Availability of a Periodically Inspected System Maintained through Several Minimal Repairs before a Replacement or a Perfect Repair

Abstract

The instantaneous availability of a periodically inspected system with several minimal repairs before a replacement or a perfect repair is studied. First, we investigate two concrete minimal repair models with periodic inspection and constant repair time and give their instantaneous availability formulas. Then the instantaneous availability of the proposed model with constant repair times is presented by a set of recursive formulas, and its piecewise monotonicity is also shown. An example is presented to illustrate its application.

1. Introduction

A system is either in the upstate or in the downstate at any instant 𝑡, where the term “up-state” means that the system still works and the term “down-state” means that the system does not work. The state of the system at instant 𝑡 can be written as, 𝑋_𝑡=1,thesystemisintheupstateatinstant𝑡,0,thesystemisinthedownstateatinstant𝑡, (1) and the instantaneous availability at instant 𝑡 can be defined as 𝐴(𝑡)=𝑃𝑋_𝑡=1. (2)

The purpose of repair is to make the failed system work well. From the viewpoint of reliability, replacement or perfect repair may be the first choice, where perfect repair means the failed system is repaired to as good as new in [1]. The availability of the system under periodic inspection with different perfect repair or replacement policy has been studied in [2, 3]. Cui et al. [4] presented an availability model for storage products under periodic inspection, and Cui et al. [5] analyzed the single-unit repairable system and gave formulas for reliability indexes by state aggregations. However, from the viewpoint of engineering or economics, replacement or perfect repair is not always desirable, since they may be too expensive or need too much time. Hence, imperfect repair policy is often adopted in practice. The imperfect repair model proposed in [6] has been studied extensively [7–14]. In the imperfect repair model, if the system fails, it will return to the good-as-new state by perfect repair with probability 𝑝, and it will return to a functioning state as bad as old by minimal repair with probability 𝑞=1−𝑝. Since the probability 𝑝 is hard to obtain, the imperfect model is not widely used in practice, and another promise model is proposed in [15], where the system is maintained by several imperfect repairs before a replacement or a perfect repair. Notice that periodic inspection is often adopted in practice, and we focus on the periodically inspected system maintained through several minimal repairs before a replacement or a perfect repair. The structure sketch of the above system can be seen in Figure 1, and some basic assumptions are presented in detail as follows:

(1)at instant 𝑡=0, the system is assumed to be normal, and from then on it is inspected regularly at time instants 𝜏₁=𝜏, 𝜏₂=2𝜏,…; (2)for each inspection, if the system is functioning, it will continue its work without any repair; if the system fails, the failure times 𝑘 will be recorded at first, and the prefixed minimal repair times 𝑚−1 between two consecutive replacements or perfect repairs can be determined from the constraints of cost, resource, and maintainability in advance;(3)if 𝑘mod𝑚≠0, the system will be repaired to as bad as old by minimal repair with constant repair time 𝑏  (0≤𝑏<𝜏); if 𝑘mod𝑚=0, the system will be replaced by a good new one or repaired to the good-as-new state by perfect repair with constant repair time 𝑐  (0≤𝑐<𝜏). The system will immediately work after repair or replacement;(4)each replacement or the completion of each perfect repair can be regarded as a renewal.

Figure 1 

The structure sketch of a periodically inspected system maintained through several minimal repairs before a replacement or a perfect repair, where “•” denotes the system failure, MR and PR denote minimal repair and perfect repair or replacement, respectively, and 𝑚 is prefixed as 2.

The time interval between two consecutive renewals is denoted by 𝑇_𝑖  (𝑖=1,2,…), and let 𝑆₀=0,𝑆_𝑗=^𝑗_𝑖=1𝑇_𝑖; (3) that is, 𝑆₁=𝑇₁ is the time of the first renewal; 𝑆₂=𝑇₁+𝑇₂ is the time of the second renewal. In general, 𝑆_𝑗 denotes the time of the 𝑗th renewal (see Figure 1). From the previous assumptions, 𝑇_𝑖  (𝑖=1,2,…) are independent, and 𝑇_𝑖  (𝑖=2,3,…) are independent and identically distributed; thus, {𝑇_𝑖,𝑖=1,2,…} forms a delayed renewal process. For convenience, the renewal times in time interval [0,𝑡) are denoted by 𝑁_𝐷(𝑡), and the last renewal time is denoted by 𝑆_{𝑁𝐷(𝑡)}.

The instantaneous availability 𝐴(𝑡) is focused on under the following two models: (A)𝑚−1 minimal repairs are allowed before replacement with a new one; (B)𝑚−1 minimal repairs are allowed before a perfect repair.

For convenience, let 𝑏_𝑘=𝜏∗𝑘+𝑏, 𝑐_𝑘=𝜏∗𝑘+𝑐, where 𝑘=1,2,…. For model B, distributions of 𝑇₁ and 𝑇_𝑖  (𝑖=2,3,…) are denoted as 𝐺(𝑡) and 𝐻(𝑡), respectively, and for model A, 𝑇₁,𝑇₂,… are independent and identically distributed, and 𝐺(𝑡)=𝐻(𝑡). From the viewpoint of rsseliability, the effect of replacement and perfect repair is identical; in other words, replacement can be considered as a special perfect repair whose repair time is 0. Hence, model A can be considered as a special case of model B in mathematics, and model B is studied in detail in this paper. If 𝑚=+∞, model B degenerates into the ordinary minimal repair model with periodic inspection and constant repair time, which is discussed in Section 2; if 𝑚=1, model B degenerates into the perfect repair model with periodic inspection and constant repair time [2, 3].

The instantaneous availability of the ordinary minimal repair model with periodic inspection and constant repair time is presented in Section 2; in Section 3, the instantaneous availability of the system under model B is given, and its monotonicity is also discussed; and some conclusions are given in the end.

2. System under Periodic Inspections with Minimal Repair

In order to study the instantaneous availability of the proposed model, a minimal repair model with periodic inspection and constant repair time is discussed first. For this minimal repair model, the system starts work at instant 0 and will be inspected at regular time instants 𝜏_𝑘=𝑘∗𝜏, 𝑘=1,2,…. At instant 𝜏_𝑘, if the system fails, it will be repaired to as bad as old by minimal repair with constant repair time 𝑏  (0≤𝑏≤𝜏); otherwise, it will continue its work without any repair. For convenience, this model is referred to as minimal repair model I.

Assume that the distribution of time to first failure of the system is 𝐹(𝑡) and its reliability 𝑅(𝑡)=1−𝐹(𝑡), and the system instantaneous availability under minimal repair model I is denoted by 𝐴₁(𝑡). In [16], the following theorem and corollary are given.

Theorem 1. For minimal repair model I, the instantaneous availability of the system is 𝐴₁(𝑡)=𝑅(𝑡),0≤𝑡≤𝜏₁,𝑅(𝑡)𝐴₁𝜏_𝑘𝑅𝜏_𝑘,𝜏_𝑘<𝑡≤𝑏_𝑘𝑅(𝑡)×𝐴,𝑘=1,2,…,₁𝜏_𝑘𝑅𝜏_𝑘+1−𝐴₁𝜏_𝑘𝑅𝑏_𝑘,𝑏_𝑘<𝑡≤𝜏_𝑘+1,𝑘=1,2,…. (4)

Corollary 2. For minimal repair model I, the availability of the system is piecewise monotonic decreasing, and it arrives at its minimum values at instant 𝑏_𝑘, 𝑘=1,2,….
The assumption of minimal repair model II is similar to that of minimal repair model I, except that the system starts work at instant (0≤𝑑<𝜏). The system instantaneous availability under minimal repair model II is denoted by 𝐴₂(𝑡). Similarly, 𝐴₂(𝑡) is expressed as follows: 𝐴₂(𝑡)=𝑅(𝑡−𝑑),𝑑≤𝑡≤𝜏₁,𝑅()𝐴𝑡−𝑑²𝜏_𝑘𝑅𝜏_𝑘,𝜏−𝑑_𝑘<𝑡≤𝑏_𝑘𝑅(𝐴,𝑘=1,2,…,𝑡−𝑑)²𝜏_𝑘𝑅𝜏_𝑘+−𝑑1−𝐴₂𝜏_𝑘𝑅𝑏_𝑘,𝑏−𝑑_𝑘<𝑡≤𝜏_𝑘+1,𝑘=1,2,…. (5) And its instantaneous availability is also piecewise monotonic decreasing, and it arrives at its minimum values at instant 𝑏_𝑘, 𝑘=1,2,….

3. Availability of a Periodically Inspected System Maintained by 𝑚−1 Minimal Repairs before a Perfect Repair

In this section, the instantaneous availability for model B is discussed. Since there is no perfect repair in time interval [0,𝑐_𝑚] from the assumptions of model B, then the system instantaneous availability of model B is the same as that of minimal repair model I, and we have the following result.

Proposition 3. For model B, 𝐴(𝑡)=𝐴₁(𝑡),𝑡∈0,𝑐_𝑚. (6)
Since model B under condition 𝑇₁>𝑡 and condition 𝑇₂>𝑡−𝑐 is equal to minimal repair model I and minimal repair model II with 𝑑=𝑐, respectively, one has 𝑃𝑋_𝑡=1∣𝑇₁>𝑡=𝐴₁(𝑡),𝑃𝑋_𝑡=1∣𝑇₂>𝑡−𝑐=𝐴₂(𝑡). (7)

Proposition 4. For model B, 𝐴(𝑡)=^𝑛_𝑘=𝑚^[𝑘/𝑚]_𝑗=1𝑃𝑆_𝑗=𝑐_𝑘𝐴₂𝑡−𝜏_𝑘𝑃𝑇₂>𝑡−𝑐_𝑘+𝐴₁(𝑡)𝑃𝑇₁>𝑡, (8) where 𝑡∈[𝑐_𝑛,𝑐_𝑛+1], 𝑛=𝑚,𝑚+1,…, [𝑥] denotes the largest integer not exceeding 𝑥, and 𝑆_𝑗 is the time of the 𝑗th renewal.

Proof. On [0,𝑡), the time of the last renewal 𝑆_𝑗 will be 𝑐_𝑘, 𝑘=𝑚,𝑚+1,…,𝑛; then the system will be renewed at most [𝑘/𝑚] times. Hence, by the total probability theorem, we have 𝐴(𝑡)=𝑃𝑋_𝑡𝑋=1=𝑃_𝑡=1,𝑇₁+>𝑡_{𝑛𝑘=𝑚[𝑘/𝑚]𝑗=1}𝑃𝑋_𝑡=1,𝑆_𝑗=𝑐_𝑘,𝑇_𝑗+1>𝑡−𝑐_𝑘=𝑃𝑋_𝑡=1∣𝑇₁>𝑡𝑃𝑇₁+>𝑡_{𝑛𝑘=𝑚[𝑘/𝑚]𝑗=1}𝑃𝑆_𝑗=𝑐_𝑘𝑃𝑋_𝑡=1,𝑇_𝑗+1>𝑡−𝑐_𝑘=𝐴₁(𝑡)𝑃𝑇₁+>𝑡_{𝑛𝑘=𝑚[𝑘/𝑚]𝑗=1}𝑃𝑆_𝑗=𝑐_𝑘𝑃𝑋_𝑡=1,𝑇₂>𝑡−𝑐_𝑘=𝐴₁(𝑡)𝑃𝑇₁+>𝑡_{𝑛𝑘=𝑚[𝑘/𝑚]𝑗=1}𝑃𝑆_𝑗=𝑐_𝑘𝐴₂𝑡−𝜏_𝑘×𝑃𝑇₂>𝑡−𝑐_𝑘, (9) where the three but last equations follow from the fact that 𝑇_𝑖, 𝑖=1,2,… are independent and the two but last equations follow from the fact that 𝑇_𝑖, 𝑖=2,3,… are independent and identically distributed. Thus, the proof is completed.

For convenience, let 𝑔_𝑘=𝑃{𝑇₁=𝑐_𝑘}, 𝑘=𝑚,𝑚+1,…. For minimal repair model I, the event that there are 𝑗  (𝑘≥𝑗≥1) failures in [0,𝜏_𝑘] and one failure in (𝜏_𝑘−1,𝜏_𝑘] is denoted by 𝐵(𝑘,𝑗,1); and the event that there are 𝑗  (𝑘≥𝑗≥1) failures in [0,𝜏_𝑘] and no failure in (𝜏_𝑘−1,𝜏_𝑘] is denoted by 𝐵(𝑘,𝑗,0). The probabilities of 𝐵(𝑘,𝑗,0), 𝑗=0,1,…,𝑘−1 and 𝐵(𝑘,𝑗,1), 𝑗=1,2,…,𝑘 can be obtained by the argument of mathematical induction 𝑃{𝐵(𝑘,0,0)}=𝑅𝜏_𝑘,𝑃{𝐵(𝑘,1,1)}=𝑅𝜏_𝑘−1−𝑅𝜏_𝑘,𝑃{𝐵(𝑘,𝑘,1)}=𝑅𝑏_𝑘−1−𝑅𝜏_𝑘{𝐵(,𝑃𝐵=𝑅𝑏×𝑃𝑘−1,𝑘−1,1)}𝑘,𝑗,1_𝑘−1−𝑅𝜏_𝑘𝐵+𝑅𝜏×𝑃𝑘−1,𝑗−1,1_𝑘−1−𝑅𝜏_𝑘𝐵,𝑃{𝐵(=𝑏×𝑃𝑘−1,𝑗−1,0𝑘,𝑘−1,0)}1−𝑅_𝑘−1+𝑅𝜏_𝑘{𝐵(,𝑃𝐵=𝑏×𝑃𝑘−1,𝑘−1,1)}𝑘,𝑗,01−𝑅_𝑘−1+𝑅𝜏_𝑘𝐵+𝜏×𝑃𝑘−1,𝑗,11−𝑅_𝑘−1+𝑅𝜏_𝑘×𝑃𝐵𝑘−1,𝑗,0, (10) where 1<𝑗≤𝑘−1.

From the definition of 𝐵(𝑘,𝑗,1) and model B, we have {𝑇₁=𝑐_𝑘}=𝐵(𝑘,𝑚,1); then 𝑔_𝑘=𝑃𝑇₁=𝑐_𝑘=𝑃{𝐵(𝑘,𝑚,1)},𝑘=𝑚,𝑚+1,…, (11) and they can be calculated by (10).

For convenience, let ℎ_𝑘=𝑃{𝑇₂=𝜏_𝑘}, 𝑘=𝑚,𝑚+1,…. For minimal repair model II with 𝑑=𝑐, the event that there are 𝑗  (𝑘≥𝑗≥1) failures in [𝑐,𝜏_𝑘] and one failure in (𝜏_𝑘−1,𝜏_𝑘] is denoted by 𝐷(𝑘,𝑗,1); and the event that there are 𝑗  (𝑘≥𝑗≥1) failures in [𝑐,𝜏_𝑘] and no failure in (𝜏_𝑘−1,𝜏_𝑘] is denoted by 𝐷(𝑘,𝑗,0). The recursive formula of probabilities of 𝐷(𝑘,𝑗,0), 𝑗=0,1,…,𝑘−1 and 𝐷(𝑘,𝑗,1), 𝑗=1,2,…,𝑘−1 can be obtained by the argument of mathematical induction 𝑃{𝐷((),𝑃{𝐷(𝜏1,1,1)}=1−𝑅𝜏−𝑐𝑘,0,0)}=𝑅_𝑘,𝑃{𝐷(𝜏−𝑐𝑘,1,1)}=𝑅_𝑘−1−𝑐−𝑅𝜏_𝑘,𝑃{𝐷(=𝑅𝑏−𝑐𝑘,𝑘,1)}_𝑘−1−𝑐−𝑅𝜏_𝑘{𝐷(,𝑃𝐷=𝑅𝑏−𝑐×𝑃𝑘−1,𝑘−1,1)}𝑘,𝑗,1_𝑘−1−𝑐−𝑅𝜏_𝑘𝐷+𝑅𝜏−𝑐×𝑃𝑘−1,𝑗−1,1_𝑘−1−𝑐−𝑅𝜏_𝑘𝐷,𝑃{𝐷(=𝑏−𝑐×𝑃𝑘−1,𝑗−1,01<𝑗≤𝑘−1,𝑘,𝑘−1,0)}1−𝑅_𝑘−1−𝑐+𝑅𝜏_𝑘{𝐷(,𝑃𝐷=𝑏−𝑐×𝑃𝑘−1,𝑘−1,1)}𝑘,𝑗,01−𝑅_𝑘−1−𝑐+𝑅𝜏_𝑘𝐷+𝜏−𝑐×𝑃𝑘−1,𝑗,11−𝑅_𝑘−1−𝑐+𝑅𝜏_𝑘𝐷,−𝑐×𝑃𝑘−1,𝑗,01<𝑗≤𝑘−1. (12)

From the definition of 𝐷(𝑘,𝑗,1) and model B, we have {𝑇₂=𝜏_𝑘}=𝐷(𝑘,𝑚,1); then ℎ_𝑘=𝑃𝑇₂=𝜏_𝑘=𝑃{𝐷(𝑘,𝑚,1)},𝑘=𝑚,𝑚+1,…, (13) and they can be calculated by (12).

For the first renewal time 𝑇₁, let 𝑄₁(𝑡)=𝑃{𝑇₁>𝑡}. In this case, the system starts work at instant 0, and from (11), we have 𝑃𝑇₁>𝑡=^∞_{[]𝑘=(𝑡−𝑐)/𝜏+1}𝑃𝑇₁=𝑐_𝑘=1−^{[(𝑡−𝑐)/𝜏]}_𝑘=𝑚𝑃𝑇₁=𝑐_𝑘=1−^{[(𝑡−𝑐)/𝜏]}_𝑘=𝑚𝑔_𝑘; (14) then 𝑄₁(𝑡)=1−^{[(𝑡−𝑐)/𝜏]}_𝑘=𝑚𝑔_𝑘. (15)

For the second renewal time, let 𝑄₂(𝑡)=𝑃(𝑇₂>𝑡). In this case, the system starts work at instant 𝑐, and from (13), we have 𝑃𝑇₂>𝑡=^∞_{[]𝑘=𝑡/𝜏+1}𝑃𝑇₂=𝜏_𝑘=1−^[𝑡/𝜏]_𝑘=𝑚𝑃𝑇₂=𝜏_𝑘=1−^[𝑡/𝜏]_𝑘=𝑚ℎ_𝑘; (16) then 𝑄₂(𝑡)=1−^[𝑡/𝜏]_𝑘=𝑚ℎ_𝑘. (17)

From the fact that 𝑇₁ and 𝑇_𝑖  (𝑖=2,3,…) follow distributions 𝐺 and 𝐻, respectively, it is easy to know that the time of the 𝑗th renewal 𝑆_𝑗=∑^𝑗_𝑘=1𝑇_𝑘 follows distribution 𝐺∗𝐻_𝑗−1, where * means the convolution of distribution functions, and the value set of 𝑆_𝑗 is {𝑐_𝑘∶𝑘=𝑚𝑗,𝑚𝑗+1,…}. For convenience, probability generating functions of distributions 𝐺 and 𝐻 are denoted by 𝑔(𝑠) and ℎ(𝑠), respectively, which can be expressed as 𝑔(𝑠)=∑^∞_𝑙=𝑚𝑔_𝑙𝑠^𝑙 and ℎ(𝑠)=∑^∞_𝑖=𝑚ℎ_𝑖𝑠^𝑖, respectively.

Let 𝑃{𝑆_𝑗=𝑐_𝑘}=𝑣_𝑘^(𝑗) and 𝐺∗𝐻_𝑗−1={𝑣_𝑘^(𝑗)∶𝑘=𝑚𝑗,𝑚𝑗+1,…}, so the probability generating function of 𝐺∗𝐻_𝑗−1 can be expressed as ∑^∞_{𝑘=𝑚𝑗}𝑣_𝑘^(𝑗)𝑠^𝑘. Since 𝑇₁,𝑇₂,… forms a delay renewal process and consequently 𝑆_𝑗  (𝑗=1,2,…) forms a delay renewal process, then ^∞_{𝑘=𝑚𝑗}𝑣_𝑘^(𝑗)𝑠^𝑘=^∞_𝑙=𝑚𝑔_𝑙𝑠^𝑙×^∞_𝑖=𝑚ℎ_𝑖𝑠^𝑖𝑗−1. (18)

Through comparison of similar items between two sides of (18), we have 𝑣_𝑘⁽¹⁾=𝑔_𝑘𝑣,𝑘=𝑚,𝑚+1,…_𝑘(2)=^𝑘−𝑚_𝑖=𝑚𝑣_𝑖⁽¹⁾ℎ_𝑘−𝑖⋮𝑣,𝑘=2𝑚,2𝑚+1,…_𝑘(𝑗)=^𝑘−𝑚_{𝑖=𝑚(𝑗−1)}𝑣_𝑖^(𝑗−1)ℎ_𝑘−𝑖,𝑘=𝑚𝑗,𝑚𝑗+1,…. (19)

From the previous results, we have the following theorem and corollary.

Theorem 5. The instantaneous availability of the system under model B is 𝐴(𝑡)=𝐴₁(𝑡),0≤𝑡≤𝑐_𝑚,𝐴₁(𝑡)𝑄₁(𝑡)+^𝑛_𝑘=𝑚^[𝑘/𝑚]_𝑗=1𝑣_𝑘^(𝑗)𝐴₂𝑡−𝜏_𝑘𝑄₂𝑡−𝑐_𝑘,𝑐_𝑛≤𝑡<𝑐_𝑛+1,𝑛=𝑚,𝑚+1,…. (20)

Corollary 6. The instantaneous availability of the system under model B is piecewise monotonic decreasing, and it arrives at its minimum values at instant 𝑏_𝑖, 𝑖=1,2,….

Proof. On [0,𝑐_𝑚), 𝐴(𝑡)=𝐴₁(𝑡), and from Corollary 2, we have that 𝐴(𝑡) is piecewise monotonic decreasing and arrives at its minimal values at instant 𝑏_𝑖, 𝑖=1,2,…,𝑚.
On [𝑐_𝑛,𝑐_𝑛+1), 𝑛=𝑚,𝑚+1,…, from (15) and (17), 𝑄₁(𝑡)=1−∑^𝑛_l=𝑚𝑔_𝑙 and 𝑄₂(𝑡−𝑐_𝑘)=1−∑^𝑛−𝑘_𝑙=𝑚ℎ_𝑙; that is, 𝑄₁(𝑡) and 𝑄₂(𝑡−𝑐_𝑘) are fixed constants on [𝑐_𝑛,𝑐_𝑛+1). Similarly, it can be proved that 𝑣_𝑘^(𝑗) is also a fixed constant. From Theorem 5, 𝐴(𝑡)=𝐴₁(𝑡)𝑄₁(𝑡)+∑^𝑛_𝑘=𝑚∑^[𝑘/𝑚]_𝑗=1𝑣_𝑘^(𝑗)𝐴₂(𝑡−𝜏_𝑘)𝑄₂(𝑡−𝑐_𝑘), and from Corollary 2, 𝐴₁(𝑡) and 𝐴₂(𝑡−𝜏_𝑘) are piecewise monotonic decreasing and arrive at their minimum values at instant 𝑏_𝑖, 𝑖=1,2,…; then 𝐴(𝑡) is piecewise monotonic decreasing and arrives at its minimum values at instant 𝑏_𝑖, 𝑖=1,2,…. Thus, the proof is completed.

For model A, the periodically inspected system is maintained by 𝑚−1 minimal repairs before replacement. As mentioned previously, model A is equal to a special case of model B whose perfect repair time is 0. Hence, the 𝑐 in (20) is replaced by 0, and the instantaneous availability of model A can be given. And notice that 𝑇_𝑖, 𝑖=1,2,… form a renewal process under model A and they are independent and identically distributed; thus, 𝐴₂(𝑡)=𝐴₁(𝑡) and 𝑄₂(𝑡)=𝑄₁(𝑡) under model A. Hence, we have the following theorem and corollary.

Theorem 7. The instantaneous availability of the system under model A is 𝐴(𝑡)=𝐴₁(𝑡),0≤𝑡≤𝜏_𝑚,𝐴₁(𝑡)𝑄₁(𝑡)+^𝑛_𝑘=𝑚^[𝑘/𝑚]_𝑗=1𝑣_𝑘^(𝑗)𝐴₁𝑡−𝜏_𝑘𝑄₁𝑡−𝑐_𝑘,𝜏_𝑛≤𝑡<𝜏_𝑛+1,𝑛=𝑚,𝑚+1,…. (21)

Corollary 8. The instantaneous availability of the system under model A is piecewise monotonic decreasing, and it arrives at its minimum values at instant 𝑏_𝑖, 𝑖=1,2,….

Example 9. Suppose the distribution of time to first failure of a system is 𝐹(𝑡)=1−exp{−(𝑡/5)^1.2}, and it is periodically inspected and maintained by 2 minimal repairs before a perfect repair, where the regular inspection interval 𝜏=1.0 and the constant repair times of minimal repair and perfect repair are 0.1 and 0.2, respectively. In other words, it is under model B with 𝜏=1.0, 𝑏=0.1, 𝑐=0.2, and 𝑚=3. From Theorem 5, its instantaneous availability 𝐴(𝑡) can be calculated, and its availability curve can be seen in Figure 2.
For the previous example, if the perfect repair is replaced by replacement, that is to say, the system is under model A with 𝜏=1.0, 𝑏=0.1, and 𝑚=3. According to Theorem 7, its instantaneous availability 𝐴(𝑡) can be calculated, and its availability curve can be seen in Figure 3.

Figure 2 

The availability curve under model B with 𝜏=1.0, 𝑏=0.1, 𝑐=0.2, and 𝑚=3.

Figure 3 

The availability curve under model A with 𝜏=1.0, 𝑏=0.1, and 𝑚=3.

The availability curves in Figures 2 and 3 show that the instantaneous availability of the periodically inspected system maintained through several minimal repairs before a replacement or a perfect repair is piecewise monotonic decreasing and arrives at its minimum values at instant 𝑏_𝑘, which provides important information for management and decisions of reliability and maintenance. For example, if the instantaneous availability requirement is considered, we can only focus on the instantaneous availability at instant 𝑏_𝑘, 𝑘=1,2,….

Conclusion

The availability of a periodically inspected system maintained through several minimal repairs before a replacement or a perfect repair is studied in this paper. The instantaneous availability of the proposed model is derived by a set of recursive formulas, and it is found that its availability is piecewise monotonic decreasing, which provides theoretical basis for management and decisions in terms of system reliability and maintenance in practice. In particular, the monotonicity of the instantaneous availability can be utilized to optimize the practical reliability work and make our attention focused on some critical time instants.

For simplicity, minimal repair and perfect repair are supposed to take different constant times in this contribution, and this assumption reflects the difference between minimal repair and perfect repair in a certain degree. Availability with random repair time and more realistic assumptions will be investigated in our future research.

Acknowledgments

This work was supported in part by China NSFC under Grant no. 11001005 and the Fundamental Research Funds for the Central Universities under Grant no. YWF-13-ZY-169.