Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 726853, 10 pages

http://dx.doi.org/10.1155/2015/726853

## Reliability Analysis of Load-Sharing -out-of- System Considering Component Degradation

^{1}School of Reliability and Systems Engineering, Beihang University, Beijing 100191, China^{2}Science and Technology on Reliability and Environmental Engineering Laboratory, Beijing 100191, China

Received 29 October 2014; Accepted 2 April 2015

Academic Editor: Enrico Zio

Copyright © 2015 Chunbo Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The -out-of- configuration is a typical form of redundancy techniques to improve system reliability, where at least -out-of- components must work for successful operation of system. When the components are degraded, more components are needed to meet the system requirement, which means that the value of has to increase. The current reliability analysis methods overestimate the reliability, because using constant ignores the degradation effect. In a load-sharing system with degrading components, the workload shared on each surviving component will increase after a random component failure, resulting in higher failure rate and increased performance degradation rate. This paper proposes a method combining a tampered failure rate model with a performance degradation model to analyze the reliability of load-sharing -out-of- system with degrading components. The proposed method considers the value of as a variable which is derived by the performance degradation model. Also, the load-sharing effect is evaluated by the tampered failure rate model. Monte-Carlo simulation procedure is used to estimate the discrete probability distribution of . The case of a solar panel is studied in this paper, and the result shows that the reliability considering component degradation is less than that ignoring component degradation.

#### 1. Introduction

Redundancy technique is widely used to improve system reliability. A typical form of redundancy is a -out-of- configuration in which at least out of components must work for normal operation of system. When using traditional methods [1–4] to analyze reliability of -out-of- system, independence is assumed within the system, which means that a component failure does not affect the failure rate or performance of surviving components. However, in the real world, many systems are load-sharing, such as electric generators sharing an electrical load in a power plant, cables in a suspension bridge, and valves or pumps in a hydraulic system. In the load-sharing system, the workload has to be shared by the remaining components, resulting in an increased load shared on each surviving component [5]. Many empirical studies of mechanical systems [6] and computer systems [7] have proved that the workload strongly affects the component failure rate. Scheuer [8] studied the reliability of -out-of- system when component failure induces higher failure rate in survivors. The method is limited in system composed of -independent and identically distributed components with exponential lifetimes. Liu [9] proposed a generalized accelerated failure-time model (AFTM) for reliability analysis of load-sharing -out-of- system with arbitrary distribution load-dependent component lifetime distributions. Amari et al. [10] provided a closed-form analytical solution for the reliability of tampered failure rate load-sharing -out-of- system, and Amari and Bergman [11] also used the cumulative exposure model to account for the effect of loading history. The mentioned reliability analysis methods are based on the assumption of binary components, that is to say the component is either failed or working in perfect state. In fact, the performance of component is degrading in lifetime. In order to meet the system performance requirement, the value of has to monotonously increase in service time. In that case, the reliability of -out-of- system considering component degradation may be less than the reliability based on assumption of binary component.

When components have several degraded performance states, the multistate system (MSS) model is introduced to analyze the reliability of -out-of- system [12–15]. Amari et al. [16] presented a fast and robust reliability evaluation algorithm for very large multistate -out-of- systems. Levitin [17] introduced a new model named multi-state vector--out-of- system, which is a generalization of existing multistate -out-of- system models. Using current MSS algorithms to calculate reliability, we should know the probability that component performance is located in each state. Many experimental and theoretical researches [18–20] indicate that the performance degradation rate is strangely affected by the load shared on components. In the load-sharing system, the shared load is increasing, and then the performance degradation rate will increase correspondingly. Consequently, it is difficult to get the probability of each performance state.

In the degradation process, the value of is increasing because of component degradation. The phased-mission system (PMS) model [21–23] is introduced to calculate the reliability of -out-of- system with variable in different phases. Xing et al. [24] proposed an efficient method for reliability evaluation of -out-of- systems subject to phased-mission requirements. Unfortunately, when using the PMS model to evaluate reliability, we need to know the specified in each phase. However, in degradation process, the value of is determined by the system performance requirement and the degraded performance of each component. As a result, the value of in each phase is variable. Considering the complexity of degradation law in load-sharing system, it is hard to know the specified in each phase.

In conclusion, there are some special characteristics of load-sharing -out-of- system with degrading components: (1) a component random failure increases the load shared on each remaining component, (2) the failure rate of surviving components will increase after component failures, (3) the rise of load will raise the degradation rate of component, (4) the duration when components have a specified degradation rate is stochastic because a component failure occurs at random time, (5) the value of is stochastic because the component performance is a random variable at any given time. Some mentioned load-sharing methods deal with characteristics (1) and (2) effectively. What is more, the MSS model is suggested to compute the reliability when component has degraded states, and the PMS model is used to deal with phase-mission requirement. However, characteristics (3), (4), and (5) make the degradation law of component performance very complicated; consequently it is very hard to get some conditions of existed methods.

This paper proposes a method combining a tampered failure rate model with a performance degradation model to analyze the reliability of load-sharing -out-of- system with degrading components. The tampered failure rate model is introduced to evaluate the reliability where the rise of load makes the failure rate of surviving components increase. The performance degradation model is derived to calculate degraded component performance in service time. Furthermore, the load-sharing effect on degradation rate and the randomness of duration are included in the performance degradation model. In this way, the discrete probability distribution of is obtained through performance degradation model coupled with load-sharing effect. Using the discrete probability distribution of and the reliability computed by the tampered failure rate model when the system has a specified , the reliability of load-sharing -out-of- system with degrading components is calculated accurately.

The remainder of this paper is divided into five sections. In Section 2, the tampered failure rate model is introduced to analyze reliability of load-sharing -out-of- system when the value of is constant. In Section 3, the performance degradation model coupled with load-sharing effect is formulated to evaluate the discrete probability distribution of . In Section 4, Monte-Carlo simulation procedure is used to estimate the discrete probability distribution of . The case of a solar panel is studied in Section 5. Conclusions are drawn in Section 6.

#### 2. Reliability of Load-Sharing -out-of- System

In a load-sharing system, the workload has to be redistributed among the remaining components after a component failure. Mostly the load is equally shared by each surviving component. Let the total workload be , and let the total number of components be . Let be the load on each surviving component when components have failed. Hence,

In order to analyze the reliability of load-sharing system, the load-sharing effect that the rise of shared load on a surviving component raises the failure rate has to be evaluated. The tampered failure rate (TFR) model proposed by Bhattacharyya and Soejoeti [25] can be applied for the load-sharing system. The acceleration of failure when load is raised from lower level to a higher level is reflected in the failure rate function.

Let the component be subject to an ordered sequence of loads, where load is applied during the time interval (). According to the TFR model, the failure rate of the component at iswhere is the baseline failure rate which has nothing to do with load, is the tampered factor at load , and is the load shared on component at .

With the assumptions that the load is equally distributed among all surviving identical components and the failure rate of a component varies as described in the TFR model, the reliability of load-sharing -out-of- system without component performance degradation could be calculated by analytical method.

##### 2.1. Exponential Case

Firstly, consider a load-sharing -out-of- system with components following the exponential lifetime distribution [8]; in other words, the baseline failure rate of the TFR model is constant.

When the system is put into operation, the failure rate of every component is denoted by . Because there are working components in the system, the first component failure occurs at failure rate . After the first failure, the remaining working components must carry the same workload of system. As a result, the failure rate of each surviving component becomes , which is commonly higher than . The second component failure occurs at rate . When components have failed, the failure rate of each remaining component is denoted by . The component failure occurs at failure rate . The system is failed when more than components are failed.

The time when the component failure occurs in the system is denoted by , and the time interval between and component failure is represented by . Since all identical components are following the exponential distributions, the follows the exponential distribution with parameter . Hence, the lifetime of system is the failure time

Then, the reliability of -out-of- system at is

In order to calculate the distribution of and the reliability function of load-sharing -out-of- system, two typical formulas which can be used are as follows.

*Case a. *All are equal (say ) [8]:This case arises when the failure rate of each surviving component is directly proportional to the load it carries, which means that in the TFR model.

*Case b. *All are distinct [8]:

##### 2.2. General Case

In the case of a load-sharing -out-of- system with components following arbitrary lifetime distributions, the baseline failure rate of the TFR model is no longer a constant. A closed-form analytical solution for TFR model with an arbitrary baseline distribution is introduced. The basic idea is to use a time-transformation to convert TFR model with an arbitrary baseline distribution into an equivalent problem with an exponential baseline distribution [10].

Lemma 1. *(1) For any failure distribution , the reliability function is .**(2) The cumulative failure rate function is**(3) The random variable follows a uniform distribution in interval . The random variable follows an exponential distribution with mean 1.**(4) For a TFR model with a standard exponential () baseline failure time distribution**(5) For a TFR model with a baseline failure rate of and a baseline cumulative failure rate of ,*(a)*under the regular scale ,*(b)*under the transformed scale .**Let **where is the cumulative failure rate in the transformed scale.*

*According to the above lemmas, it becomes obvious that if the load-sharing effect on the failure rate of an individual component follows a TFR model with , the reliability of a load-sharing system at is equivalent to the reliability of corresponding exponential load-sharing model at time , where the failure rate of a component is when components have failed for .*

*3. Degradation Effect in Load-Sharing System*

*3. Degradation Effect in Load-Sharing System*

*In a load-sharing -out-of- system with degrading components, a random component failure raises the load shared on remaining components, leading to the rise of the failure rate and performance degradation rate. Besides, the duration when component has a specified degradation rate is variable because a component failure occurs at random time. Therefore, the component performance is a random variable at any given time, thus the value of is stochastic in service time.*

*3.1. Independent Degradation Effect*

*3.1. Independent Degradation Effect*

*At the first step, consider the component degradation rate to be independent of load-sharing effect, which means that the load-sharing effect just makes the failure rate higher but the degradation rate has nothing to do with shared load. When the system is just put into operation, the value of is determined by , where is the system performance requirement, is the initial performance of each component, and is the minimum integer that is greater than or equal to the quotient. When the system has operated for a period of time , the degraded performance of component is denoted by , where is degradation law of component performance. is a monotone decreasing function, and , . For example, the common degradation laws are exponential law denoted by and power law denoted by . At time , the value of is determined by . Since is a monotone decreasing function, it is quite clear that . Thus, , while the condition for equality is . Hence, the reliability of -out-of- system with degrading component is calculated in a general expression as*

*Correspondingly, using the TFR model to analyze the load-sharing -out-of- system with degrading components, the value of should be replaced by .*

*Obviously, because of , meaning that the reliability of -out-of- system considering component degradation is less than the reliability ignoring component degradation.*

*3.2. Degradation Coupled with Load-Sharing Effect*

*3.2. Degradation Coupled with Load-Sharing Effect*

*When the component performance degradation is related to load-sharing effect, a random failure of component will raise the degradation rate of surviving component. Since a component failure occurs at random time, the duration when components have a specified degradation rate is variable. Therefore, the variable denoted by is stochastic at any given time.*

*In a load-sharing system, the degradation law of component performance is not only a function of time, but also a function of the shared load. The generalization of degradation law is expressed as a function of time and load, denoted by . The degradation rate of component performance with load-sharing effect is . After component failure, the performance degradation rate of surviving components is denoted by .*

*In order to analyze the reliability of load-sharing -out-of- system with degrading components at , should be evaluated previously, which is determined by the degraded performance of component at . With the understanding of degradation rate after component failure, the degraded performance at is estimated by using the duration when the component has a specified degradation rate, denoted by . The duration is the interval between component failure and component failure. In other words, is the minimum order statistic of the components failure time of the surviving components under the condition that the components do not fail at .*

*Lemma 2.
(1) Conditional Distribution Function(a)The reliability of component at is(b)The reliability at is(c)The cumulative distribution function of component failure time under the condition that the component did not fail at is(2) Minimum Order Statistic(a)Let the cumulative distribution function of a population be , and probability distribution function is . For samples with size , probability distribution function of order statistic is(b)The probability distribution function of the minimum order statistic is(c)The cumulative distribution function of the minimum order statistic is*

*Hence, the cumulative distribution function of is expressed as*

*In a special situation that the baseline distribution of TFR is exponential distribution, the distribution of is an exponential distribution with parameter , which agrees with the conclusion in previous paper Section 2.1.*

*As components have failed in the system, the degradation rate of each surviving component is . Moreover, the time interval between component failure and failure is . Therefore, the component performance degradation in time interval is*

*The degraded component performance at is calculated as*

*According to , () and for , is determined by ,*

*Hence,where the distribution of is and the degradation rate at each phase is .*

*According to (22), the distribution of the degraded component performance is calculated on basis of the distribution of denoted by and degradation rate . Then, the discrete probability distribution of could be obtained through*

*Using the probability distribution of estimated by the performance degradation model and the reliability calculated by the TFR model when the system has a specified , the reliability of load-sharing -out-of- system with degrading components is computed in formula aswhere is the probability when the is equal to and is the reliability of -out-of- system when the is equal to . Using the proposed formula to evaluate the reliability of load-sharing -out-of- system with degrading components, the load-sharing effect on component failure rate and the degradation effect coupled with load-sharing effect are all included in the model. Therefore, the reliability is calculated more accurately.*

*4. Monte-Carlo Simulation*

*4. Monte-Carlo Simulation**In the load-sharing system, the failure rate and degradation rate are variable during the lifetime, and the duration in each phase is stochastic because a component failure occurs at random time. The analytic expression of degraded performance could be obtained only if the failure rate or degradation rate is subject to some specified formats, such as components following the exponential lifetime distribution or degradation rate being constant. This paper employs Monte-Carlo simulation to estimate the discrete probability distribution of . The simulation procedure is shown in Figure 1.*