Stochastic modeling and cost-benefit evaluation of consecutive k/n: F repairable retrial systems with two-phase repair and vacation

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Highlights

  • Two-phase repair and vacation are considered in consecutive k/n: F retrial system.

  • Phase-type distribution governs the repair time of failed components in both phases.

  • The state set analysis method is used to analyze the state probability of the system.

  • Reliability indexes and cost-benefit ratio of the system are evaluated.

  • System models with and without retrial or vacation are compared numerically.

Abstract

In this paper, a reliability model of consecutive k/n: F linear retrial system with two-phase repair and Bernoulli vacation is investigated, where failure may occur to all non-failed components even though the system has failed. The failed component joins retrial space and attempts repeatedly for repair if the repairman is not idle. The repairman provides two-phase repair for each component that has failed, namely the first essential repair and the second optional repair. After completing the repair of a failed component, the repairman may leave for a vacation at a probability p. The developed system model is investigated as a Markov process. Various reliability indexes are derived by means of matrix analysis, state set analysis method and the technique of solving differential equations. Numerical experiments are executed to discuss how performance indexes vary as system parameters. Moreover, the system models with and without retrial or vacation in terms of cost-benefit ratio are compared numerically. A relevant potential application is provided to illustrate the proposed model as well.

Introduction

The reliability design and evaluation of the system, which is a hot issue in practical engineering problems, has attracted considerable attention in the research. As one of the means to improve system reliability, k/n redundancy design has been applied in the recent work of Gao and Wang (2021), Singh et al. (2021) and Singh and Poonia (2022). The consecutive k/n: F (G) system (Chiang and Niu, 1981, Wu and Guan, 2005, Yusuf et al., 2016), as a typical form of redundancy, is extensively encountered in the reliability modeling and evaluation of spatially distributed systems, for instance telecommunications systems, vacuum systems in electronic accelerators and oil pipeline systems.

Several scholars have contributed significantly to the consecutive k/n: F system models in terms of reliability analysis. Lambiris and Papastavridis (1985) derived the accurate formulas of reliability for consecutive k/n: F systems including n linearly or circularly arranged components. Peng and Xiao (2018) provided a nonrecursive closed-form expression of mean time to failure and dynamic survival function for the consecutive k/n: F systems containing three different kinds of components. At present, various extension works (Wang et al., 2021b, Wu et al., 2021, Yin and Cui, 2021) on non-repairable consecutive k/n: F systems and other related systems have also been carried out. Cheng and Zhang (2001) and Yam et al. (2003) utilized the concept of generalized transition probability to analyze the repairable consecutive k/n: F system. There are numerous more works available on such repairable systems, as detailed in Villén-Altamirano (2010), Yuan and Cui (2013) and Gurcan and Gokdere (2018). Most of the previous studies only provide one-phase repair for failed components, which does not address all practical situations often arising in engineering.

The second optional repair, as a special case of two-phase repair, has wide applications in manufacturing systems, communication engineering and service industries. In some practical systems, the repaired components may start up operation in one of two ways: (1) To deal with emergencies and reduce the risk of system operation, the failed component will start up operation directly after completion of a main body repair; (2) To reach the highest standards required by the industry, the failed component will be further externally mended and decorated after the main body repair completion. In this case, the two repair phases are the first essential repair and the second optional repair (Madan, 2000), respectively. Most of the previous studies (called second optional service in queuing theory) mainly provided queueing performance indexes (Chakravarthy and Parthasarathy, 1989, Laxmi and Jyothsna, 2020). In aspect of reliability point of view, El-Said and El-Sherbeny (2010) conducted the research work of two-phase repair in the two-unit standby system. Assumed in Gupta et al. (2014), the repair of each failed unit is accomplished in two stages. Gao (2021) introduced two-phase repair of failed components into the retrial system with warm standbys, where the second phase is optional.

For system reliability models, retrial is characterized by the feature that an arriving failed component enters a virtual retrial space when the repairman is not available. After a random period of time, the failed component in retrial space makes repeated attempts for repair. Krishnamoorthy and Ushakumari (1999) initially incorporated retrial into the k/n repairable system and reliability theory. Some studies have found that working vacation (Yang & Tsao, 2019) and Bernoulli vacation (Liu et al., 2021) were combined with retrial system, respectively. Chen and Wang (2018) and Yen et al. (2020) took into consideration the N-policy and F-policy for machine repair problems with retrial, respectively. More recently, a comparative analysis among four retrial systems with imperfect coverage (Yen & Wang, 2020), preventive maintenance and service breakdowns (Wang et al., 2022) in terms of cost/benefit was provided. A circular consecutive k/n: F retrial system model was proposed by Li et al. (2021), where the model may be employed to evaluate the reliability of intelligent repairable systems with limited maintenance resources.

In many real-world repairable systems, managers often assign repairmen to perform some other work. Keeping this fact in view, the research work involving repairman vacation has been conducted by Zhang et al. (2017) and Wang et al. (2021a). However, the case is often met in practice that as soon as a failed component is completed even if there are still components that have failed need for repair, the repairman may choose to take a vacation for some factors. This is known as the Bernoulli vacation schedule, its applications appear naturally in gate control systems, communication systems and computer networks due to the flexibility. Some related literature from queueing theory point of view should be mentioned, details of which can be seen in Choudhury (2008) and Wu and Lian (2013). Recently, Liu et al. (2021) introduced Bernoulli vacation into k/n retrial systems, and evaluated several indexes relevant to system reliability.

From the literature review, various repair policies are often adopted to satisfy different requirements, and the waste of resources is reduced by delegating repairmen to perform additional tasks in some real systems. It follows that the second optional repair policy and Bernoulli vacation schedule are two significant factors that are taken into account in practice. But there has been no research that incorporates these characteristics for consecutive k/n: F retrial systems. This motivates us to investigate the reliability and cost-benefit of consecutive k/n: F linear retrial systems involving Bernoulli vacation and second optional repair. Compared with Cheng and Zhang (2001), Yam et al. (2003) and Li et al. (2021), this study is an expansion from two aspects: repair behavior and vacation mechanism. In addition, the most relevant researches to this paper are Gao (2021) and Liu et al. (2021), but there are some salient features in comparison: (1) this paper investigates a consecutive k/n: F retrial system, while the above literature discusses a k/n: G retrial system with standbys; (2) Gao (2021) focuses on the consideration of second optional repair service for failed components and Liu et al. (2021) mainly concentrates on studying the repairman’s vacation schedule, while two aspects are considered simultaneously in our work; (3) the failure may occur to all non-failed components at a lower failure rate even though the system has failed in this paper, while the feature is not involved in the study of Gao (2021) and Liu et al. (2021); (4) this paper makes the assumption that phase-type (PH) distribution (He, 2014) governs the repair time in both phases, which differs from the exponential distribution repair time in Gao (2021) and Liu et al. (2021). To summarize, the main objectives and contributions are briefly listed as follows:

Two-phase repair for failed components and Bernoulli vacation schedule of the repairman both are embedded into consecutive k/n: F linear retrial system, where non-failed components may still fail when the system fails.

The repair time in both phases is governed by PH distribution, which makes the proposed model have wider application and stronger universality.

The state set analysis method is used to analyze the system state probability. Further, several exact expressions of reliability indexes for this complex system are deduced.

On the one hand, numerical experiments are presented to illustrate how performance indicators vary with key parameters. A numerical comparison of cost-benefit ratio among system reliability models with and without retrial or vacation is presented as well. On the other hand, an application example is displayed to illustrate the developed system model.

The remainder of the work is divided into several sections. The general model is clearly outlined in Section 2, including the model description, system state analysis and system performance indicators analysis. Section 3 provides numerical examples and an application example for the developed system model. Section 4 presents the result discussion and conclusions, as well as possible further research orientation.

Section snippets

PH distribution

The probability distribution of non-negative random variable X is called the m-order PH distribution with denotation (β,R), if and only if it is an absorption time distribution of finite state Markov process. The distribution function is denoted as F(x)=1βexp(Rx)em,x0.

The Markov process has state set 1,2,,m,m+1, in which states 1,2,,m are all non-recurrent and state m+1 is absorbing. The initial probability is (β,0), where β is represented as (β1,,βm). The infinitesimal generator of this

Numerical example

In this section, numerical simulation and analysis on a linear consecutive 2/4: F retrial system considering two-phase repair and Bernoulli vacation are conducted, in which Fig. 1 depicts the state transitions of the system model (taking m=2 as an example).

Firstly, the basic parameters of the system are fixed as follows: λ=0.03,λ=0.02,γ=2,q=0.3,p=0.4,v=1. The 2-order PH distribution governs the repair time of every phase, which is described as (θ,T) and (α,S), respectively, where θ=0.50.5,T=21

Result discussion and conclusions

This paper develops a novel stochastic model for linear consecutive k/n: F retrial system with two-phase repair and vacation schedule. Suppose that the component lifetime, retrial time and vacation time are all distributed exponentially, while the PH distribution governs the repair time of each phase. Based on the divided state sets, the system state transition rate matrix is presented. Several system reliability indexes including transient availability, steady-state availability and state

Acknowledgments

The authors thank the editor and reviewers for their positive and constructive comments and suggestions, which improved and enriched this paper. This work was supported by the National Natural Science Foundation of China [grant number 72071175], the Project of Hebei Key Laboratory of Software Engineering [grant number 22567637H] and the Postgraduate Innovation Project of Hebei Province of China [grant number CXZZSS2022128].

CRediT authorship contribution statement

Yan Wang: Methodology, Software, Writing – original draft, Visualization. Linmin Hu: Conceptualization, Writing – review & editing, Funding acquisition. Bing Zhao: Validation, Writing – review & editing. Ruiling Tian: Validation, Data curation.

References (36)

Cited by (2)

This work was supported by the National Natural Science Foundation of China [grant number 72071175], the Project of Hebei Key Laboratory of Software Engineering, China [grant number 22567637H] and the Postgraduate Innovation Project of Hebei Province of China [grant number CXZZSS2022128].

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