# A System with Two Spare Units, Two Repair Facilities, and Two Types of Repairers

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. System Description and Mathematical Framework

- 1.
- Three identical units comprise a single-unit system. At the beginning, only one unit is operational, while the remaining two units stay on cold standby.
- 2.
- Two repair facilities are respectively serviced by a regular and an expert repairer.
- 3.
- The operational unit’s failure is noted instantaneously; the dead unit is dispatched to a repairer, while a spare is promptly activated.
- 4.
- The regular repairer must finish repair within a random patience time (RPT) T.
- 5.
- The system goes down when all three units fail.
- 6.
- When either the regular repairer’s patience time runs out or when the system dies, whichever occurs earlier, the expert is alerted to come at once.
- 7.
- The regular repairer works on the failed unit until his patience time is over or until the expert is freed up to take over, whichever comes later.
- 8.
- Lifetime, repair time, and patience time are independently and exponentially distributed with arbitrary parameters. This assumption being restrictive ought to be eliminated in a future work.
- 9.
- When the expert comes in, the progress made by the regular repairer is lost. Specifically, it is a consequence of the previous assumption due to the memoryless property of the exponential distribution.
- 10.
- We consider two options for the expert repairer, resulting in a multiple repair by expert (MRE) model and single repair by expert (SRE) model.
- 11.
- Repair by either repairer is perfect, rendering a unit brand new after repair is complete.

- 1.
- Beginning at State 1, the system stays in State 1 for a period X, before going to State 2.
- 2.
- In State 2, the system remains for $min(X,Y,T)$; if Y is the smallest, then the system goes back to State 1, if T is the minimum, then it moves to State 3, and if X is the smallest, then it moves to State 4.
- 3.
- In State 3, the system stays for $min(X,Z)$; if $Z<X$, then the system goes to State 1; otherwise, it moves to State 5.
- 4.
- In State 4, the time spent equals $min({X}^{\prime},Y,{T}^{\prime})$; if Y is the smallest, then the system goes to State 2, if ${T}^{\prime}$ is the minimum, then it goes to State 5, and if ${X}^{\prime}$ happens to be the minimum, then it moves to State 7.
- 5.
- The time spent in State 5 equals $min(X,Y,Z,T)$; if Z is the smallest, then the system goes to State 2, if Y turns out to be the minimum, then it moves to State 3, if T happens to be the minimum, then it goes to State 6, and if X is the smallest, then it goes to State 8.
- 6.
- The time spent in State 6 equals $min(X,Y,Z)$; if either Y or Z is the smallest, then the system goes to State 3, but if X is the minimum, the system goes to State 9.
- 7.
- The time spent in State 7 equals $min(Z,Y,{T}^{\u2033})$; if ${T}^{\u2033}$ is the minimum, then the system moves to State 9, if either Y or Z is the minimum, then it goes to State 5 (under the MRE policy), but under the SRE policy, if Z is the smallest, then the system goes to State 4, and if Y is the smallest, then it goes to State 5.
- 8.
- The time spent in State 8 equals $min(Z,Y,{T}^{\prime})$; if ${T}^{\prime}$ is the smallest, the system goes to State 9. Under both SRE and MRE policies, transitions from State 8 to States 4 and 5 are identical to those from State 7.
- 9.
- The sojourn time in State 9 is $min(Z,Y)$; and as soon as either the expert or the regular repairer repairs one of the failed units in State 9, the system moves to State 5 under both SRE and MRE policies.

## 4. Computing Limiting Availability and Limiting Profit

**Theorem**

**1.**

#### 4.1. The MRE-RPT Model

#### 4.2. The SRE-RPT Model

## 5. Comparison of Models

- 1.
- Both ${A}_{\infty}$ and $\omega $ are greater for the MRE model than for the SRE model regardless of the number of spare units and the number of repair facilities.
- 2.
- Adding a second spare when a system currently has one spare improves both ${A}_{\infty}$ and $\omega $. As an example, ${A}_{\infty}$ is below 80% when $S=1$, but it is more than 80% when $S=2$. See [3] for further details.
- 3.
- Including one more spare unit causes ${\Theta}_{r}>{\Theta}_{e}$, implying that we utilize the regular repairer more than the expert. Likewise, adding a second repair facility makes the regular repairer busier than the expert, resulting in even less cost and higher limiting profit per unit time $\omega $.
- 4.
- Adding a second repair facility to the system with two spare units raises both ${A}_{\infty}$ and $\omega $ further. For example, ${A}_{\infty}$ is increased to almost 90% under the MRE policy.

## 6. Concluding Remarks

- Keeping our focus on building repairable models, we have assumed exponentially distributed lifetime and repair time random variables. While it may pose additional challenges since the stochastic process will no longer be an SMP, extension beyond exponential distribution is highly desired.
- While we assumed the units are identical, a more realistic model would admit non-identical units with different lifetime and repair rates. Specifically, when there are multiple such units, we must determine at each decision epoch which unit should be prioritized for operation and which should be prioritized for repair.
- While we studied patience time as a random variable, a logistically more desirable option is to permit a predetermined constant patience time. Again, we cannot use SMP under a deterministic patience time policy, as the Markovian property is violated in some states. This is a fertile ground for developing a new mathematical theory.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Transition diagrams for SRE (

**left**) and MRE (

**right**) models. The symbol ∧ denotes minimum; a red arrow denotes a failed system is revived.

**Figure 2.**Limiting profit per unit time as a function of ${C}_{e}$ for system with $S=2$ and $RF=2$.

Criteria | SRE | MRE | ||||
---|---|---|---|---|---|---|

$\mathit{S}=1$ | $\mathit{S}=2$ | $\mathit{S}=2$ | $\mathit{S}=1$ | $\mathit{S}=2$ | $\mathit{S}=2$ | |

$\mathit{RF}=1$ | $\mathit{RF}=1$ | $\mathit{RF}=2$ | $\mathit{RF}=1$ | $\mathit{RF}=1$ | $\mathit{RF}=2$ | |

${A}_{\infty}$ | 0.736 | 0.801 | 0.884 | 0.788 | 0.844 | 0.896 |

$\omega $ | 11.919 | 13.640 | 15.143 | 12.484 | 14.068 | 15.236 |

${\Theta}_{r}$ | 0.320 | 0.442 | 0.605 | 0.174 | 0.227 | 0.572 |

${\Theta}_{e}$ | 0.342 | 0.327 | 0.307 | 0.426 | 0.457 | 0.331 |

**Table 2.**Cost–profit analysis under different cost parameters ${C}_{r}$ and ${C}_{e}$, when ${R}_{p}-{C}_{p}=20$, ${C}_{l}=3$, $\alpha =0.3$, $\lambda =0.5$, $\beta =0.4$, $\gamma =0.8$.

System | C | SRE | MRE | |||||
---|---|---|---|---|---|---|---|---|

${\mathit{C}}_{\mathit{r}}$ | ${\mathit{C}}_{\mathit{e}}$ | ${\mathit{A}}_{\mathit{\infty}}$ | Cost | $\mathit{\omega}$ | ${\mathit{A}}_{\mathit{\infty}}$ | Cost | $\mathit{\omega}$ | |

$S=1,RF=1$ | 2 | 5 | 0.758 | 2.770 | 12.180 | 0.800 | 2.770 | 12.805 |

2 | 6 | 0.758 | 3.168 | 11.864 | 0.800 | 3.168 | 12.407 | |

3 | 6 | 0.758 | 3.345 | 11.545 | 0.800 | 3.345 | 12.230 | |

3 | 7 | 0.758 | 3.743 | 11.233 | 0.800 | 3.743 | 11.832 | |

3 | 8 | 0.758 | 4.142 | 10.917 | 0.800 | 4.142 | 11.434 | |

4 | 8 | 0.758 | 4.319 | 10.602 | 0.800 | 4.319 | 11.257 | |

$S=2,RF=1$ | 2 | 5 | 0.825 | 2.650 | 13.848 | 0.862 | 2.892 | 14.349 |

2 | 6 | 0.825 | 2.946 | 13.551 | 0.862 | 3.313 | 13.928 | |

3 | 6 | 0.825 | 3.384 | 13.113 | 0.862 | 3.548 | 13.692 | |

3 | 7 | 0.825 | 3.681 | 12.817 | 0.862 | 3.970 | 13.271 | |

3 | 8 | 0.825 | 3.977 | 12.520 | 0.862 | 4.390 | 12.850 | |

4 | 7 | 0.825 | 4.415 | 12.082 | 0.862 | 4.626 | 12.615 | |

$S=2,RF=2$ | 2 | 5 | 0.901 | 2.924 | 15.103 | 0.912 | 3.042 | 15.196 |

2 | 6 | 0.901 | 3.201 | 14.827 | 0.912 | 3.340 | 14.898 | |

3 | 6 | 0.901 | 3.775 | 14.253 | 0.912 | 3.884 | 14.354 | |

3 | 7 | 0.901 | 4.051 | 13.977 | 0.912 | 4.182 | 14.056 | |

3 | 8 | 0.901 | 4.327 | 13.700 | 0.912 | 4.480 | 13.758 | |

4 | 8 | 0.901 | 4.901 | 13.126 | 0.912 | 5.024 | 13.215 |

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**MDPI and ACS Style**

Andalib, V.; Sarkar, J.
A System with Two Spare Units, Two Repair Facilities, and Two Types of Repairers. *Mathematics* **2022**, *10*, 852.
https://doi.org/10.3390/math10060852

**AMA Style**

Andalib V, Sarkar J.
A System with Two Spare Units, Two Repair Facilities, and Two Types of Repairers. *Mathematics*. 2022; 10(6):852.
https://doi.org/10.3390/math10060852

**Chicago/Turabian Style**

Andalib, Vahid, and Jyotirmoy Sarkar.
2022. "A System with Two Spare Units, Two Repair Facilities, and Two Types of Repairers" *Mathematics* 10, no. 6: 852.
https://doi.org/10.3390/math10060852