# Optimal Task Abort and Maintenance Policies Considering Time Redundancy

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## Abstract

**:**

## 1. Introduction

- Dynamic preventive maintenance and task abort policies are designed that vary with the number task attempts;
- TSP and SSP are derived under proposed preventive maintenance and task abort policies considering ITR and IITR;
- The optimal preventive maintenance and abort thresholds minimizing the expected cost of preventive maintenance, task failure and system failure are studied.

## 2. Literature Review

## 3. Problem Formulation

- $Y\left(0\right)=0$ with probability one.
- For $u<v$, the degradation increment in time interval $(u,v)$, $Y\left(v\right)-Y\left(u\right)$ follows Gamma distribution with the distribution function ${G}_{\left(\alpha \right(v-u),\beta )}\left(y\right)$ given as$${G}_{\left(\alpha \right(v-u),\beta )}\left(y\right)=1-\frac{\mathsf{\Gamma}\left(\alpha \right(v-u),\beta y)}{\mathsf{\Gamma}\left(\alpha \right(v-u\left)\right)},$$$${g}_{\left(\alpha \right(v-u),\beta )}\left(y\right)=\frac{{\beta}^{\alpha (v-u)}{y}^{\alpha (v-u)-1}{e}^{-\beta y}}{\mathsf{\Gamma}\left(\alpha \right(v-u\left)\right)}.$$Here, $\mathsf{\Gamma}\left(u\right)$ and $\mathsf{\Gamma}(u,v)$ are the Gamma function and incomplete Gamma function, which are, respectively, defined as$$\left\{\begin{array}{c}{\displaystyle \mathsf{\Gamma}\left(u\right)={\int}_{0}^{\infty}{v}^{u-1}{e}^{-v}dv,}\hfill \\ {\displaystyle \mathsf{\Gamma}(u,v)={\int}_{v}^{\infty}{z}^{u-1}{e}^{-z}dz.}\hfill \end{array}\right.$$
- $Y\left(t\right)$ has independent increments over disjoint intervals.

- Task success under ITR: the continuous operating time should exceed a threshold $\tau $$(\tau <\widehat{\tau})$;
- Task success under IITR: the cumulative operating time should exceed a threshold $\tau $$(\tau <\widehat{\tau})$.

## 4. TSP and SSP under ITR

#### 4.1. TSP Evaluation under ITR

#### 4.2. SSP Evaluation under ITR

## 5. TSP and SSP under IITR

#### 5.1. TSP Evaluation under IITR

#### 5.2. SSP Evaluation under IITR

## 6. Optimal Abort and Maintenance Policies

## 7. Case Study

#### 7.1. Background

#### 7.2. Evaluation of TSP and SSP

#### 7.3. Optimal Task Abort and Maintenance Policies

## 8. Conclusions, Limitations, and Future Research

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

Notation | Description |

$Y\left(t\right)$ | system degradation at time t |

D | failure threshold |

$\varphi \left(t\right)$ | rescue duration at time t |

$\tau $ | task duration |

${d}_{i}$ | abort limit at attempt i |

${T}_{{d}_{i}}$ | task abort time |

ITR | type I time redundancy |

IITR | type II time redundancy |

TSP | task success probability |

SSP | system survival probability |

${R}_{I}$ | TSP under ITR |

${R}_{II}$ | TSP under IITR |

${S}_{I}$ | SSP under ITR |

${S}_{II}$ | SSP under IITR |

${c}_{m}$ | task failure cost |

${c}_{s}$ | system failure cost |

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(Allowable Time $\widehat{\mathit{\tau}}$, Task Duration $\mathit{\tau}$) | Allowed Attempts K = 2 | Allowed Attempts K = 3 | Allowed Attempts K = 4 |
---|---|---|---|

(30 h, 15 h) | $(13.2,0;13.2)$ | $(15.1,0.23;13.7;0;12.8)$ | $(15.7,0.21;14.0,0.30;13.7,0;12.6)$ |

(35 h, 15 h) | $(13.9,0;13.9)$ | $(15.2,0.27;14.4;0;13.5)$ | $(16.0,0.26;15.0;0.31;14.2;0;13.4)$ |

(40 h, 15 h) | $(14.2,0;14.2)$ | $(15.5,0.29;14.7;0;14.1)$ | $(16.1,0.27;15.3,0.35;14.6,0;14.0)$ |

(45 h, 15 h) | $(14.7,0;14.7)$ | $(15.8,0.32;15.0;0;14.3)$ | $(16.5,0.30;15.7;0.38;15.0;0;14.2)$ |

(50 h, 15 h) | $\left(15.3,0;15.3\right)$ | $\left(16.9,0.35;15.6;0;14.5\right)$ | $\left(17.7,0.32;16.8,0.43;15.4,0;14.3\right)$ |

(30 h, 10 h) | $(13.7,0;13.7)$ | $(15.4,0.28;13.8;0;13.3)$ | $(16.3,0.26;14.5,0.15;14.1,0;12.9)$ |

(30 h, 15 h) | $(13.2,0;13.2)$ | $(15.1,0.23;13.7;0;12.8)$ | $(15.7,0.21;14.0,0.15;13.7,0;12.6)$ |

(30 h, 20 h) | $(12.9,0;12.9)$ | $(14.3,0.21;13.4;0;12.5)$ | $(15.1,0.15;13.5,0.12;13.2,0;12.1)$ |

(30 h, 25 h) | $(12.2,0;12.2)$ | $(14.0,0.10;13.1;0;11.9)$ | $(14.6,0.11;13.1,0.10;12.8,0;11.1)$ |

(Allowable Time $\widehat{\mathit{\tau}}$, Task Duration $\mathit{\tau}$) | Allowed Attempts K = 2 | Allowed Attempts K = 3 | Allowed Attempts K = 4 |
---|---|---|---|

(30 h, 15 h) | $(12.7,0.15;12.5)$ | $(14.6,0.28;13.3;0;12.3)$ | $(15.1,0.45;13.6,0.38;13.3,0.31;12.1)$ |

(35 h, 15 h) | $(13.2,0.20;12.9)$ | $(15.0,0.30;13.7;0.11;13.2)$ | $(15.7,0.43;14.5;0.40;13.5;0.36;12.6)$ |

(40 h, 15 h) | $(14.0,0.25;13.3)$ | $(15.3,0.39;14.3;0.24;13.8)$ | $(15.8,0.47;15.1;0.42;14.0;0.37;12.9)$ |

(45 h, 15 h) | $(14.3,0.41;13.8)$ | $(15.1,0.43;14.8;0.37;14.0)$ | $(16.4,0.48;15.6;0.43;14.5;0.39;13.2)$ |

(50 h, 15 h) | $\left(14.7,0.44;14.1\right)$ | $\left(16.4,0.51;15.0;0.44;14.2\right)$ | $\left(17.2,0.52;16.3,0.51;15.0,0.46;13.9\right)$ |

(30 h, 10 h) | $(13.5,0.20;13.1)$ | $(15.2,0.33;13.5;0;13.0)$ | $(16.1,0.49;14.2,0.40;13.6,0.31;12.5)$ |

(30 h, 15 h) | $(12.7,0.15;12.5)$ | $(14.6,0.28;13.3;0;12.3)$ | $(15.1,0.45;13.6,0.34;13.3,0.30;12.1)$ |

(30 h, 20 h) | $(12.3,0;12.1)$ | $(14.1,0.22;13.4;0;12.1)$ | $(14.7,0.30;13.2,0.21;13.0,0;11.4)$ |

(30 h, 25 h) | $(12.0,0;11.8)$ | $(13.7,0.18;13.0;0;11.6)$ | $(14.3,0.27;12.8,0.17;12.5,0;10.8)$ |

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## Share and Cite

**MDPI and ACS Style**

Chen, K.; Zhao, X.; Qiu, Q.
Optimal Task Abort and Maintenance Policies Considering Time Redundancy. *Mathematics* **2022**, *10*, 1360.
https://doi.org/10.3390/math10091360

**AMA Style**

Chen K, Zhao X, Qiu Q.
Optimal Task Abort and Maintenance Policies Considering Time Redundancy. *Mathematics*. 2022; 10(9):1360.
https://doi.org/10.3390/math10091360

**Chicago/Turabian Style**

Chen, Ke, Xian Zhao, and Qingan Qiu.
2022. "Optimal Task Abort and Maintenance Policies Considering Time Redundancy" *Mathematics* 10, no. 9: 1360.
https://doi.org/10.3390/math10091360