Reliability of three-dimensional consecutive k-type systems

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Highlights

  • Reliabilities of several three-dimensional consecutive k-type systems are studied.

  • Overlapping and non-overlapping failed blocks are also considered for these systems.

  • Finite Markov chain imbedding approach is developed to study these systems in a novel way.

Abstract

Three-dimensional consecutive k-type systems are widely found in reliability practice such as in sensing systems, but it is not an easy task to evaluate reliability of these systems. In this paper, several three-dimensional consecutive k-type systems, namely, linear connected-(k1,k2,k3)-out-of-(n1,n2,n3):F system, linear connected-(k1,k2,k3)!-out-of-(n1,n2,n3):F system, and linear l-connected-(k1,k2,k3)-out-of-(n1,n2,n3):F system, without/with overlapping, are studied. Reliability of these systems is determined by using finite Markov chain imbedding approach (FMCIA), and some specific techniques are employed to reduce the state space of the involved Markov chain. Some numerical illustrative examples are then provided to demonstrate the accuracy and efficiency of the proposed method, and finally some possible applications and generalizations are pointed out.

Introduction

Redundant systems, such as consecutive k-out-of-n systems and their related systems, have found key applications in reliability theory and practice [1], [2], [3], [4], [5], [6], [7]. For the calculation of reliability of these systems, an efficient method, finite Markov chain imbedding approach (FMCIA), was first discussed by Fu and Koutras [8], and the method got its name subsequently from Koutras [9]. For more detailed discussions on the method and its application to different types of redundant systems, one may refer to [10], [11], [12], [13], [14], [15], [16].

Most work on redundant systems focuses on one-dimensional systems. However, these one-dimensional systems are not sufficient to model many practical systems. For example, a redundant group of connector pins fails only if a square of pins fail, a disease can be diagnosed by X-ray only if a square or cluster of cells are not healthy, a supervision system fails only if a square of TV cameras fail, and a sensing system fails only if a square of sensors fail with sensors [17,18]. In this regard, some work on two-dimensional redundant systems has been carried out, such as the k2/n2:Fsystem by Salvia and Lasher [17], linear/circular connected-X-out-of-(m,n):F lattice system by Boehme et al. [18], k-within-(r,s)-out-of-(m,n):F lattice system by Malinowski and Preuss [19], connected-(r1,s1)-or-(r2,s2)--(rk,sk)-out-of-(m,n):F lattice system by Yamamoto [20], consecutive-kr-out-of-nr:F linear zigzag structure and circular polygon structure by Lin et al. [21], and toroidal connected-(r,s)-out-of-(m,n):F lattice system by Nakamura et al. [22]. Some pertinent discussions in this direction can be found in [23], [24], [25], [26].

Reliability calculation of these two-dimensional redundant systems has received considerable attention in the literature. Specifically, there are mainly three different types of methods—efficient bounds [17,19,27,28], recursive algorithms [20,[29], [30], [31]] and FMCIA methods [32], [33], [34], [35]. Efficient bounds reduce the reliability calculation process by providing efficient upper/lower bounds for the reliability function. This method is simple and direct, and is often efficient timewise. However, the errors involved in the approximation may not always be small and may not be even computable. Unlike efficient bounds, recursive methods provide exact reliability for redundant systems, which are needed in many practical situations like system optimization, component importance analysis, component allocation and so on [33]. However, they do not provide analytic expressions for reliability function, and may often be quite time-consuming. FMCIA methods can be regarded as improved versions of recursive methods as they have a unified explicit form of exact reliability function for systems of different sizes and also require less memory and computational time. FMCIA will be quite useful in reliability calculation of high-dimensional consecutive-k type systems if suitable Markov chains with small state spaces can be defined for them.

Even though much work exists on one/two-dimensional consecutive k-type systems as detailed above, the study of three/high-dimensional consecutive k-type systems is somewhat lacking. For example, Psillakis and Makri [36] studied 3-dimensional consecutive-k-out-of-n:F systems by a simulation method, while Boushaba and Ghoraf [37] proposed upper and lower bounds and a limit theorem for them. Godbole et al. [38] presented upper bounds for the reliability of d-dimensional consecutive-k-out-of-n:F systems, for d3, and Cowell [39] derived a formula for the reliability of such systems. Akiba and Yamamoto [40] proposed upper and lower bounds for the reliability of 3-dimensional k-within-consecutive-(r1,r2,r3)-out-of-(n1,n2,n3):F systems. No further work on exact reliability calculation has been done for three/high-dimensional consecutive k-type systems mentioned above; but, this is essential in many practical applications in image analysis, pattern recognition, reliability theory, and minefield detection via remote sensing [41]. For example, consider a sensing system for cold storage consisting of temperature and humidity sensors in a rectangular solid, and the system is considered to have failed if a rectangle of sensors have failed, which means the sensing system can be regarded as a three-dimensional consecutive-k type system. There are some other important practical scenarios that necessitate the study of reliability of three-dimensional consecutive-type systems such as a three-dimensional integrated circuit manufactured by stacking as many as 16 or more integrated circuits and interconnecting them vertically using Cu-Cu connections (see [42]) and three-dimensional shipboard electrical power systems (see [43]). Based on these important applications, in this work, we discuss the determination of the exact reliability of three-dimensional consecutive k-type systems by the use of FMCIA in a novel way. In fact, the adopted method is logical and easy to understand as it is analogous to their one-dimensional and two-dimensional counterparts. The method developed here is not only efficient for large systems, but also can be used for many different types of three/high-dimensional consecutive-k systems.

Primary contributions of this work can be highlighted as follows: (1) Reliability of several three-dimensional consecutive k-type systems are studied; (2) Overlapping and non-overlapping failed blocks are also considered for these systems; and (3) Finite Markov chain imbedding approach is developed to study these systems in a novel way. Theoretical results provided in this paper can be applied to describe many practical systems. For example, a human organ consisting of cells can be regarded as a three-dimensional consecutive-k type system when we consider the occurrence of cancer or nodule and associated medical treatments. Similar examples can also be found in machines; for example, a liquid crystal display consisting of liquid crystal molecules can be regarded as a three-dimensional consecutive-k type system since its failure is caused by abnormal condition of clusters of molecules. Besides, a cooling system consisting of radiators in a large chemical factory or warehouse can be regarded as a three-dimensional consecutive-k type system since failures of clusters of radiators will cause emergency alarm or even disasters like a fire.

The rest of this paper proceeds as follows. In Section 2, the reliability of linear connected-(k1,k2,k3)-out-of-(n1,n2,n3):F system and linear connected-(k1,k2,k3)!-out-of-(n1,n2,n3):F system are studied by using FMCIA. Similar discussions for linear l-connected-(k1,k2,k3)-out-of-(n1,n2,n3):F system without/with overlapping are presented in Section 3. In Section 4, a number of illustrative examples are presented to demonstrate the method of reliability calculation developed here. Finally, some concluding remarks are made in Section 5.

Section snippets

Linear connected-(k1,k2,k3)-out-of-(n1,n2,n3):F system

Consider a linear connected-(k1,k2,k3)-out-of-(n1,n2,n3):F system that would fail if and only if there exist consecutive k1×k2×k3 failed components among its n1×n2×n3 components. As shown in Fig. 1, components in the system can be denoted by xi1,i2,i3(i1=1,,n1,i2=1,,n2,i3=1,,n3) according to their row number i1, column number i2 and layer number i3. (For example, see Fig. 2 for a linear connected-(1,2,2)-out-of-(3,4,3):F system and a possible failure case of it.)

To obtain the reliability of

Linear l-connected-(k1,k2,k3)-out-of-(n1,n2,n3):F system

Consider a linear l-connected-(k1,k2,k3)-out-of-(n1,n2,n3):F system without overlapping that would fail if and only if there exist l blocks of consecutive k1×k2×k3 failed components without overlapping among its n1×n2×n3 components. Here, without overlapping means the l blocks do not have any shared component. (See Fig. 6 for a possible failure case of a linear 2-connected-(1,2,2)-out-of-(3,4,3):F system without/with overlapping). Note that in this work, blocks are searched in a given order of

Illustrative examples

In this section, seven examples of three-dimensional consecutive systems are presented to illustrate the results in Sections 2 and 3, where Examples 4.1 and 4.2 are for Section 2, Examples 4.3 and 4.4 are for Section 3, and Examples 4.5-4.7 are for Remark 2.2.

Example 4.1

Consider a linear connected-(1,2,2)-out-of-(2,2,n):F sensing system that would fail if and only if there exist consecutive 1×2×2 failed sensors among its 2×2×n sensors. We now assume that all the sensors are independent and identically

Concluding remarks

In this work, we have applied FMCIA in the reliability calculation of three-dimensional consecutive k-type systems—linear connected-(k1,k2,k3)-out-of-(n1,n2,n3):F system, linear connected-(k1,k2,k3)!-out-of-(n1,n2,n3):F system, and linear l-connected-(k1,k2,k3)-out-of-(n1,n2,n3):F system, without/with overlapping. The reliability calculation is efficient for large systems, especially for systems with large n3. Practical applications of this work could be in the reliability analysis of

CRediT authorship contribution statement

He Yi: Conceptualization, Methodology, Software, Validation, Writing – original draft, Writing – review & editing. Narayanaswamy Balakrishnan: Conceptualization, Validation, Writing – original draft, Writing – review & editing, Visualization. Xiang Li: Writing – original draft, Project administration, Funding acquisition.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 72001016, No. 71931001 and No. 71722007), the Fundamental Research Funds for the Central Universities (buctrc202102) and the Funds for First-class Discipline Construction (XK1802-5), and the Natural Sciences and Engineering Research Council of Canada (to the second author) through an Individual Discovery Grant (RGPIN-2020-06733). Our sincere thanks also go to the Editor-in-Chief and the anonymous reviewers for

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