# Maximum Entropy Approach to Reliability of Multi-Component Systems with Non-Repairable or Repairable Components

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## Abstract

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## 1. Introduction

## 2. Modeling the Degradation and the Recovery Processes

## 3. The Non-Repairable System

#### 3.1. MaxEnt for Double-Component Non-Repairable Model: Independent Degradation

#### 3.2. MaxEnt for Double-Component Non-Repairable Model: Correlated Degradation Case

## 4. System Hierarchy by Reliability Block Diagram

#### 4.1. System-Level Observation and Coarse-Grained Information

#### 4.2. Tree-Type Networks

#### 4.3. Homogeneous Hazard Assumption

#### 4.4. Loop Networks and Parallel-Series Type Diagram

## 5. The Repairable System

#### 5.1. Double-Component Model

#### 5.2. Degradation Propagation on Star Graph and Complex Networks

#### 5.3. Illustration of the Method

## 6. Conclusions and Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The illustrations of the system networks. (

**a**) the double-component model of independent degradation, (

**b**) the double-component model of correlated degradation (

**c**) the chain-type graph model (

**d**) the graph model contains loop. The filled and the numbered circles represent the degradation source and normal components, respectively.

**Figure 3.**The reduction of the reliability block diagram. The right arrow means that if the n components are on one path, the parallel type diagram can be further reduced for tree graphs.

**Figure 6.**The sub-graph of the generated Watts-Strogatz small world where 14 nodes are shown for illustration purposes.

**Figure 7.**The performance of the MaxEnt method. (

**a**) The inference of the bare hazard rate. (

**b**) The forecast of the number of normal and failed components by MaxEnt. The parameters of simulation are ${\alpha}_{s}=-2,{\beta}_{s}=-1/3,{\alpha}_{r}=1/4,{\beta}_{r}=0$. The simulation result shows that the total spreading last about $T=15\left({\alpha}_{r}^{-1}\right)$, the data with an initial $1.5\left({\alpha}_{r}^{-1}\right)$. The data of 27 single-component samples are used to estimate the moments.

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**MDPI and ACS Style**

Du, Y.-M.; Chen, J.-F.; Guan, X.; Sun, C.P.
Maximum Entropy Approach to Reliability of Multi-Component Systems with Non-Repairable or Repairable Components. *Entropy* **2021**, *23*, 348.
https://doi.org/10.3390/e23030348

**AMA Style**

Du Y-M, Chen J-F, Guan X, Sun CP.
Maximum Entropy Approach to Reliability of Multi-Component Systems with Non-Repairable or Repairable Components. *Entropy*. 2021; 23(3):348.
https://doi.org/10.3390/e23030348

**Chicago/Turabian Style**

Du, Yi-Mu, Jin-Fu Chen, Xuefei Guan, and C. P. Sun.
2021. "Maximum Entropy Approach to Reliability of Multi-Component Systems with Non-Repairable or Repairable Components" *Entropy* 23, no. 3: 348.
https://doi.org/10.3390/e23030348