝑙=1𝜆_𝑙Λ^𝑡₀₀^𝑡−𝑥𝐼_𝑙(𝑡−𝑥−𝑢)1−𝑒^−Λ𝑢𝑑𝑉(𝑢)𝑒^−Λ𝑥𝑑𝑉^(𝑘−1)(𝑥)+^∞_𝑗=1𝑐_𝑗^𝑛_𝑙=1^𝑡₀𝐼_𝑙(𝑡−𝑥)𝜆_𝑙𝑒^−Λ𝑥𝑉^(𝑗)(𝑥)𝑑𝑥. (3.24) Similarly, by the definitions of 𝐽_𝑙(𝑡) and 𝑀(𝑡), the sum of the third and fourth terms of (3.23) is decomposed as ^∞_𝑗=1𝑐_𝑗^𝑗𝑛_𝑘=1𝑙=1𝐸thereplacementnumberofthefacilityduring0,^𝑘_𝑖=1𝑉_𝑖+𝑌_𝑙,𝜉₁=𝑋_𝑙,^𝑘−1_𝑖=1𝑉_𝑖<𝑋_𝑙≤^𝑘_𝑖=1𝑉_𝑖,𝑡≥^𝑘_𝑖=1𝑉_𝑖+𝑌_𝑙+𝐸thereplacementnumberofthefacilityduring^𝑘_𝑖=1𝑉_𝑖+𝑌_𝑙,𝜉,𝑡₁=𝑋_𝑙,^𝑘−1_𝑖=1𝑉_𝑖<𝑋_𝑙≤^𝑘_𝑖=1𝑉_𝑖,𝑡≥^𝑘_𝑖=1𝑉_𝑖+𝑌_𝑙+^∞_𝑗=1𝑐_𝑗^𝑛_𝑙=1𝐸thereplacementnumberofthefacilityduring0,𝑋_𝑙+𝑌_𝑙,𝜉₁=𝑋_𝑙,𝑋_𝑙>^𝑗_𝑖=1𝑉_𝑖,𝑡≥𝑋_𝑙+𝑌_𝑙,+𝐸thereplacementnumberofthefacilityduring𝑋_𝑙+𝑌_𝑙,𝜉,𝑡₁=𝑋_𝑙,𝑋_𝑙>^𝑗_𝑖=1𝑉_𝑖,𝑡≥𝑋_𝑙+𝑌_𝑙=^∞_𝑗=1𝑐_𝑗^𝑗𝑛_𝑘=1𝑙=1𝜆_𝑙Λ₀^𝑡−𝑥𝐽_𝑙(𝑡−𝑥−𝑢)1−𝑒^−Λ𝑢𝑑𝑉(𝑢)𝑒^−Λ𝑥𝑑𝑉^(𝑘−1)(𝑥)+^𝑡₀₀^𝑡−𝑥₀^{𝑡−𝑥−𝑢}𝑀𝑡−𝑥−𝑢−𝑦𝑑𝐺_𝑙𝑦1−𝑒^−Λ𝑢×𝑑𝑉(𝑢)𝑒^−Λ𝑥𝑑𝑉^(𝑘−1)(𝑥+_∞𝑗=1)𝑐_𝑗^𝑛_𝑙=1^𝑡₀𝐽_𝑙(𝑡−𝑥)𝜆_𝑙𝑒^−Λ𝑥𝑉^(𝑗)(𝑥)+𝑑𝑥_{𝑡00𝑡−𝑥}𝑀𝑡−𝑥−𝑦𝑑𝐺_𝑙𝑦𝜆_𝑙𝑒^−Λ𝑥𝑉^(𝑗)(𝑥)𝑑𝑥. (3.25) Inserting (3.24), (3.25), and (3.22) in (3.23), taking the Laplace-Stieltjes transform of (3.23) and simplifying lead to 𝑚(𝑠)=^∞_𝑗=1𝑐_𝑗^𝑗𝑛_𝑘=1𝑙=1𝜆_𝑙Λ𝑚(𝑠)𝑔_𝑙(𝑠)[𝑣(𝑠)−𝑣(𝑠+Λ)][𝑣(𝑠+Λ)]^𝑘−1+𝑚₀(𝑠)1−𝑔_𝑙(𝑠)[𝑣(𝑠)−𝑣(𝑠+Λ)][𝑣(𝑠+Λ)]^𝑘−1+^∞_𝑗=1𝑐_𝑗^𝑛_𝑙=1𝑚₀(𝑠)1−𝑔_𝑙(𝑠𝜆)^𝑙[𝑣(𝑠+Λ𝑠+Λ)]^𝑗+𝑚(𝑠)𝑔_𝑙(𝑠)𝜆_𝑙[𝑣(𝑠+Λ𝑠+Λ)]^𝑗. (3.26) Solving (3.26) and using (3.2) we get (3.19). By the Tauberian theorem, we have lim_𝑡→∞𝑀(𝑡)𝑡=lim_𝑠→0𝑠𝑚(𝑠). (3.27) Substituting (3.19) into the above equation and applying Lemma 3.2 and (3.3) give (3.20).

Remark 3.5. Taking the Laplace and Laplace-Stieltjes inverse transforms of (3.13) and (3.19), respectively, we obtain two convolution relations as follows: Φ(𝑡)=Φ₀(𝑡)∗𝐴(𝑡)=^𝑡₀Φ₀()(𝑥)𝑀(𝑡)𝑡−𝑥𝑑𝐴,𝑡≥0,=𝑀₀(𝑡)∗𝐴(𝑡)=^𝑡₀𝑀₀(𝑡−𝑥)𝑑𝐴(𝑥),𝑡≥0. (3.28)
Clearly, (3.28) turn studying the unavailability and the replacement number during (0,𝑡] of the repair facility into studying the corresponding indices in the classical one-unit system and the probability that the repair facility is during its generalized busy period presented in this paper. It is important that the above results are not derived by the supplementary variable method in [13, 14].

Remark 3.6. It is seen from (3.3), (3.14), and (3.20), that the steady-state relation equations lim_𝑡→∞Φ(𝑡)=lim_𝑡→∞Φ₀(𝑡)lim_𝑡→∞𝐴(𝑡) and lim_𝑡→∞(𝑀(𝑡)/𝑡)=lim_𝑡→∞(𝑀₀(𝑡)/𝑡)lim_𝑡→∞𝐴(𝑡) also hold. The two steady-state relations are also new, which are not obtained in [13, 14].

Remark 3.7. By the decomposition method used in this paper, we can also analyze the reliability indices of the series system presented by this paper.

4. Special Cases

Case 1. If 𝐻=∞,𝐻(𝑧)=0, that is, the repairman in the system takes multiple vacations, then (3.3), (3.14), and (3.20), can be simplified as lim_𝑡→∞𝐴(𝑡)=1𝛿Λ^𝑛_𝑙=1𝜆_𝑙𝐸𝑌_𝑙(1+𝛼𝐸(𝐵)),lim_𝑡→∞Φ(𝑡)=𝛼𝛿Λ^𝑛_𝑙=1𝜆_𝑙𝐸𝑌_𝑙𝐸(𝐵),lim_𝑡→∞𝑀(𝑡)𝑡=𝛼𝛿Λ^𝑛_𝑙=1𝜆_𝑙𝐸𝑌_𝑙, (4.1) where 𝛿=(1/Λ)∑^𝑛_𝑙=1𝜆_𝑙𝐸(𝑌_𝑙)(1+𝛼𝐸(𝐵))+𝐸(𝑉)/(1−𝑣(Λ)).
In this case, the steady-state unavailability lim_𝑡→∞Φ(𝑡) and steady-state replacement frequency lim_𝑡→∞((𝑀(𝑡))/𝑡) are identical to those in [13], which are analyzed with the help of the supplementary variable method. Also, the steady-state probability lim_𝑡→∞𝐴(𝑡)is not obtained in [13].

Case 2. If 𝐻=1, 𝐻(𝑧)=𝑧, that is, the repairman in the system takes a single vacation, then lim_𝑡→∞𝐴(𝑡)=1𝛿Λ^𝑛_𝑙=1𝜆_𝑙𝐸𝑌_𝑙(1+𝛼𝐸(𝐵)),lim_𝑡→∞Φ(𝑡)=𝛼𝛿Λ^𝑛_𝑙=1𝜆_𝑙𝐸𝑌_𝑙𝐸(𝐵),lim_𝑡→∞𝑀(𝑡)𝑡=𝛼𝛿Λ^𝑛_𝑙=1𝜆_𝑙𝐸𝑌_𝑙, (4.2) where 𝛿=(1/Λ)∑^𝑛_𝑙=1𝜆_𝑙𝐸(𝑌_𝑙)(1+𝛼𝐸(𝐵))+𝐸(𝑉)+(1/Λ)𝑣(Λ).
In this case, lim_𝑡→∞Φ(𝑡) and lim_𝑡→∞(𝑀(𝑡)/𝑡) are in accordance with the results obtained by Liu et al. [14], which are discussed by means of the supplementary variable method. In addition, lim_𝑡→∞𝐴(𝑡) is a new result.

Case 3. If 𝐻=1, 𝐻(𝑧)=𝑧 and 𝑉(𝑡)=1−𝑒^−𝑟𝑡, 𝑡≥0, then letting 𝑟→∞, we get the following indices of the repair facility of the series system without vacation: lim_𝑡→∞𝐴(𝑡)=1𝛿Λ^𝑛_𝑙=1𝜆_𝑙𝐸𝑌_𝑙(1+𝛼𝐸(𝐵)),lim_𝑡→∞Φ(𝑡)=𝛼𝛿Λ^𝑛_𝑙=1𝜆_𝑙𝐸𝑌_𝑙𝐸(𝐵),lim_𝑡→∞𝑀(𝑡)𝑡=𝛼𝛿Λ^𝑛_𝑙=1𝜆_𝑙𝐸𝑌_𝑙, (4.3) where 𝛿=(1/Λ)∑^𝑛_𝑙=1𝜆_𝑙𝐸(𝑌_𝑙)(1+𝛼𝐸(𝐵))+(1/Λ).
In this special case, lim_𝑡→∞Φ(𝑡) and lim_𝑡→∞(𝑀(𝑡)/𝑡) coincide with the results obtained in [13] or [14], which are derived using the supplementary variable method.

5. Numerical Examples

In this section, some numerical examples are presented to analyze the effects of various parameters on the derived steady-state results, including the probability that the repair facility is during its generalized busy period lim_𝑡→∞𝐴(𝑡), the unreliability lim_𝑡→∞Φ(𝑡) and the replacement frequency lim_𝑡→∞((𝑀(𝑡))/𝑡). For convenience, we consider an 𝑛-identical-component series system with a nonreliable repair facility and multiple adaptive vacations. Let(1)𝑛=3, 𝜆₁=𝜆₂=𝜆₃=0.3, 𝐸(𝑌₁)=𝐸(𝑌₂)=𝐸(𝑌₃)=𝐸(𝑌);(2)𝐻 follows a geometric distribution with probability generating function 𝐻(𝑧)=0.4𝑧/(1−0.6𝑧), |𝑧|≤1;(3)𝑉 obeys an exponential distribution with a mean 𝐸(𝑉)=1.

Numerical results are reported in Tables 1–3. Table 1 shows that the effects of varying mean repair time 𝐸(𝑌) on the reliability indices of the facility for the set of parameters (𝛼,𝐸(𝐵))=(0.2,0.6). We observe that the probability lim_𝑡→∞𝐴(𝑡), the unreliability lim_𝑡→∞Φ(𝑡) and the replacement frequency lim_𝑡→∞(𝑀(𝑡)/𝑡) all increase monotonously as the value 𝐸(𝑌) increases. The effects of varying failure rate 𝛼 on the reliability indices of the facility are shown in Table 2, where we set (𝐸(𝑌),𝐸(𝐵))=(0.5,0.6). As is to be expected, the probability lim_𝑡→∞𝐴(𝑡), the unreliability lim_𝑡→∞Φ(𝑡), and the replacement frequency lim_𝑡→∞((𝑀(𝑡))/𝑡) all decrease monotonously as the rate 𝛼 decreases. Table 3 reports the effects of varying mean replacement time 𝐸(𝐵) on the reliability indices of the facility for the set of parameters (𝐸(𝑌),𝛼)=(0.5,0.2). We note that when 𝐸(𝐵) increases, both lim_𝑡→∞𝐴(𝑡) and lim_𝑡→∞Φ(𝑡) increase, while lim_𝑡→∞(𝑀(𝑡)/𝑡) decreases. The trends shown by the tables are as expected.

Table 1 

Reliability indices for varying value of 𝐸(𝑌) with (𝛼,𝐸(𝐵))=(0.2,0.6).

Table 2 

Reliability indices for varying value of 𝛼 with (𝐸(𝑌),𝐸(𝐵))=(0.5,0.6).

Table 3 

Reliability indices for varying value of 𝐸(𝐵) with (𝐸(𝑌),𝛼) = (0.5, 0.2).

6. Conclusions

In this paper, based on the renewal process theory we develop a new approach—the decomposition method. According to the method, we analyze the unavailability and average replacement number during (0,𝑡] of the repair facility in a series vacation system with an unreliable repair facility. On this basis, we derive two convolution relations, which make us see clearly the structures of the facility indices in our system. Special cases and numerical examples show that our method is valid in studying the reliability indices of the repair facility of complex vacation systems with a nonreliable repair facility.

Notations

𝑔(𝑠):	The Laplace-Stieltjes transform of 𝐺(𝑡),  that is,  𝑔(𝑠)=∫∞0𝑒−st𝑑𝐺(𝑡)
𝑔∗(𝑠):	The Laplace transform of 𝐺(𝑡),  that is,  𝑔∗(𝑠)=∫∞0𝑒−st𝐺(𝑡)𝑑𝑡
𝐺(𝑘)(𝑡):	The 𝑘-fold convolution of 𝐺(𝑡) with itself (𝑘≥1) and 𝐺(0)(𝑡)=1,  𝑡≥0
𝐸(𝑋):	The mean of random variable 𝑋
Λ:	Λ=𝜆1+𝜆2+⋯+𝜆𝑛
ℜ(𝑠):	The real part of complex number 𝑠
𝜉1:	The first operating time length of the system
𝜂1:	The first repair time length of the system
𝑃(𝑄):	The probability of event 𝑄
𝐻(𝑧),|𝑧|≤1:	The probability generating function of 𝐻. That is,  𝐻(𝑧)=∑∞𝑖=0𝑐𝑖𝑧𝑖.

Acknowledgments

The authors would like to thank the anonymous referees and editor for their valuable comments and suggestions. This work is supported by the Youth Foundation of Chongqing University of Technology of China (no. 2010ZQ14).