# Optimizing Availability of a Framework in Series Configuration Utilizing Markov Model and Monte Carlo Simulation Techniques

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## Abstract

**:**

## 1. Introduction and Literature Review

## 2. Markov Model and Monte Carlo Simulation for Deteriorating Frameworks

#### 2.1. Markov Model

#### 2.2. MC Simulation

## 3. Description of the Framework

#### 3.1. Multi-Level Deterioration Modeling

- State 1—Fully Operational State;
- State 2—Partially Operational State; and,
- State 3—Failed State.

#### 3.2. Repair Procedure

#### 3.3. Developing the Model

- The opportunistic maintenance process will finish before the completion of repair process for the failed unit.
- The unit can perform its intended task in fully operational and partially operational states only.

## 4. Implementation of Analytical and Simulation Approaches for Problem Solving

#### 4.1. Analytical Approach—Markov Model

#### 4.2. Simulation Approach—MC Simulation

^{−5}. The algorithm used for developing the MC Simulation codes is depicted using a flow chart in Figure 4, as below:

## 5. Case Study

#### 5.1. Problem Formulation

#### 5.2. Numerical Solution to the Problem

#### 5.3. Graphical Solution to the Problem

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. System of DEs Corresponding to the State Diagram Illustrated as Figure 5

## Appendix B. System of DEs Corresponding to the State Diagram Illustrated as Figure 6

## References

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**Figure 9.**Optimal values of mean time to failure (MTTF) and mean time to repair (MTTR) for the framework under an opportunistic maintenance strategy.

Units | Failure Rate | Average Repair Time | ||
---|---|---|---|---|

Temporary | Sustained | Average | ||

Transformer (33–110 kV) | 0.4 year per 100 transformers | 0.6 year per 100 transformers | 0.5 year per 100 transformers | 115.983 ≈ 116 h |

Safety unit for transformer (33–110 kV) | 0.9 year per 100 transformers | 0.6 year per 100 transformers | 0.75 year per 100 transformers | 14.8 ≈ 15 h |

Serial | Unit A | Unit B | ||
---|---|---|---|---|

Failure Rates $\left(\mathit{\lambda}\right)$ | Repair Rates $\left(\mathit{\mu}\right)$ | Failure Rates $\left(\mathit{\lambda}\right)$ | Repair Rates $\left(\mathit{\mu}\right)$ | |

1 | ${\lambda}_{12}=1.143\text{}\times {10}^{-4}$ | ${\mu}_{31}=8.622\text{}\times {10}^{-3}$ | ${\lambda}_{12}=1.714\text{}\times {10}^{-4}$ | ${\mu}_{31}=6.763\text{}\times {10}^{-2}$ |

2 | ${\lambda}_{23}=1.714\text{}\times {10}^{-4}$ | ${\mu}_{32}=17.25\text{}\times {10}^{-3}$ | ${\lambda}_{23}=2.566\text{}\times {10}^{-4}$ | ${\mu}_{32}=13.57\text{}\times {10}^{-2}$ |

Types of Corrective Repairs | Availability of the Framework | |||
---|---|---|---|---|

Using Markov Model | Using MC Simulation | |||

Without Opportunistic Maintenance | With Opportunistic Maintenance | Without Opportunistic Maintenance | With Opportunistic Maintenance | |

Perfect Repair | 97.71% | 98.73% | 97.71% | 98.74% |

Imperfect Repair | 97.95% | 98.91% | 97.94% | 98.89% |

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**MDPI and ACS Style**

Siddiqui, M.A.; Butt, S.I.; Gilani, O.; Jamil, M.; Maqsood, A.; Zhang, F.
Optimizing Availability of a Framework in Series Configuration Utilizing Markov Model and Monte Carlo Simulation Techniques. *Symmetry* **2017**, *9*, 96.
https://doi.org/10.3390/sym9070096

**AMA Style**

Siddiqui MA, Butt SI, Gilani O, Jamil M, Maqsood A, Zhang F.
Optimizing Availability of a Framework in Series Configuration Utilizing Markov Model and Monte Carlo Simulation Techniques. *Symmetry*. 2017; 9(7):96.
https://doi.org/10.3390/sym9070096

**Chicago/Turabian Style**

Siddiqui, Mansoor Ahmed, Shahid Ikramullah Butt, Omer Gilani, Mohsin Jamil, Adnan Maqsood, and Faping Zhang.
2017. "Optimizing Availability of a Framework in Series Configuration Utilizing Markov Model and Monte Carlo Simulation Techniques" *Symmetry* 9, no. 7: 96.
https://doi.org/10.3390/sym9070096