Reliability modelling for linear and circular k-out-of-n: F systems with shared components

https://doi.org/10.1016/j.ress.2021.108172Get rights and content

Highlights

  • Linear and circular k-out-of-n: F systems with shared components are modelled.

  • A method combining the finite Markov chain imbedding approach with some theoretical analysis is proposed.

  • Reliability calculation rules for both two considered systems are described based on our method.

  • Numerical examples are presented to illustrate the efficiency of the proposed reliability evaluation method.

Abstract

Reliability analysis for k-out-of-n systems has received much attention in the field of reliability due to its practical importance. In this work, models for linear and circular k-out-of-n: F systems with shared components, consisting of m subsystems, are discussed. The finite Markov chain imbedding approach cannot be used directly to derive reliability of the considered systems because of the common area existing between adjacent subsystems. For this reason, we first derive the probability that the number of failed components is a given value in the m common areas. Then, all disjoint cases, each of which is regarded as a series system consisting of m subsystems with different initial state probability distributions, are obtained, with which the nith step transition rate matrix is calculated by using the Markov chain technique, for i=1,2,,m. Finally, the reliability of the system is obtained by summing the reliabilities of all cases along with the finite Markov chain imbedding approach. Some numerical examples are presented to illustrate the efficiency of the proposed reliability evaluation method.

Introduction

The k-out-of-n system is a common type of redundancy for improving system reliability since Birnbaum, Esary, and Saunders [1] first discussed it. It is quite useful in describing fault-tolerant systems. In the reliability literature, many authors have studied k-out-of-n systems with many different types of redundancy. Often, k-out-of-n systems consider active redundancy, and a general k-out-of-n system consisting of n identical components is discussed with a binary-state assumption, that is, working and failed. A k-out-of-n: F system ordered in a line or in a circle is in a failure state if and only if at least k of the n components have failed. In the circular structure, the first component is considered to be adjacent to (and follows) the nth component in the system.

Because of great practical importance, the reliability modelling and analysis of k-out-of-n systems have been discussed rather extensively in the reliability literature. Several extensions of k-out-of-n systems have also been considered. For example, Emil et al. [2] defined a generalized multi-performance weighted multi-state k-out-of-n system and studied its reliability evaluation. Bibartiu et al. [3] modelled a scalable k-out-of-n voting gate representation for Bayesian networks, based on the temporal noisy adder. Eryilmaz and Devrim [4] considered the reliability and optimal replacement policy for a k-out-of-n system that is subjected to shocks. Multi-state k-out-of-n systems with different k in different system states, have been studied in [5] with the number of components required in each state being different. Various types of extensions of k-out-of-n systems based on multiple failure criteria have also been considered in [6], [7], linear and circular (n,f,k) systems by Chang et al. [8], and its dual system: n,f,k system by Cui et al. [7].

Recently, some authors have focused on analysis of systems with shared components. Coolen-Maturi et al. [9] introduced a joint survival signature for multiple systems with multiple types of components and with some components being shared by the systems. Considering several coherent systems with some shared components, with independent and identically distributed lifetimes, Zarezadeh et al. [10] made use of joint signature to study the joint reliability of the two coherent systems and specifically obtained a pseudo-mixture representation. Lin et al. [11] studied consecutive-kr-out-of-nr: F linear zigzag structure and circular polygon structure. That is, a system is composed of m subsystems, each of which is a consecutive-kr-out-of-nr: F subsystem (r=1,2,,m), with two adjacent subsystems overlapping with one shared node. Yin and Cui [12] modelled consecutive-k-out-of-n: F systems with shared components between adjacent subsystems, extending the work of Lin et al. [11] by considering two adjacent subsystems overlapping with d shared components (d can be unequal for different adjacent subsystems).

The finite Markov chain imbedding approach (FMCIA) has been used widely in the reliability literature. For example, new system reliability models such as n,f,k systems [7] and sparse-d consecutive-k systems [13] could pose a great computational challenge without FMCIA. FMCIA was introduced originally by Fu and Koutras [14], and developed further by [7], [8], [12], [13], [15], [16]. This method can be used as an excellent tool for system reliability evaluation. The FMCIA is closely related to the problems of runs and scans as can be seen in the books by Balakrishnan and Koutras [17], Fu and Lou [18], and Balakrishnan et al. [19]. A survey paper on the developments and applications of FMCIA in the field of reliability is due to Cui et al. [20].

In this paper, we focus on k-out-of-n: F linear and circular systems, consisting of m subsystems with shared components. Such a k-out-of-n: F linear system is similar to a series-kr-out-of-nr: F system. However, they are different because of the “shared components” presented in the former. Due to the common area existing between adjacent subsystems, we compute the reliabilities of the considered linear and circular systems by using the FMCIA on different disjoint cases and then combining these results.

The new systems studied here are useful in the actual engineering cases such as unmanned aerial vehicle (UAV) system design. UAV is a form of self-powered, radio remote control or autonomous flight, that can perform a variety of tasks and can be used many times [21]. There are many situations in which UAVs carry out such tasks as heavy air transportation, bombing and even passenger transportation. Such UAVs have different performance, function and key technology, and we now detail an actual UAVs performing mission example to demonstrate the use of a linear k-out-of-n: F system with shared components.

Consider UAVs performing a disaster rescue mission by transporting goods such as food and water as the rescue team was unable to get there in time. The mission is divided into three sub-tasks: monitoring, locating, and delivering sub-tasks as shown in Fig. 1. For the monitoring sub-task, UAVs are used to detect where crowds gather in the area; for the locating sub-task, UAVs get used to identify the specific location of crowds accurately when the monitoring sub-task finds them; finally, for the delivering sub-task, UAVs get employed to deliver goods after the monitoring and locating sub-tasks get completed. Of course, the goal of the rescue mission gets achieved if and only if all three sub-tasks get accomplished. Success rates of the monitoring, locating, and delivering tasks get affected by internal instrument accuracy and the external environment such as weather, location, etc. Thus, for example, for the monitoring sub-task with five UAVs, the sub-task fails if and only if at least two of the five UAVs do not find crowds. Some UAVs are multifunctional as they may be involved in both monitoring and locating functions, and some UAVs may be involved in both locating and delivering functions. This type of UAVs can then be set as shared UAVs for both monitoring and locating sub-tasks, and locating and delivering sub-tasks, for the purpose of maximizing the resource utilization and also minimizing the cost of the mission.

In this example, for the monitoring sub-task, it could be considered as a 2-out-of-5: F subsystem; for the locating sub-task, it could be considered as 3-out-of-7: F subsystem; and finally, for the delivering sub-task, it could be considered as 3-out-of-6: F subsystem. In addition, there are two shared UAVs for the monitoring and locating sub-tasks, and for the locating and delivering sub-tasks, respectively.

It is evident that some UAVs have both monitoring and delivering functions, and so these UAVs can be set as shared UAVs for the monitoring and delivering sub-tasks. In this case, we can use a circular k-out-of-n: F system with shared components to model the actual UAVs performing the mission. The studied systems are also applicable for some other real situations such as the system of railroad tracks.

The rest of this paper is organized as follows. In Section 2, we present some preliminaries and the description of the considered systems and then provide some basic methods that will get used in subsequent sections. In Section 3, we discuss how the reliability of the linear k-out-of-n: F system with shared components can be obtained. By using the FMCIA, we obtain the reliability of the whole system by summing the reliabilities of all distinct cases. In Section 4, we decompose the circular k-out-of-n: F system with shared components into a linear k-out-of-n: F system with shared components from any one of its common areas and then use it to compute the reliability of the circular system. Some numerical examples are finally presented in Section 5 to illustrate the usefulness of the established results for the computation of reliabilities of the considered systems. Finally, some conclusions are made in Section 6.

Section snippets

Preliminaries

The expression of the reliability of a linear k-out-of-n: F system can be derived by using FMCIA [15]. Define a Markov chain {Yi,t=0,1,}, with a finite state space S={0,1,,k} and state k representing an absorbing state as it corresponds to the failure of the system, by Yi=j,system with components 1, 2, …,i has j components failed(0j<k);k,at least k components of system with components1, 2, …, i have failed.The FMCIA technique adds the components into the system one after another. Based on

Reliability calculation for the SC/Lin/k/n:F system

In this section, we describe the reliability calculation rule for the SC/Lin/k/n:F system. It is well known that the reliability of a linear k-out-of-n system can be computed by employing FMCIA directly. However, for the scenario considered here, we cannot use FMCIA to calculate the reliability value directly since there is a common area between adjacent subsystems. For this reason, for the shared components in the common areas, we first derive the probabilities of having 0, 1, , fi,i+1

Reliability calculation for the SC/Cir/k/n:F system

For the SC/Cir/k/n:F system, we can decompose it into a SC/Lin/k/n:F system from any one of its common areas for computing the reliability. First, all disjoint cases need to be determined by the relationship between min{ki,ki+1} and the number of shared components, i.e., di,i+1, for i=1,2,,m. Particularly, we specify k1km+1 and dm,1dm,m+1. Then, we obtain the reliability of each case, which can be regarded as a series system consisting of m subsystems and each subsystem having two common

Numerical results

In this section, we use some examples to demonstrate the calculation of the reliabilities of SC/Lin/k/n:F and SC/Cir/k/n:F systems, by making use of reliability formulae derived in the preceding sections.

The numerical examples are based on real-life examples. In many real-life situations, there are overlaps in subsystems, such as street lights at intersections and railway platforms wherein railroad tracks enter and exit the station. The railroad tracks can be seen as redundant voting systems

Conclusions

Due to their prominent presence in many real-life applications, k-out-of-n systems have been studied extensively in reliability theory. In this work, we have considered linear and circular k-out-of-n: F systems with shared components, consisting of m subsystems, i.e., SC/Lin/k/n:F and SC/Cir/k/n:F systems, which are extensions of k-out-of-n: F systems. Though FMCIA can be used to compute reliabilities of k-out-of-n system, it cannot be used directly to derive the reliabilities of the systems

CRediT authorship contribution statement

Juan Yin: Conceptualization, Methodology, Software, Writing – original draft. Lirong Cui: Formal analysis, Supervision, Writing – review & editing. Yudao Sun: Conceptualization, Writing – review & editing. Narayanaswamy Balakrishnan: Supervision, Writing – review & editing.

Acknowledgements

This work has been supported by the National Natural Science Foundation of China under grants 71631001 & 71871021. The authors appreciate the valuable comments provided by the editor and the anonymous reviewers which improved the quality and presentation of this paper.

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