Multi-state balanced systems with multiple failure criteria

https://doi.org/10.1016/j.ress.2020.106888Get rights and content

Highlights

  • Three types of competing risk models are developed for balanced systems.

  • New failure criteria of multi-state balanced systems are presented.

  • Formulas for some reliability indexes of proposed systems are given.

Abstract

The concept of balanced systems has been evolved with many meanings recently, and the research on this topic is a hot and significant issue in the reliability field. In this paper, the system consisting of two components does not fail immediately when it is unbalanced. An imminent unbalanced state is defined that the system experienced before reaching its final failure. According to the sojourn time of the system in the imminent unbalanced state and the number of transitions from the balanced to imminent unbalanced states, two new failure criteria are proposed. Combined with the situation where the system components may be permanently damaged, three types of competing risk models are developed for multi-state balanced systems with multiple failure criteria, which are time threshold models, count threshold models and the mixed threshold model. The closed-form formulas of reliability indexes of the multi-state balanced system under five proposed models are obtained by using the theory of aggregated stochastic processes, such as the probability density function of the system lifetime. The extensions to systems with multiple components are discussed. Finally, numerical examples are provided to illustrate the proposed model and the obtained results.

Introduction

Many new reliability models are required to be presented with the rapid development of science and technology. The combination of the concept of “balance” and the theory of system modeling has injected new vitality into the analysis of reliability systems. As is well known, there are many meanings of the concept of balance, such as the equilibrium state in thermodynamic systems, the axial symmetry and plane symmetry in geometry, etc. In engineering practice, especially in aerospace and military industries, the balanced systems have wide applications, such as unmanned aerial vehicles and deep space vehicles.

The research on the balanced reliability system has begun since 2016, when Hua and Elsayed [1], [2] first proposed a k-out-of-n pairs: G balanced system with spatially distributed units, which was assumed to have n pairs of units distributed evenly on a circular configuration and must keep the remaining operating units symmetric. The reliability estimation of the k-out-of-n pairs: G balanced system was given in Hua and Elsayed [1], [2], but the computational time increases dramatically and the reliability estimation becomes extremely difficult when the number of units increases. Hence, a computationally efficient method for reliability approximation of the k-out-of-n pairs: G balanced system was provided in Hua and Elsayed [3], by assuming that the lifetime of each individual unit followed an independent identical exponential distribution. Based on their work, Endharta et al. [4] and Cui et al. [5] considered a circular k-out-of-n pairs: G balanced system under a new balanced condition and a k-out-of-n pairs: F balanced system with m sectors respectively.

A new group of balanced systems has attracted much attention since it was developed by Cui et al. [6]. They proposed a balanced system where the evolution of each component follows an irreducible homogeneous continuous-time Markov process, and balance is measured by the distance between states of specified components. The formulas for system reliability, probability density functions and moments of the lifetimes of balanced systems under two proposed models were obtained in Cui et al. [6]. However, in the practical engineering, the system usually does not fail immediately when it enters an unbalanced state. On the contrary, the system alternates between balanced and unbalanced states. The system is considered to have failed when it cannot adjust quickly from the unbalanced state to the balanced state, or when it switches between the balanced state and the unbalanced state too frequently. Hence, in the present paper, we define the unbalance of the system experienced before reaching its final failure as an imminent unbalance. A balanced reliability system consisting of components whose behaviors are described by Markov processes as in Cui et al. [6] is considered in this paper. However, what distinguishes this paper from Cui et al. [6] is that the failure of the developed balanced system is caused by multiple failure criteria.

The competing risk model refers to a system with multiple failure criteria whose failure is caused by the failure criterion which occurs first [7]. It originated in the field of biostatistics in the 18th century, and has been widely used in the reliability filed these years, including accelerated life tests [8], mixed soft and hard failure processes [9], [10], maintenance planning [11] and so on. In our paper, five developed competing risk models are categorized into three types according to different failure criteria, which are called time threshold models, count threshold models and the mixed threshold model respectively. In time threshold models, the system will fail when either of two events occurs: 1) the system enters the absorbing subset; 2) it stays in the imminent unbalanced subset for longer than a specified threshold. In count threshold models, the system failure is caused by either of two events: 1) the system enters the absorbing subset; 2) the system has moved from the balanced subset to the imminent unbalanced subset more than a given number of times. The mixed threshold model is developed based on time threshold models and count threshold models.

The present paper focuses on the balance problem of multi-state systems. Since Barlow and Wu [12] first considered coherent systems composed of multi-state components as an extension of binary coherent systems, multi-state systems have subsequently attracted the attention of numerous researchers and culminated in the significant monograph by Lisnianski and Levitin [13]. There are mainly four kinds of methods for reliability modeling and evaluation of complex multi-state systems: extensions of binary models to multi-state cases [14], [15], [16], the universal generating function (UGF) technique [17], [18], [19], Monte-Carlo simulation [20], [21], [22] and stochastic process approach [23], [24], [25]. In the present paper, the behavior of each component is governed by a finite state Markov process and the system composed of multiple components is verified to still follow the Markov process. In order to avoid the state explosion problem, we introduce the theory of aggregated stochastic process to group the states with similar performance levels. In this way, not only the impact of each component on the behavior of the system can be observed, but also the computational time can be simplified.

Since the theory of aggregated stochastic processes was presented and further developed to describe the behaviors of ion channels [26], [27], it has gained a significant amount of attention in the field of reliability engineering. The most extensive application of the aggregated stochastic process in the reliability field is for multi-state systems, whose state space can be aggregated by a partitioning into classes. The theory of aggregated stochastic process is powerful due to its diversity in usage, which mainly consists of the proposition and derivation of new reliability indexes [28], and the birth of new complex system models [29]. The first paper which applied the theory of aggregated stochastic processes to competing risk models is by Cui and Wu [30], where two extended Phase-type models were developed. The two models were combined and extended to random threshold cases in Wu et al. [31], and the optimization problem under semi-Markov processes was discussed in Wu et al. [32]. The theory of aggregated stochastic processes will be applied to derive the formulas for reliability indexes of the balanced system with multiple failure criteria, such as the probability density function and expectation of the system lifetime in the present paper.

The balanced system with multiple failure criteria which is discussed in our paper has some real applications. Before going to the detailed modeling description, two examples are provided for the potential applications.

The unbalance of batteries in a battery pack system matters in the battery system life, because without balancing, the capacity of the battery pack will decrease more quickly during operation due to the drift apart of battery voltages, and then the battery pack system will fail [33]. There are two categories that lead to the unbalance of batteries, which are internal and external. Internal sources refer to manufacturing differences in the state of charge (SOC), self-discharge rate, capacity, impedance, and temperature characteristics. External sources consist of thermal difference across the pack which leads to different self-discharge rates of batteries, and some multi-rank pack protection ICs draining unequally from the different series ranks in the pack [34]. In other words, the unbalance of batteries can appear both in the beginning and during the operation of the battery pack.

Battery balancing which refers to the techniques that improve the available capacity of a battery pack with multiple batteries plays an important role in battery management system. To equalize the voltage among batteries when they are at full charge is the fundamental solution of battery balancing. Hence, a brief unbalance does not cause the battery pack to fail. A 2% capacity unbalance from a 2200-mAh cell can be balanced within a charge cycle or two [35]. However, the balancing may be overpowered by the rate of battery unbalance [35].

Therefore, in this paper, the failure rate of the battery pack system is characterized by the dwell time of the system in the state of battery unbalance, and the number of times from battery balance to battery unbalance. The battery voltage can be discretized so that the battery pack system can be regarded as a multi-state system.

Another example is the balance of vehicle tires. It is not easy to detect the process of gradually losing the balance of tireo over time by visual inspection alone, which may be caused by failure to keep tires properly inflated. It will bring many problems with the vehicle and lead to the unsafety of drivers. For example, the vibration increases as tires get out of balance and becomes even worse when tires are worn out. Certain areas of the tires will wear down quicker than they should due to the unbalanced tires. Worst, uneven tread wear combined with too much vibration from the lack of balance can make the vehicle harder to steer properly, and increase the risk of sudden blowouts.

Vehicles will not break down and become undrivable immediately once tires lose balance. Instead, vehicle tires will switch between balance and unbalance many times, and the final failure of the vehicle is caused by the cumulative wear due to unbalanced tires. Therefore, there is an upper bound for the time when the vehicle is in a state where its tires are out of balance. Sometimes, it is hard to accurately measure the sojourn time of the vehicle in the unbalanced state, so the number of transition times from the balanced state to the unbalanced state of the vehicle can be specified with an upper limit.

The rest of the paper is organized as follows. In Section 2, five competing risk models are defined based on the given basic assumptions. The theory of aggregated stochastic processes is introduced to derive the formulas for the system reliability indexes under time threshold models, count threshold models and the mixed threshold model in Section 3, Section 4 and Section 5 respectively. Section 6 discusses the balanced systems with multiple components as an extension of the previous derived results. Section 7 shows some numerical examples to illustrate the obtained results. Finally, conclusions are presented in Section 8.

Throughout this paper vectors and matrices are rendered in bold; all vectors are column vectors. I denotes an identity matrix and 0 denotes a matrix or vector of zeros.

Section snippets

Definition and modeling

Suppose a balanced system which consists of two components is governed by a stochastic process {Z(t), t ≥ 0}. The state evolution of component 1 and component 2 can be described by two irreducible finite state Markov process {X(t), t ≥ 0} and {Y(t), t ≥ 0} respectively. The two Markov processes X(t) and Y(t) are independent of each other, and have the transition rate matrices Q1 and Q2 respectively. The initial probabilities of X(t) and Y(t) are assumed to be α1 and α2 respectively. In general,

System reliability analysis under time threshold models

In this section, the formulas for lifetime distributions of balanced systems under TTM-I and TTM-II will be discussed respectively.

Let Td(t) denote the sojourn time of the system at time t, starting from the last moment when the distance between X(t) and Y(t) exceeds d. Let TdX(t) denote the sojourn time of component 1 at time t, starting from the last time when X(t) exceeds dX, and let TdY(t) denote the sojourn time of component 2 at time t, starting from the last time when Y(t) exceeds dY.

System lifetime distribution under count threshold models

In this section, we will discuss the formulas for lifetime distribution of the balanced system under CTM-I and CTM-II. In order to simplify, in CTM-I and CTM-II, we choose the same symbols as in TTM-I and TTM-II regarding the distance threshold between two components of the system and the state thresholds for components 1 and 2, i.e., d, dX and dY respectively.

Let nd(t) represent the number of occurrences that the distance between states of components 1 and 2 exceeds d from time 0 up to time t.

System lifetime distribution under mixed threshold model

In this section, we will discuss the formula for lifetime distribution of the balanced system under MTM. Since MTM is a mixed model of TTM-I and CTM-I, the lifetime of the balanced system under MTM, say, TV, can be given byTV=inf{t:Td(t)τorTdX(t)τorTdY(t)τornd(t)+ndX(t)+ndY(t)>NorX(t)=n1orY(t)=n2}.

The state space of the Markov process Z(t) is divided into four subsets which are BI, UI, AI, and DI, similarly to TTM-I and CTM-I.

Let PMTM(BI → AI) and PMTM(UI → AI) denote the probabilities that

Extensions to multi-dimensional situations

In this section, we consider a system which is composed of M components (M ≥ 3). It is necessary to extend the two-dimensional situation to multi-dimensional situation because it is common for a system to be composed of multiple components in practical engineering, such as a battery pack composed of multiple batteries.

Suppose the behavior of component i (i=1,2,...,M) is described by an irreducible finite state Markov process {Xi(t), t ≥ 0} with transition rate matrix Qi. The initial probability

Numerical examples: battery pack systems

A numerical example of battery pack systems is presented in this section to demonstrate the results obtained above. MATHEMATICA programs have been exploited to implement the previously derived formulas.

The battery pack system consists of two batteries, which are Battery I and Battery II. Each battery voltage can be discretized into multiple states. Battery I, i.e., component 1 is assumed to have four states which means S1={1,2,3,4} where state 4 is an absorbing state. Battery II, i.e.,

Conclusions

Two new failure criteria of the balanced system are proposed in the present paper, with regard to the sojourn time of the system in the imminent unbalanced state and the number of transitions from the balanced state to the imminent unbalanced state. Five competing risk models are developed based on the new failure criteria to describe the behavior of various balanced reliability systems. In addition, the circumstance where the component cannot be repaired is discussed. Formulas for reliability

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.

Acknowledgment

This work was supported by the National Natural Science Foundation of China under grant 71631001.

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