Renewable k-Out-of-n System with the Component-Wise Strategy of Preventive System Maintenance

Abstract

At the SMARTY-22 conference, a review of the regenerative methods development was presented, including its application to the study of a non-renewable k-out-of-n system. This paper develops the previous study for the renewable k-out-of-n system, including an investigation different preventive maintenance strategies based on the system state observation. We also include the review of Smith’s regeneration idea development. Some new results are presented that form the basis for an algorithm for comparing preventing maintenance strategies with respect to the maximization of the availability factor. A numerical study was conducted for the 4-out-of-6 and 4-out-of-8 models. The study demonstrates the sensitivity of decision making to the shape of the repair time distribution.

1. Introduction

At the SMARTY-22 conference, the history of Smith’s regenerative idea development was presented, and the methods of the theory of decomposable semi-regenerative processes (DSRPr) was applied to the investigation of a non-renewable k-out-of-n system. Creating reliable products, systems, and processes, and maintaining during their functioning is the cornerstone of the development of the industry. The redundancy and timely preventive maintenance of elements of complex engineering systems contribute to reliability and stability. A k-out-of-n system (a system of n elements running in parallel that fails as soon as at least k of them fail) is widely used in a variety of applications. There is much research for this kind of models based on the mathematical methods of probability theory and the analysis of stochastic processes; for example, Trivedi [1] and Chakravarthy [2].

k-out-of-n systems with dependent failures in the framework of the Marshall—Olkin failure model and copula analysis were studied in [3,4]. A research of a k-out-of-n system’s reliability function with general repair time distribution, and its applications in telecommunication and transportation were presented in [5]. The algebraic analysis of variants and the reliability of multistate k-out-of-n systems were studied in [6,7]. The potential application of k-out-of-n systems in oil and gas industry objects was investigated in [8]. A mathematical model of a k-out-of-n system for the preventive maintenance of an unmanned underwater vehicle for monitoring underwater pipelines was presented in [9]. The high-altitude module modelling of a telecommunications base station was analyzed as a heterogeneous k-out-of-n system, taking into account the load redistribution between operable components, in [10]. In [11], the degradation modeling and condition-based maintenance of boiler heat exchangers was described. In [12], a preventive maintenance strategy for a photovoltaic plant was discussed with the aim of achieving maximal system availability. The model of a closed homogeneous redundant hot-standby system for data links was also presented in [13].

Another way to maintain the reliability of products, systems, and processes is to organize their preventive maintenance (PM). PM models are a branch of a large field of research related to controllable stochastic systems and processes, which has been developed by such eminent scientists as Bellman, Blackwell, Derman, Dynkin, Shiryaev.

A series of surveys in different directions of stochastic systems have appeared in recent decades. There was review of mathematical methods for controlling stochastic systems with applications to queueing and reliability models from the very beginning to the current time [14]. A fairly detailed overview of PM models based on different system state observation possibilities can be found in Gertsbach’s monograph [15]. A survey of the application of gamma processes in the maintenance of reliability systems was proposed in [16]. The article in [17] focused on optimal preventive maintenance and examined two failure scenarios for system components: due to aging, and due to fatal shocks occurring outside or inside the system.

The authors in [18] investigated preventive maintenance for an $(n-k+1)$-out-of-n system due to the component lifetime and cost per unit time. Two system failure scenarios were chosen for that study. In the first scenario, the failure of a single component occurred, and the process was of the Marshall–Olkin type. In the second scenario, an extended type of the shock process was investigated in which the shock leads to the failure of any n components at random points in time. Within each scenario, an optimal age-based PM policy that minimizes the mean cost per unit time was found.

A new approach to PM based on the prediction of system failures has become very popular over the past few decades; see [19] and the references therein, and a survey of the recent literature in [20].

In the works of Cha, Finkelstein, Gertsbakh, Levitin, and others [21,22,23,24,25,26,27,28,29,30,31], different aspects of PM strategy optimization were considered. The problem of decision making and preliminary prophylactic analysis was considered in [32,33,34]. In some of our previous works [9,35,36], the problem of the sensitivity analysis of output model characteristics to the distributions of the input characteristics was considered. The papers of Z. Kala presented a general approach to the problems of sensitivity analysis [37,38].

In [39], DSRPr methods were applied to investigate the main reliability characteristics of a k-out-of-n system. The authors in [35,36] presented a comparison of different PM strategies with respect to the availability factor maximization for a non-renewable k-out-of-n system, and the cost-type criterion was proposed. This paper prolongs this study for a renewable k-out-of-n system. This study is based on DSRPr theory methods; therefore, we start with a brief description of Smith’s regenerative idea development.

The paper is organized as follows. In the next section, we give a short overview of the main steps of Smith’s regenerative theory development, which is necessary for further investigations. In Section 3, a new problem statement for comparing PM strategies for a renewable k-out-of-n system and choosing the best one with respect to the system availability factor is proposed. The main notations and assumptions are also presented. Then, in Section 4, the DSRPr approach is used for a new analytical solution of the stated problem. Section 5 presents the calculation procedure and algorithm for implementing the proposed analytical solution. Some examples are numerically studied in Section 6. Section 7 proposes directions for further investigations and concludes the paper.

The novelty of the paper consists of the new problem statement in Section 3, its new analytical solution in Section 4, the development of a new algorithm for its implementation in Section 5, and the numerical examples in Section 6.

2. Regeneration Idea Development

In this section, we briefly recall the development of Smith’s regeneration idea.

2.1. Regenerative Processes

The well-known regeneration notion from Smith (1955) [40] assumed that a random process has random moments in time that break its trajectory into independent parts. The presence of independence allows for obtaining a number of interesting theoretically and practically useful results for such processes. Let $X=\left\{X\right(t),t\in R\}$ be a stochastic process (SPr) with measurable states space $(E,\mathcal{E})$, filtration ${\mathcal{F}}_t^X=\sigma \{X\left(v\right),v\le t\}$, and a sequence of Markov times $S_n,S_{n-1}<S_n,S_0=0$ with respect to it.

Definition 1. 

Pair $X=\{(X\left(t\right),S_n),t\in R,n\in N\}$ is called a (homogeneous) regenerative process (RPr) with regeneration times (RT) $S_n$, if it “forgets” its past at any RT $S_n$, and its behavior is stochastic equivalent after any RT $S_n$,

$$\begin{array}{ccc}\hfill \mathbf{P}\{X(S_n+t)\in \Gamma |{\mathcal{F}}_{S_n}^X\}& =& \mathbf{P}\{X(S_n+t)\in \Gamma \}=\mathbf{P}\{X(S_1+t)\in \Gamma \}.\hfill \end{array}$$

(1)

Intervals $[S_n,S_{n+1})$ and their length $T_n=S_{n+1}-S_n,n=0,1,2,\dots S_0=0$ are called regeneration periods (RPs), and functional random elements $W_n=\{(X(S_n+t),T_n),t\le T_n,$$n=1,2,\dots \}$ are called regeneration cycles (RCs).

$F\left(t\right)=\mathbf{P}\{T_n\le t\}$ denotes the cumulative distribution function (CDF) of RP $T_n$, and

$$H\left(t\right)=\mathbf{E}\left[\sum _{n\ge 1}{1}_{\{S_n\le t\}}\right]=\sum _{n\ge 1}\mathbf{P}\{S_n\le t\}=\sum _{n\ge 1}{F}^{*n}\left(t\right)$$

denotes the renewal function (RF) of the renewal process $N\left(t\right)={\sum }_{n\ge 1}{1}_{\{S_n\le t\}}$, which satisfies the Wiener–Hopf equation:

$$H\left(t\right)=F\left(t\right)+\underset{0}{\overset{t}{\int }}H\left(du\right)F(t-u)\equiv F\left(t\right)+H*F\left(t\right).$$

(2)

With the help of the complete probability formula, RPr distribution

$$\pi (t,\Gamma )=\mathbf{P}\left\{X\right(t)\in \Gamma \}$$

can be represented in terms of its distributions at separate RPs:

$${\pi }^{\left(1\right)}(t,\Gamma )=\mathbf{P}\{X(S_n+t)\in \Gamma ,t<T_n\}$$

and the RF of the process in the following form:

$$\begin{array}{c}\hfill \pi (t;\Gamma )={\pi }^{\left(1\right)}(t;\Gamma )+\underset{0}{\overset{t}{\int }}{\pi }^{\left(1\right)}(t-u,\Gamma )H\left(du\right).\end{array}$$

(3)

Since distribution at a separate RP is more available in many applications, the above formula enables calculating the process distribution. The key renewal theorem [40] allows for proving the existence of the stationary probability distribution of the RPr, which it represents with

$$\pi (\Gamma )=\underset{t\to \infty }{lim}\pi (t;\Gamma )=\frac{1}{\mathbf{M}\left[T_n\right]}\underset{0}{\overset{\infty }{\int }}{\pi }^{\left(1\right)}(t,\Gamma )dt.$$

(4)

2.2. Semi-Regenerative Processes

In applications, RTs are usually some states’ (regeneration states (RSs)) destination times. If there exist several RSs, the notion of RPr is generalized to a semi-regenerative process (SRPr). The joining of the regeneration idea with the notion of Markov’s dependency led to the introduction of semi-regenerative processes (SRPr) that first appeared under different names:

• Semi-Markov processes with additional trajectories G. Klimov (1966) [41].
• Regenerative processes with several types of regeneration points V. Rykov, M.Yastreenetsky (1971) [42].

Further, J. Jacod (1971) [43] and E. Numellin (1978) [44] gave this notion the term SRPr.

Consider SP $X=\left\{X\right(t),t\in R\}$ with measurable state space $(E,\mathcal{E})$, filtration ${\mathcal{F}}_t^X=\sigma \{X\left(v\right),v\le t\}$ and a sequence of Markov times $S_n,S_{n-1}<S_n,S_0=0$ with respect to it.

Definition 2. 

Pair $X=\{(X\left(t\right),S_n),t\in R,n=1,2,\dots \}$ is called a (homogeneous) semi-regenerative process (SRPr) with RTs $S_n$, if for any subset $\Gamma \in \mathcal{E}$ and for all $S_n(n=1,2,\dots )$ the process forgets its past up to the present state and behaves stochastically equivalent,

$$\begin{array}{ccc}\hfill \mathbf{P}\{X(S_n+t)\in \Gamma |{\mathcal{F}}_{S_n}\}& =& \mathbf{P}\{X(S_n+t)\in \Gamma |X\left(S_n\right)\}\hfill \\ & =& \mathbf{P}\{X(S_1+t)\in \Gamma |X\left(S_1\right)\}.\hfill \end{array}$$

(5)

Intervals $T_n=(S_n,S_{n+1}]$ and their lengths $T_n=S_{n+1}-S_n$ are called RPs, functional random elements $W_n=\{(X(S_n+t),T_n),t\le T_n,n=1,2,\dots \}$ are called RCs, and the random values (RVs) $X_n=X\left(S_n\right)$ are called the RSs of the process.

Due to the monotony of the sequence $\left\{S_n\right\}$, there exists a limit $\underset{n\to \infty }{lim}{S}_n$. SRPr $\{X\left(t\right),S_n\}$ is regular if it exists on the entire time axis with probability 1:

$$\mathbf{P}\{\underset{n\to \infty }{lim}{S}_n=\infty \}=1.$$

In SRPr investigation, so-called auxiliary processes play an essential role:

• Embedded Markov Chain (EMC) $X_n=X\left(S_n\right)$.
• Embedded semi-Markov Chain (ESMC) $Y_n=(X_n,T_n)$.
• Markov renewal process (MRPr) $N\left(t\right)=max\{n:S_n\le t\}={\sum }_{n\ge 1}{1}_{\{S_n\le t\}}.$

$P=P(x,y)$ denotes the transitional matrix (TM) of MC $\left\{X_n\right\}$, and $Q=Q(x,t,y)$ denotes the semi-Markov matrix (SMM) of semi-Markov chain (SMC) $\left\{Y_n\right\}$,

$$\begin{array}{ccc}\hfill P(x,y)& =& \mathbf{P}\{X_{n+1}=y|X_n=x\},\hfill \\ \hfill Q(x,t,y)& =& \mathbf{P}\{X_{n+1}=y,T_{n+1}\le t|X_n=x\}.\hfill \end{array}$$

The behavior of a SRPr $\{X\left(t\right),S_n\}$ is mainly determined via the properties of these chains. For Markov renewal matrix (MRM)

$$H(x,t,y)={\mathbf{E}}_x\sum _{n\ge 1}{1}_{\left\{\right([0,t],y\left)\right\}}(S_n,X_n)$$

Equation (2) is generalized as follows:

$$H(x,t,y)=Q(x,t,y)+\sum _{j\in E_1}\underset{0}{\overset{t}{\int }}H(x,du,j)Q(j,t-u,y)\equiv Q(x,t,y)+H★Q(x,t,y),$$

(6)

where ★ denotes the matrix-functional convolution given by this formula.

For SRPr distribution, the following generalization of Formula (3) occurs:

$$\begin{array}{ccc}\hfill \pi (x,t,\Gamma )& =& {\pi }^{\left(1\right)}(x,t,\Gamma )+\underset{0}{\overset{t}{\int }}\sum _{y\in E_1}H(x,du,y){\pi }^{\left(1\right)}(y,t-u,\Gamma )\hfill \\ & \equiv & {\pi }^{\left(1\right)}(x,t,\Gamma )+H★{\pi }^{\left(1\right)}(x,t,\Gamma ),\hfill \end{array}$$

(7)

where

$$\begin{array}{ccc}\hfill \pi (x,t,\Gamma )& =& \mathbf{P}\left\{X\right(t)\in \Gamma |X\left(0\right)=x\},\hfill \\ \hfill {\pi }^{\left(1\right)}(x,t,\Gamma )& =& \mathbf{P}\{X(S_n+t)\in \Gamma ,t<T_n|X\left(S_n\right)=x\},n\ge 1.\hfill \end{array}$$

The well-known key renewal theorem is generalized for SRPrs as follows (see Rykov, Yastrebenetsky (1971) [42] for the proof of the discrete set of RSs $E_1$, and see Jacod (1971) [43] and Nummelin (1978) for the general RS space [44]).

For (i) a regular SRPr $\{X\left(t\right),S_n\}n\in N$ with (ii) non-decomposable, positively recurrent EMC $\left\{X_n\right\}$, (iii) absolutely continuous function $Q(x,[t,\infty ),E)$ and (iv) integrable on $R^{+}=[0,\infty )$ function $g(y,t)$, the following relation holds:

$$\begin{array}{ccc}\hfill \underset{t\to \infty }{lim}H★g(t,x)& \equiv & \underset{t\to \infty }{lim}\underset{0}{\overset{t}{\int }}\sum _{y\in E_1}H(x,du,y)g(y,t-u)dt\hfill \\ & =& m^{-1}\underset{0}{\overset{\infty }{\int }}\sum _{y\in E_1}\alpha \left(y\right)g(y,t)dt,\hfill \end{array}$$

where $\alpha =\left\{a\right(x),x\in E\}$ is an invariant measure of EMC $\left\{X_n\right\}$, and

$$m=\sum _{x\in E_1}a\left(x\right)\underset{0}{\overset{\infty }{\int }}Q(x,[t,\infty ),E)dt$$

is a stationary mean of RP length. The last statement provides the calculation of SRPr steady state probabilities (SSPs):

$$\pi (\Gamma )=\underset{t\to \infty }{lim}\pi (x,t,\Gamma )=m^{-1}\underset{0}{\overset{\infty }{\int }}\sum _{x\in E}\alpha \left(x\right){\pi }^{\left(1\right)}(x,t,\Gamma )dt$$

(8)

in terms of distributions at a separate RP and invariant probabilities of EMC.

However, if the behavior of a process in a separate RP is complex enough, calculating function ${\pi }^{\left(1\right)}(x,t,\Gamma )$ is not a simple problem. Its solution can be simplified if, for any RP $(S_n^{\left(1\right)},S_{n+1}^{\left(1\right)}]$, another embedded RT $S_{n,k}^{\left(2\right)}$ can be found.

2.3. Decomposable Semi-Regenerative Process

The next step in the development of Smith’s regenerative idea was proposed by V. Rykov (1975) [45], who introduced the concept of decomposable semi-regenerative processes. For a random interval $T=\left[\underline{S},\overline{S}\right)$,

$$\begin{array}{c}{\mathcal{F}}_t^X\left(T\right)=\sigma \left\{X\left(v\right),\underline{S}\le v\le t\wedge \overline{S}\right\},\end{array}$$

denotes the flow of the $\sigma$-algebras of events associated with the process $\left\{X\right(t\left)\right\}$ behavior at interval T. Consider SRPr $\{(X\left(t\right),S_n),t\in R,n\in N\}$ with the sequence of main RTs $S_n$ and main RPs $T_n=S_n-S_{n-1}$.

Definition 3. 

For a fixed n of a SRPr X, triplet $\left\{X\left(t\right),S_{n,k}^{\left(1\right)}{T}_n\right\}$ is called

• an embedded semi-regenerative process (ESRPr) if the analogous relation to (5) holds for all k and any measurable set $\Gamma \subset E$

  $$\begin{array}{ccc}\hfill & & \mathbf{P}\left\{X\left(S_{n,k}^{\left(1\right)}+t\right)\in \Gamma \mid {\mathcal{F}}_{S_{n,k}^{\left(1\right)}}^X\left(T_n\right)\right\}=\mathbf{P}\left\{X\left(S_{n,k}^{\left(1\right)}+t\right)\in \Gamma \mid X\left(S_{n,k}^{\left(1\right)}\right)\right\}\hfill \\ & =& \mathbf{P}\left\{X\left(S_{n,1}^{\left(1\right)}+t\right)\in \Gamma \mid X\left(S_{n,1}^{\left(1\right)}\right)\right\}.\hfill \end{array}$$

  (9)
• It is called the DSRPr (of level 1) if it is holds for all n.
• RVSs $S_{n,k}^{\left(1\right)}$ are called embedded regeneration times (ERTs).
• Intervals $T_{n,k}^{\left(1\right)}=\left(S_{n,k}^{\left(1\right)},S_{n,k+1}^{\left(1\right)}\right]$ and their lengths $T_{n,k}^{\left(1\right)}=S_{n,k+1}^{\left(1\right)}-S_{n,k}^{\left(1\right)}$ are called embedded regeneration periods (ERPs).
• RVSs $X_{n,k}^{\left(1\right)}=X\left(S_{n,k}^{\left(1\right)}\right)$ are called embedded regeneration states (ERSs).

As above, it is possible to define second-level ERTs. The maximal value of the decompositional levels of a DSRPr is called its rank r. The set of regeneration states of the k-th level is denoted by $E_k$.

Remark 1. 

Considering only homogeneous processes, we simplified the notation by omitting one sub-index; thus, the super-index denotes the level of embedded process, and the sub-index denotes the number of ERTs in the upper ERP. With this remark, the notation is as follows:

$$S_n^{\left(k\right)},T_n^{\left(k\right)},X_n^{\left(k\right)}.$$

In DSRPr analysis, the role of the ordinary renewal process is played by the embedded renewal process (ERPr), which, for the k-th level and $y\in E_k$, is given by

$$\begin{array}{c}{N}^{\left(k\right)}(t,y)=\sum _{n\ge 1}{1}_{\left\{\right[0,t),y\}}\left(S_n^{\left(k\right)},X_n^{\left(k\right)}\right)1_{\{S_n^{\left(k\right)}<T^{(k-1)}\}}.\end{array}$$

An appropriate embedded renewal matrix (ERM) is given by

$$\begin{array}{c}{H}^{\left(k\right)}(x,t,y)={\mathbf{E}}_x\left[N^{\left(k\right)}(t,y)|X\left(S^{(k-1)}\right)=x\right].\end{array}$$

The type of embedded regeneration points significantly affects the research process. If the embedded regeneration points occur as $min[\left\{{max}_n{S}_n^{\left(k\right)}\right\},T^{k-1}],$ then, unlike Equations (2) and (6), for ERM, the following equation holds (for a proof of the corresponding theorem for a denumerable set of states, see [45]).

The family of ERM satisfies the following (modified Wiener–Hopf) system of equation for $k=1,2,\dots r$

$$\begin{array}{ccc}\hfill & & H^{\left(k\right)}(x,t,y)=Q_1^{\left(k\right)}(x,t,y)\hfill \\ & +& \sum _{j\in E_k}\underset{0}{\overset{t}{\int }}{H}^{\left(k\right)}(x,du,j)Q^{\left(k\right)}(j,t-u,y)-Q^{(k-1)}(x,t,y)\hfill \\ & \equiv & Q_1^{\left(k\right)}(x,t,y)+H^{\left(k\right)}★Q^{\left(k\right)}(x,t,y)-Q^{(k-1)}(x,t,y),\hfill \end{array}$$

(10)

where $Q^{\left(k\right)}(x;t,y)$ is the SMM of the process at external ERP, and $Q^{(k-1)}((x;t,y)$ is the SMM of the process at internal 1.

In the majority of practical scenarios, however, both internal and external regeneration times coincide with the time moments of the process entering the appropriate regeneration states. In this instance, the transition matrix for the embedded moments of regeneration $Q^{\left(k\right)}\left(t\right)={\left[Q_{ij}^{\left(k\right)}\left(t\right)\right]}_{ij\in E_k}$ is a sub-matrix of the transition matrix of the previous level $Q^{k-1}\left(t\right)$ with components from the subset of the embedded set of states $E_k$; hence, it is a non-degenerative one. Then, the equation for ERM $H^{\left(k\right)}\left(t\right)$ is as follows:

$$H^{\left(1\right)|}\left(t\right)=Q^{\left(k\right)}\left(t\right)+Q^{\left(k\right)}★H^{(k}\left(t\right)$$

with the solution

$$H^{\left(k\right)}\left(t\right)={(I-Q^{\left(k\right)})}^{-1}{Q}^{\left(k\right)}\left(t\right)=\sum _{n\ge 1}\left[Q^{\left(k\right)★n}\left(t\right)\right].$$

It is finite for all t when $t\to \infty$, and tends to the mean number of arrivals to subset $E_k.$

Analogously to (7), different characteristics of the DSRPr of the $(k-1)$-th level can be represented by the corresponding parameters of the k-th level. In particular, for one-dimensional distributions

$$\begin{array}{c}{\pi }^{\left(k\right)}(x,t,\Gamma )=\mathbf{P}\left\{X\left(S_n^{\left(k\right)}+t\right)\in \Gamma ,t<T_n^{\left(k\right)}\mid X_n^{\left(k\right)}=x\right\}(k=\overline{1,r})\end{array}$$

the following equation holds:

$${\pi }^{(k-1)}(x,t,\Gamma )={\pi }^{\left(k\right)}(x,t,\Gamma )+Q^{\left(k\right)}★{\pi }^{(k-1)}(x,t,\Gamma ),$$

(11)

whose solution is

$${\pi }^{(k-1)}(x,t,\Gamma )={\pi }^{\left(k\right)}(x,t,\Gamma )+H^{\left(k\right)}★{\pi }^{\left(k\right)}(x,t,\Gamma ),$$

(12)

where ERM $H^{\left(k\right)}(x,t,y)$ satisfies Equation (10). These equations allow for recovering the process distribution via its distribution on separate minimal periods of embedded regeneration.

The limit theorem for SRPrs enables calculating their stationary distributions, and the system of embedded regeneration periods gives the possibility to calculate them using distributions on smaller regeneration periods.

For a more detailed review of the development of the regenerative idea and applications of the DSRPrs, including a non-renewable k-out-of-n system, see [39,46]. Further, we applied the theory of DSRPrs to the study of the renewable k-ouf-of-n system, and propose a procedure for comparing different PM strategies with respect to maximizing its availability.

3. Problem Set, Notations, and Assumptions

Consider a renewable k-out-of-n system that consists of n components running in parallel that fails if at least k of them fail [47]. After each component failure (before the k-th one), it is repaired by a unique repair unit within a random repair time. Assuming the possibility of a PM based on the observation of system states, let $\mathcal{L}=\{1,2,\dots ,k\}$ be the set of possible strategies for the starting PM. The l-th strategy $(l<k)$ means that PM starts when l components fail. Strategy $l=k$ means that the decision maker does not use PM, and after k elements fail, the repair and maintenance of all elements of the system begin.

Further, we use the following notations:

• $A_i(i\in N)$ is the series of components lifetimes.
• $B_i(i\in N)$ is the series of components repair times,
• $G_i^{\left(l\right)}(i\in N,l\in \mathcal{L})$ is a series of the system PM duration (system repair times for $l=k$).
• $E=\{j,j\in \{0,1,\dots ,k\left\}\right\}$ is the set of the system states where state j means the number of failed components.
• ${\overline{S}}_n^{\left(l\right)}$ is the time to the n-th PM beginning under the l-th PM strategy.
• $S_n^{\left(l\right)}$ is the time to the n-th PM completion under the l-th PM strategy.
• $T_n^{\left(l\right)}=S_n^{\left(l\right)}-S_{n-1}^{\left(l\right)}$ is the duration of the interval between two successive system renovation times under the l-th PM strategy.
• $F_n^{\left(l\right)}={\overline{S}}_n^{\left(l\right)}-S_{n-1}^{\left(l\right)}$ is the duration of the system’s n-th working cycle under the l-th PM strategy.
• $F_l\left(t\right)=\mathbf{P}\{F_n^{\left(l\right)}\le t\}$ are their CDFs with mean values

  $$f_l=\mathbf{E}\left[F_n^{\left(l\right)}\right]={\int }_0^{\infty }(1-F_l\left(t\right))dt.$$

The following assumptions were used here:

• Variables $A_i(i\in N)$ are supposed to be independent identically distributed (i.i.d.) random variables (RVSs) with yje exponential distribution of parameter $\alpha$ and mean value $a=\mathbf{E}\left[A_i\right]={\alpha }^{-1}$;
• Variables $B_i(i\in N)$ are supposed to be i.i.d. RVSs with a common CDF, $B\left(t\right)=\mathbf{P}\{B_i\le t\}$, and mean value

  $$b=\mathbf{E}\left[B_i\right]={\int }_0^{\infty }(1-B\left(t\right))dt;$$
• Variables $G_i^{\left(l\right)}(i\in N,l\in \mathcal{L})$ are supposed to be i.i.d. RVSs with a common CDF, $G_l\left(t\right)=\mathbf{P}\{G_i^{\left(l\right)}\le t\}$, and mean values

  $$g_l=\mathbf{E}\left[G_i^{\left(l\right)}\right]={\int }_0^{\infty }(1-G_l\left(t\right))dt;$$
• Mean PM times $g_l(l<k)$ are supposed to be less than the mean repair time $g_k,(g_l\le g_k)$, and they may or may not depend on the type of maintenance.
• Initially, all system components were in the operating state, which means that the initial state of the system was zero, $j=0$.
• The immediate repairs of components, and the maintenance of the whole system for any $l<k$ and repair after its failure for $l=k$ are impossible, and both mean times are finite:

  $$B\left(0\right)=G\left(0\right)=0,\underset{0}{\overset{\infty }{\int }}(1-B\left(t\right))dt<\infty ,\underset{0}{\overset{\infty }{\int }}(1-G\left(t\right))dt<\infty$$
• After any system repair and its PM completion, the system becomes like new, i.e., returns to the zero state.

Our aim was to present a method for comparing different PM strategies $l\in \mathcal{L}$ (including running to system failure for $l=k$) and choosing the best one with respect to system availability factor $K_{\mathrm{av}.,l}$,

$$K_{\mathrm{av}.,l}=\underset{t\to \infty }{lim}\frac{1}{t}\left\{\mathrm{the}\mathrm{system}\mathrm{working}\mathrm{time}\mathrm{during}\mathrm{time}t\mathrm{under}\mathrm{strategy}l\right\}.$$

(13)

For this purpose, we defined a random process $J=\left\{J\right(t):t\ge 0\}$ with the set of space E as follows:

$$J\left(t\right)=j,\mathrm{if}\mathrm{at}\mathrm{time}t\mathrm{system}\mathrm{is}\mathrm{in}\mathrm{the}\mathrm{state}j\in E.$$

(14)

J is used instead of X, as in Section 2, since its set of states E is finite. Employing this process, we explore the key reliability characteristics of a renewable k-out-of-n system and propose a procedure for comparing different PM strategies as a recommendation for the decision maker (DM). These are:

• System reliability function

  $$R_k\left(t\right)=\mathbf{P}\{F_1^{\left(k\right)}>t\}\mathrm{and}\mathrm{its}\mathrm{mean}\mathrm{value}{M}_k=\underset{0}{\overset{\infty }{\int }}{R}_k\left(t\right)dt.$$

  (15)
• The distributions of different PM start times and their mean values

  $$F_l\left(t\right)=\mathbf{P}\{F_1^{\left(l\right)}\le t\},M_l=\underset{0}{\overset{\infty }{\int }}(1-F_l\left(t\right))dt.$$

  (16)
• System availability $K_{\mathrm{av}.,l}$, given by (13), for different PM strategies $l\in \mathcal{L}$.
• The calculation algorithm for the preference indicator of PM strategies is given below.

Statistics on the lifetime of any component within a complex engineering system usually include mean values, and information on the second moments is rarely available, so, the sensitivity of the choice of PM strategies to the initial information regarding the shape of the lifetime distribution and its parameters deserves attention.

4. J Process and Strategy Comparison

4.1. J Process and the Ergodic Theorem

For any PM strategy $l\in \mathcal{L}$, the J process defined by (14) is regenerative; Figure 1 demonstrates its trajectory. Its regenerative epochs $S_n^{\left(l\right)}(n\in N,S_0^{\left(l\right)}=0$ for any $l\in \mathcal{L})$ are the PM and system repair completion times. Regeneration periods $T_n^{\left(l\right)}=S_n^{\left(l\right)}-S_{n-1}^{\left(l\right)}$ of the J process are composed of two terms: the system lifetimes (times to PM beginning for $l<k$ or time to system failure for $l=k$) $F_n^{\left(l\right)}$ and the system PM or repair times $G_n^{\left(l\right)}:T_n^{\left(l\right)}=F_n^{\left(l\right)}+G_n^{\left(l\right)}$ (see Figure 1). $F_l\left(t\right)=\mathbf{P}\{F_n^{\left(l\right)}\le t\}$ and ${\Gamma }_l\left(t\right)=\mathbf{P}\{T_n^{\left(l\right)}\le t\}$ denote the common distribution function of rvs $F_n^{\left(l\right)}$ and $T_n^{\left(l\right)}(n\in N)$, respectively.

Remark 2. 

Further, considering an embedded semi-regenerative process, we also added the number of embedded process ranks to the upper index of the strategy type. Therefore, $S_n^{(l,1)}$ is the n-th embedded semi-regeneration point of the ESRPr of rank 1 under strategy $l\in \mathcal{L}$.

For any controllable aspect in the regenerative process of regeneration epochs, the following ergodic theorem holds.

Theorem 1. 

For any admissible strategy $l\in \mathcal{L}$ of a controllable (in regeneration epochs) regenerative process $J=\left\{J\right(t),t\ge 0\}$ with finite expectation of its RP $\mathbf{E}\left[T_n^{\left(l\right)}\right]<\infty$ and any integrable function g, defined on the process set of states E, the following limit property holds:

$$\underset{t\to \infty }{lim}\frac{1}{t}\underset{0}{\overset{t}{\int }}g\left(J\left(u\right)\right)du=\frac{1}{{\mathbf{E}}_0\left[T_1^{\left(l\right)}\right]}{\mathbf{E}}_0\left[\underset{0}{\overset{T_1^{\left(l\right)}}{\int }}g\left(J\left(u\right)\right)du\right],$$

(17)

where ${\mathbf{E}}_0$ means the expectation provided that the initial state is equal to zero $i=0$.

The proof, which we omit here, can be found in [36], where the law of large numbers for the sum of a random number of random variables had to be used [48].

Theorem 1 shows that, to compare different strategies $l\in \mathcal{L}$, it is only necessary to study the behavior of the process in a particular regeneration period, $T_n^{\left(l\right)}=S_n^{\left(l\right)}-S_{n-1}^{\left(l\right)}$ (suppose the first one, $T_1^{\left(l\right)}=S_1^{\left(l\right)}$).

4.2. Strategy Comparison

To compare the different PM strategies with respect to their availability factors, let us consider function g as an indicator function of the set of states $E_l=\{1,\dots ,l\},g\left(j\right)=1_{E_l}\left(j\right)$. Then, the last integral in Formula (17) takes the following value:

$$\underset{0}{\overset{S_1^{\left(l\right)}}{\int }}{1}_{E_l}\left(J\left(u\right)\right)du=inf\{t:J\left(t\right)=l,t\le S_1^{\left(l\right)}\}\equiv F_1^{\left(l\right)}.$$

The above equals to the time to the first l-th type PM beginning (the time to the l-th type PM beginning in a separate regeneration period, time to the system lifetime for $l=k$). Its mean value is denoted by $f_l=\mathbf{E}\left[F_1^{\left(l\right)}\right].$ To further simplify the notations, we omitted the subscript of the number of the regeneration period, and present the index of the strategy number. Then, the system availability for different PM strategies $l\in \mathcal{L}$ becomes

$$K_{\mathrm{av}.,l}=\frac{f_l}{\mathbf{E}\left[T_1^{\left(l\right)}\right]}.$$

(18)

Therefore, due to the properties of regenerative processes, to calculate availability $K_{\mathrm{av}.,l}$, we only need to find the mean value $\mathbf{E}\left[T_1^{\left(l\right)}\right]$ of any (say the first one) regeneration period $T_1^{\left(l\right)}$ and the mean value $f_l=\mathbf{E}\left[F_1\left(l\right)\right]$ of the working time $F_1\left(l\right)$ in it. Since, for any PM strategy $l\in \mathcal{L}$, the regeneration period equals $T_1^{\left(l\right)}=F_1^{\left(l\right)}+G_1^{\left(l\right)},$ and the mean repair and PM times $g_l=\mathbf{E}\left[G_1^{\left(l\right)}\right]$ presumably must be known to a decision maker (DM), to solve the problem, it suffices to calculate the distributions (or their mean values only) of the system operation times $F_1^{\left(l\right)}$ for the case when it runs to failure (for $l=k$) and for the system operating under the l-th PM strategy. Hence, from Theorem 1:

Theorem 2. 

The l-th strategy is preferable to the j-th one ($l\succ j$) iff

$$\frac{g_l}{g_j}<\frac{f_l}{f_j}.$$

(19)

Proof. 

A comparison of different PM strategies in terms of the system availability (18) maximization criterion results in a comparison of expressions:

$$K_{\mathrm{av}.,l}=\frac{f_l}{f_l+g_l}$$

Since the l-th preventive maintenance strategy is superior to the j-th one iff $K_{\mathrm{av}.,l}>K_{\mathrm{av}.,j}$, it follows from inequality

$$\frac{f_l}{f_l+g_l}>\frac{f_j}{f_j+g_j},$$

that the l-th strategy is preferable to the j-th one ($l\succ j$) if and only if $f_l{g}_j>f_j{g}_l$, which, after algebraic rearrangements, can be written as (19). This ends the proof. □

The above discussion shows that, for the problem solution, we need to study the distribution or only the mean working time $M_l$ of the system for any strategy $l\in \mathcal{L}$ during regeneration period $T_1^{\left(l\right)}.$

5. Calculation of Mean Time to the PM Beginning

Since the process behavior within a separate regeneration period $T_1^{\left(l\right)}$ is rather complicated, for its calculation, process J during this time for any strategy PM $l\in \mathcal{L}$ should be considered as an ESRP.

5.1. Embedded Semi-Regenerative Process and Calculation of Its Characteristics

Thus, for any $l\in \mathcal{L}$, consider an ESRP $J_n^{\left(1\right)}=\{J_n^{\left(1\right)}\left(t\right):t\ge 0\}$ (see Figure 1) with

$$J_n^{\left(1\right)}\left(t\right)=J(S_{n-1}^{\left(l\right)}+t),t\le T_n^{\left(l\right)}.$$

Its semi-regeneration times $S_n^{(l,1)}$ of type j are the completion times of the repair. When the system passes to state $j\le l-2,J(S_n^{(l,1)}+0)=j$, semi-regeneration periods are $T_n^{(l,1)}=S_n^{(l,1)}-S_{n-1}^{(l,1)}$; since it is impossible to enter state $l-1$ from state l after repair completion, the set of embedded regeneration states is $E_1=\{j:(j=\overline{0,l-2})\}$. According to the above results, it is sufficient to only study process $J_n^{\left(1\right)}$ at a separate regeneration period, for example, the first one $J_1^{\left(1\right)}\equiv J^{\left(1\right)}$. In order to study the behavior of process $J^{\left(1\right)}$, we introduce the following notations where we omit dependence on symbol strategy l for simplicity.

• $Q^{\left(1\right)}\left(t\right)={\left[Q_{ij}^{\left(1\right)}\left(t\right)\right]}_{ij\in E_1}$ is the embedded semi-Markov matrix (ESMM) of which the components are the process transition probabilities between semi-regeneration times:

  $$Q_{ij}^{\left(1\right)}\left(t\right)=\mathbf{P}\{J^{\left(1\right)}(S_n^{(l,1)}+0)=j,T_n^{(l,1)}\le t|J^{\left(1\right)}(S_{n-1}^{(l,1)}+0)=i\}.$$
• ${\overrightarrow{Q}}^{\left(1\right)}\left(t\right)=\left[Q_{il}^{\left(1\right)}\left(t\right)\right]$ is the vector function of which the components are the CDFs of the first passage time from state i to absorbing state l by the ESRPr along a monotone trajectory.
• $\overrightarrow{F}\left(t\right)=\left[F_{il}\left(t\right)\right]$ is the vector function of which the components are the CDFs of the destination time of absorbing state l via the ESRPr, starting from state $i(i=\overline{0,l-2})$.
• $H^{\left(1\right)}\left(t\right)={\left[H_{ij}^{\left(1\right)}\left(t\right)\right]}_{ij\in E_1}$ is the embedded Markov renewal matrix whose components are the conditional embedded renewal functions in the separate lifetime period:

  $$H_{ij}^{\left(1\right)}\left(t\right)=\mathbf{E}\left[\sum _{n\ge 1}{1}_{\{S_n^{(l,1)}\le t\le T_1^{\left(l\right)},J_n^{\left(\right(1)}(S_n^{(l,1)}+0)=j\}}|J_n^{\left(1\right)}\left(0\right)=i\right].$$

First, we calculate ESMM $Q^{\left(1\right)}\left(t\right)={\left[Q_{ij}^{\left(1\right)}\left(t\right)\right]}_{ij\in E_1}$ of the ESRPr $J^{\left(1\right)}$ and vector function ${\overrightarrow{Q}}^{\left(1\right)}\left(t\right)=\left[Q_{il}^{\left(1\right)}\left(t\right)\right]$. To calculate matrix $H_{ij}^{\left(1\right)}\left(t\right)$, we denote with $p_{ij}\left(t\right)$ the probability that the non-repairable k-out-of-n system in time t passes from state i to state j, i.e., in time t, $j-i$ components of the system out of $n-i$, which are running at the beginning of the interval, fail. $P_{il}\left(t\right)$ denotes the probability that, starting from state i, the system reaches state l in time t, i.e., the probability that at least $l-i$ system components out of $n-i$, operating at the beginning of the time interval, fail during time t. Due to the assumptions about the components of the system failures, these probabilities are calculated as follows:

$$p_{ij}\left(t\right)=\left(\genfrac{}{}{0pt}{}{n-i}{j-i}\right){(1-e^{-\alpha t})}^{j-i}{e}^{-(n-j)\alpha t},$$

and:

$$P_{il}\left(t\right)=\sum _{j\ge l}{p}_{ij}\left(t\right)=1-\sum _{i\le j\le l-1}{p}_{ij}\left(t\right).$$

$\Pi \left(t\right)$ also denotes a Poisson process with set of states $E=\{0,1\dots ,k\}$ and variable intensity ${\lambda }_i=(n-i)\alpha$, and by ${\Lambda }_{ij}$ the time of its transition from the state i to the state j. Thus, it is possible to represent the probabilities $p_{ij}\left(t\right)$ as follows:

$$p_{ij}\left(t\right)=\mathbf{P}\{{\Lambda }_{ij}\le t<{\Lambda }_{ij+1}\}$$

Moreover, probability $P_{il}\left(t\right)=\mathbf{P}\{{\Lambda }_{il}\le t\}$ is the time that a non-repairable k-out-of-n system needs to reach state l from state i in time t.

To further simplify the calculations, we present these probabilities according to Newton’s binomial formula as a sum. With substitution ${\lambda }_i=(n-i),\alpha$ we have:

$$p_{ij}\left(t\right)=\left(\genfrac{}{}{0pt}{}{n-i}{j-i}\right)\sum _{m=0}^{j-i}{(-1)}^m\left(\genfrac{}{}{0pt}{}{j-i}{m}\right)e^{-{\lambda }_{j-m}t},$$

(20)

and:

$$P_{il}\left(t\right)=\sum _{j\ge l}\left(\genfrac{}{}{0pt}{}{n-i}{j-i}\right)\sum _{m=0}^{j-i}{(-1)}^m\left(\genfrac{}{}{0pt}{}{j-i}{m}\right)e^{-{\lambda }_{j-m}t}.$$

(21)

Using these notations for the differentials of the ESMM, the following lemma holds.

Theorem 3. 

1. The ESMMs $Q^{\left(1\right)}\left(t\right)$ of process $J^{\left(1\right)}$ are:

$$\begin{array}{ccc}\hfill Q_{0j}^{\left(1\right)}\left(t\right)& =& \underset{0}{\overset{t}{\int }}{\lambda }_0{e}^{-{\lambda }_{0}u}du\underset{0}{\overset{t-u}{\int }}{p}_{1j+1}\left(v\right)dB\left(v\right),j=\overline{0,l-2};\hfill \\ \hfill Q_{ij}^{\left(1\right)}\left(t\right)& =& \underset{0}{\overset{t}{\int }}{p}_{ij+1}\left(v\right)dB\left(v\right),(i=\overline{1,l-2},j=\overline{i-1,l-2}\hfill \end{array}$$

(22)

The components $Q_{il}^{\left(1\right)}\left(t\right)$ of vector ${\overrightarrow{Q}}^{\left(1\right)}\left(t\right)$ are:

$$\begin{array}{ccc}\hfill Q_{0l}^{\left(1\right)}\left(t\right)& =& \underset{0}{\overset{t}{\int }}{\lambda }_0{e}^{-{\lambda }_{0}u}du\underset{0}{\overset{t-u}{\int }}{P}_{1l}\left(v\right)dB\left(v\right);\hfill \\ \hfill Q_{il}^{\left(1\right)}\left(t\right)& =& \underset{0}{\overset{t}{\int }}{P}_{il}\left(v\right)dB\left(v\right),(i=\overline{1,l-2}).\hfill \end{array}$$

(23)

2. Appropriate Laplace–Stiltjes transform (LST) ${\tilde{q}}_{ij}^{\left(1\right)}\left(s\right)={\int }_0^{\infty }{e}^{-st}d{Q}_{ij}^{\left(1\right)}\left(t\right)$ equals:

$$\begin{array}{ccc}\hfill {\tilde{q}}_{0j}^{\left(1\right)}\left(s\right)& =& {\displaystyle \frac{{\lambda }_0}{s+{\lambda }_0}}\left(\genfrac{}{}{0pt}{}{n-1}{j}\right)\sum _{m=0}^j{(-1)}^m\left(\genfrac{}{}{0pt}{}{j}{m}\right)\tilde{b}(s+{\lambda }_{j+1-m}),(j=\overline{0,l-2});\hfill \\ \hfill {\tilde{q}}_{ij}^{\left(1\right)}\left(s\right)& =& \left(\genfrac{}{}{0pt}{}{n-i}{j+1-i}\right)\sum _{m=0}^{j+1-i}{(-1)}^m\left(\genfrac{}{}{0pt}{}{j+1-i}{m}\right)\tilde{b}(s+{\lambda }_{j+1-m}),\hfill \\ & & (i=\overline{1,l-2},j=\overline{i-1,l-2});\hfill \end{array}$$

(24)

and

$$\begin{array}{ccc}\hfill {\tilde{q}}_{0l}^{\left(1\right)}\left(s\right)& =& \frac{{\lambda }_0}{s+{\lambda }_0}\sum _{j\ge l}\left(\genfrac{}{}{0pt}{}{n-1}{j-1}\right)\sum _{m=0}^{j-1}{(-1)}^m\left(\genfrac{}{}{0pt}{}{j-1}{m}\right)\tilde{b}(s+{\lambda }_{j-m}),\hfill \\ \hfill {\tilde{q}}_{il}^{\left(1\right)}\left(s\right)& =& \sum _{j\ge l}\left(\genfrac{}{}{0pt}{}{n-i}{j-i}\right)\sum _{m=0}^{j-i}{(-1)}^m\left(\genfrac{}{}{0pt}{}{j-i}{m}\right)\tilde{b}(s+{\lambda }_{j-m}),(i=\overline{1,l-2});\hfill \end{array}$$

(25)

Proof. 

For the system to be in state $j\le l-2$ at some regeneration epoch when in state 0 in the previous regeneration epoch, it is necessary and sufficient that, at some time u and before time t, one of the system components fails with probability ${\lambda }_0{e}^{-{\lambda }_{0}u}du$. Then, during the remaining time $t-u$, other j components fail in some time v with probability $p_{1j+1}\left(v\right)$ before the originally failed component is repaired in time point v with probability $dB\left(v\right)$.

$$\begin{array}{ccc}\hfill Q_{0j}^{\left(1\right)}\left(t\right)& =& \mathbf{P}\{J^{\left(1\right)}(S_n^{(l,1)}+0)=j,T_n^{(l,1)}\le t|J^{\left(1\right)}(S_{n-1}^{(l,1)}+0)=0\}\hfill \\ & =& \underset{0}{\overset{t}{\int }}{\lambda }_0{e}^{-{\lambda }_{0}u}du\underset{0}{\overset{t-u}{\int }}\mathbf{P}\{{\Lambda }_{1j}\le B<{\Lambda }_{1j+1},B\in dv\}\hfill \\ & =& \underset{0}{\overset{t}{\int }}{\lambda }_0{e}^{-{\lambda }_{0}u}du\underset{0}{\overset{t-u}{\int }}{p}_{1j+1}\left(v\right)dB\left(v\right).\hfill \end{array}$$

Similarly, for the system to be in state j at some regeneration epoch while having been in state i at the previous epoch, it is necessary and sufficient that the repair process of the component being repaired is completed at some time v before t with probability $dB\left(v\right)$, and the other $j+1-i$ components fail during time v with probability $p_{ij+1}\left(v\right)$.

$$\begin{array}{ccc}\hfill Q_{ij}^{\left(1\right)}\left(t\right)& =& \mathbf{P}\{J^{\left(1\right)}(S_n^{(l,1)}+0)=j,T_n^{(l,1)}\le t|J^{\left(1\right)}(S_{n-1}^{(l,1)}+0)=i\}\hfill \\ & =& \underset{0}{\overset{t}{\int }}\mathbf{P}\{{\Lambda }_{ij}\le B<{\Lambda }_{ij+1},B\in dv\}=\underset{0}{\overset{t}{\int }}{p}_{1j+1}\left(v\right)dB\left(v\right).\hfill \end{array}$$

The proof of Expression (24) follows from the transition to the LST in Formula (22) using Expression (20):

$$\begin{array}{ccc}\hfill {\tilde{q}}_{0j}^{\left(1\right)}\left(s\right)& =& \underset{0}{\overset{\infty }{\int }}{e}^{-st}d{Q}_{0j}^{\left(1\right)}\left(t\right)=\underset{0}{\overset{\infty }{\int }}{e}^{-st}\left[\underset{0}{\overset{t}{\int }}{\lambda }_0{e}^{-{\lambda }_{0}u}du\right]p_{1j+1}(t-u)d_{t}B(t-u)\hfill \\ & =& \underset{0}{\overset{\infty }{\int }}{e}^{-s(t-u)}du\left[\underset{0}{\overset{t}{\int }}{\lambda }_0{e}^{-(s+{\lambda }_0)u}\right]p_{1j+1}(t-u)d_{t}B(t-u)\hfill \\ & =& \underset{0}{\overset{\infty }{\int }}{\lambda }_0{e}^{-(s+{\lambda }_0)u}du\underset{0}{\overset{\infty }{\int }}{p}_{1j+1}\left(v\right)dB\left(v\right)\hfill \\ & =& {\displaystyle \frac{{\lambda }_0}{s+{\lambda }_0}}\left(\genfrac{}{}{0pt}{}{n-1}{j}\right)\sum _{m=0}^j{(-1)}^m\left(\genfrac{}{}{0pt}{}{j}{m}\right)\tilde{b}(s+{\lambda }_{j+1-m});\hfill \end{array}$$

Analogously, for $i\ne 0$, it holds

$$\begin{array}{ccc}\hfill {\tilde{q}}_{ij}^{\left(1\right)}\left(s\right)& =& {\int }_0^{\infty }{e}^{-st}d{Q}_{ij}^{\left(1\right)}\left(t\right)={\int }_0^{\infty }{e}^{-st}{p}_{ij+1}\left(t\right)dB\left(t\right)\hfill \\ & =& \left(\genfrac{}{}{0pt}{}{n-i}{j+1-i}\right)\sum _{m=0}^{j+1-i}{(-1)}^m\left(\genfrac{}{}{0pt}{}{j+1-i}{m}\right)\tilde{b}(s+{\lambda }_{j+1-m}).\hfill \end{array}$$

Similarly, for components ${\tilde{q}}_{il}^{\left(1\right)}\left(s\right)$ of vector ${\tilde{\overrightarrow{q}}}^{\left(1\right)}\left(s\right)$, after some calculations and the substitution of Expression (21), it holds:

$$\begin{array}{ccc}\hfill {\tilde{q}}_{0l}^{\left(1\right)}\left(s\right)& =& \underset{0}{\overset{\infty }{\int }}{e}^{-st}d{Q}_{0l}^{\left(1\right)}\left(t\right)=\underset{0}{\overset{\infty }{\int }}{e}^{-st}\left[\underset{0}{\overset{t}{\int }}{\lambda }_0{e}^{-{\lambda }_{0}u}du\right]P_{1l}(t-u)d_{t}B(t-u)\hfill \\ & =& \underset{0}{\overset{\infty }{\int }}{\lambda }_0{e}^{-(s+{\lambda }_0)u}du\left[\underset{t\ge u}{\int }{e}^{-s(t-u)}{P}_{1l}(t-u)d_{t}B(t-u))\right]\hfill \\ & =& \frac{{\lambda }_0}{s+{\lambda }_0}\sum _{j\ge l}\left(\genfrac{}{}{0pt}{}{n-1}{j-1}\right)\sum _{m=0}^{j-1}{(-1)}^m\left(\genfrac{}{}{0pt}{}{j-1}{m}\right)\tilde{b}(s+{\lambda }_{j-m}),\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {\tilde{q}}_{il}^{\left(1\right)}\left(s\right)& =& {\int }_0^{\infty }{e}^{-st}d{Q}_{il}^{\left(1\right)}\left(t\right)={\int }_0^{\infty }{e}^{-st}{P}_{il}\left(t\right)dB\left(t\right)\hfill \\ & =& \sum _{j\ge l}\left(\genfrac{}{}{0pt}{}{n-i}{j-i}\right)\sum _{m=0}^{j-i}{(-1)}^m\left(\genfrac{}{}{0pt}{}{j-i}{m}\right){\int }_0^{\infty }{e}^{-(s+{\lambda }_{j-m})t}dB\left(t\right)\hfill \\ & =& \sum _{j\ge l}\left(\genfrac{}{}{0pt}{}{n-i}{j-i}\right)\sum _{m=0}^{j-i}{(-1)}^m\left(\genfrac{}{}{0pt}{}{j-i}{m}\right)\tilde{b}(s+{\lambda }_{j-m})\hfill \end{array}$$

That ends the proof. □

5.2. Mean Time to the PM Beginning Calculation

For vector column $\overrightarrow{F}\left(t\right)$, the following representation holds.

Lemma 1. 

Column vector function $\overrightarrow{F}\left(t\right)$ satisfies the following equation:

$$\overrightarrow{F}\left(t\right)={\overrightarrow{Q}}^{\left(1\right)}\left(t\right)+Q^{\left(1\right)}★\overrightarrow{F}\left(t\right),$$

(26)

whose unique solution in terms of its LST is:

$$\tilde{\overrightarrow{f}}\left(s\right)={(I-{\tilde{q}}^{\left(1\right)}\left(s\right))}^{-1}{\tilde{\overrightarrow{q}}}^{\left(1\right)}\left(s\right).$$

(27)

Proof. 

Equation (26) follows from Formula (11) if we use in it the set of absorption states $E_1$ instead of any state $\Gamma$. Absorption in state l during the regeneration period from any state i can occur either along a monotonous trajectory (first term in this formula) or after a transition from it to some other state j with a probability $d{Q}_{ij}^{\left(1\right)}\left(u\right)$ during time u followed by absorption from the last one. The proof the next formula follows directly via the transition to LST in Formula (26) and the remark that the matrix $(I-{\tilde{q}}^{\left(1\right)}\left(s\right))$ is not degenerate. □

Remark 3. 

The first component $F_{0l}\left(t\right)$ of vector $\overrightarrow{F}\left(t\right)$ constitutes the CDF of the time to the PM beginning (time to the system failure for $l=k$) for the l-th strategy. Its moment generating function (MGF) is the first component ${\tilde{f}}_{0l}\left(s\right)$ of vector $\tilde{\overrightarrow{f}}\left(s\right)$.

Lemma 1 allows calculating the LST of the time to reach any state, and via inverse transformation, the corresponding CDF. However, in order to calculate the average time to reach the absorbing state from zero, which is required to solve the stated problem, one could only directly calculate the corresponding average values.

• $r_{il}={\tilde{f}}_{il}\left(0\right)$ denotes the probability of absorption during the regeneration period from state $i,\overrightarrow{r}=(r_{0l}\dots r_{l-2,l})$.
• $f_{il}=-{\tilde{f}}_{il}^{\prime }\left(0\right)$ is the mean time to absorption during the regeneration period from state $i,\overrightarrow{f}=(f_{0l},\dots f_{l-2,l})$.
• $m_{il}=-{\tilde{q}}_{il}^{\prime }\left(0\right)$ is the mean time to absorption during repair time B from state i along monotone trajectory $\overrightarrow{m}=(m_{0l}\dots m_{l-2,l})$.
• ${\pi }_{il}={\tilde{q}}_{il}^{\left(1\right)}\left(0\right)$ denotes the probability of absorption from the state i along a monotone trajectory during repair time B, $\overrightarrow{\pi }=({\pi }_{0l}\dots {\pi }_{l-2,l})$.
• $q_{ij}={\tilde{q}}_{ij}^{\left(1\right)}\left(0\right)$ denotes the probability to transition from state i to state j for ESRPr during repair time B, $Q={\left[q_{ij}\right]}_{i,j\in E_1}$.
• $m_{ij}=-\frac{d}{ds}{\tilde{q}}_{ij}^{\left(1\right)}\left(0\right)$ is the mean time of transition from state i to state j during repair time B for ESRPr, $M={\left[m_{ij}\right]}_{i,j\in E_1}$.

Theorem 4. 

Mean time $f_l$ to the PM beginning under strategy l (mean time to system repair for $l=k$) is the first component $f_l=f_{0l}$ of vector ${\overrightarrow{f}}_l$

$$\overrightarrow{f}={(I-Q)}^{-1}(\overrightarrow{m}+M{(I-Q)}^{-1}\overrightarrow{\pi })={(I-Q)}^{-1}\overrightarrow{v},$$

(28)

where $\overrightarrow{v}=\overrightarrow{m}+M{(I-Q)}^{-1}\overrightarrow{\pi }$.

Proof. 

System (26), in terms of LST, has the following form:

$$\tilde{\overrightarrow{f}}\left(s\right)={\tilde{\overrightarrow{q}}}^{\left(1\right)}\left(s\right)+{\tilde{q}}^{\left(1\right)}\left(s\right)\tilde{\overrightarrow{f}}\left(s\right).$$

(29)

By substituting the value $s=0$ into this equation, we obtain the following equation:

$$\begin{array}{ccc}\hfill \overrightarrow{r}& =& \overrightarrow{\pi }+Q\overrightarrow{r}.\hfill \end{array}$$

(30)

By differentiating Equation (29) with respect to s at point $s=0$, and using the above-introduced notations, we find the following equation:

$$\begin{array}{ccc}\hfill \overrightarrow{f}& =& \overrightarrow{m}+M\overrightarrow{r}+Q\overrightarrow{f}.\hfill \end{array}$$

(31)

Thus, due to the non-degeneration of matrix Q, via the solution of Equation (30) and substituting it into Equation (31), we can find its solution in the form of (28). □

Algorithm 1 indicates the comparison of PM strategies.

Algorithm 1 PM strategy comparison.

Beginning. Determine integers: $n,k$. Real: $\alpha ,b,g_l(l=\overline{1,k})$. Functions: distributions $B_l\left(x\right)(l=\overline{1,k}).$

Work.

Step 1. Calculate the parameters of distributions B.

Step 2. In accordance to the above notations, calculate the components

$m_{il},{\pi }_{il},q_{ij},m_{ij}(i,j=\overline{0,l-2})$

of vectors $\overrightarrow{m},\overrightarrow{\pi }$ and matrices $Q,M$.

Step 3. By solving Equations (30) and (31), calculate components $r_{il}$ and $f_{il}$ of vectors $\overrightarrow{r}$ and $\overrightarrow{f}$.

Step 4. Considering the first component $f_{0l}$ of vector $\overrightarrow{f}$ as $f_l,f_l=f_{0l}$ compare the different

strategies $l\in \mathcal{L}$ according to Formula (19) and choose the best one.

Step 5. Propose the best strategy to the DM.

Stop.

Remark 4. 

The algorithm also allows for investigating the sensitivity of the decision making to the distribution of components’ repair times $B\left(t\right)$ and their parameters.

6. Numerical Study

The algorithm was implemented in a MATLAB environment. We considered two variants of the 4-out-of-6 and 4-out-of-8 systems. The mean failure time of each system component was supposed to be exponentially distributed with mean value ${\alpha }^{-1}=1$ (this means that we considered it a time-scale unit). We assumed that the mean repair time for each system component b could take values in the interval of [0.05 0.3]. We investigated the performance of algorithm for the exponential, gamma, Gnedenko–Weibull (GW), and lognormal distributions of repair time B for each system component.

Figure 2 shows a comparison of the PM strategies for systems 4-out-of-6 (left) and 4-out-of-8 (right).The surfaces were plotted for the gamma distribution of the repair time, and demonstrate the $f_3/f_4$ ratio calculated for different values of the mean repair time for each system component b and the coefficient of variation c. The red line shows the case of the exponential distribution. The plane was drawn at the level corresponding to the $g_3/g_4=0.6$ ratio, where $g_3$ is the duration of system PM after 3 failures, and $g_4$ is the duration of system recovery after 4 failures. According to (19), for an area where the surfaces are higher than the plane, the strategy of conducting the PM after 3 failures is better than working up to 4 failures followed by system recovery. Figure 2 shows that the strategy to conduct a PM after 3 failures had an advantage with large coefficients of variation and with large mean repair time b. The area where the strategy of performing PM after 3 failures was preferable was wider in the case of the 4-out-of-8 system. For the exponential distribution with the $g_3/g_4=0.6$ ratio, the repair strategy after 4 failures is preferred for any value of the mean repair time $b<0.3$ for 4-out-of-6 system, and for the mean repair time $b<0.2$ for the 4-out-of-8 system.

Figure 2 and Figure 3 show that, for large values of variation and a longer mean repair time b, the PM strategy after 3 failures had an advantage. For the lognormal distribution, the area of preference for PM after 3 failures was smaller than that in the case of gamma distribution.

Figure 4 presents the surface-level lines for different values of the $g_3/g_4$ ratio. Red circles correspond to the exponential distribution. Solid lines present gamma distribution, dashed lines demonstrate the lognormal distribution, and dotted lines correspond to GW distribution. For variation coefficients $c<1$, gamma and GW distributions practically coincided when $b<0.2$. A difference can be seen for the red ($g_3/g_4=0.6$) and magenta ($g_3/g_4=0.5$) level lines.The lognormal distribution differed from the other two distributions when the coefficient of variation $c>0.5$ or $b>0.1$. The difference increased as the variation coefficient or mean repair time increased.

7. Conclusions and Further Investigations

In this paper, a brief review of the theory of decomposable semi-regenerative processes was presented. The methods of DSRPr not only represent a significant contribution to the study of any particular system, but also expand and enrich the methods for studying stochastic models, which is a novelty of this paper. This theory was used to investigate the PM strategies of a renewable k-out-of-n system. We proposed a new problem statement for comparing PM strategies for a renewable k-out-of-n system and choosing the best one with respect to the system availability factor. The DSRPr approach was used for a new analytical solution of the stated problem. The calculation procedure and a new algorithm for implementing the proposed analytical solution were presented. The problem solution provides information for the decision maker on whether to initiate preventive maintenance to maximize system availability.

Moreover, the proposed approach opens up the possibility to analyze decision-making sensitivity to the shape of system components’ repair time distribution. As a future investigation, this approach could be applied to a wide class of the engineering applications of stochastic systems.