Reliability Analysis and Optimal Replacement Policy for Systems with Generalized Pólya Censored δ Shock Model
Abstract
:1. Introduction
2. Preliminaries and Model Description
Model Description
- (a)
- ;
- (b)
- .
- (a)
- The GPP with the set of parameters , where , and , is an HPP with intensity λ.
- (b)
- The GPP with the set of parameters , where and , is an NHPP with intensity .
- (c)
- The GPP with the set of parameters , where , and , is a Pólya process with a set of parameters .
3. Reliability of a System with a Generalized Pólya Censored Shock Model
- (1)
- The distribution of is given by
- (2)
- The conditional joint probability density function of in , given that , is
- (a)
- The probability mass function of M is given byProof. Note that
- (b)
- The survival function of M is given by
- (c)
- The mean of M is given by
4. Optimal Replacement Policy under
- (1)
- A new system is installed at time . The system is repaired immediately upon its failure. The system is immediately replaced with a new system when the system is observed to fail for the Nth time.
- (2)
- The system suffers from external shocks. The arrival of the external shocks follows a generalized Pólya process with a set of parameters , , . The system fails when the interval time of successive shocks is larger than . After the nth repair, the system failure threshold decreases as . Denote as the first failure time of the system and let be the operating time of the system after the th repair to the nth failure, where .
- (3)
- Let represent the repair time of the system after the nth failure, where . The repair time sequence forms an increasing geometric process; then, . In particular, when , the maintenance time is ignored.
- (4)
- The system repair cost per unit time is , the operating reward rate per unit time is , the replacement cost is , and the replacement time is negligible.
- (5)
- The GPP, , and are independent.
5. Case Study
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
Notations | |
T | Life of the system |
Total shocks number occurred by time t | |
Stochastic intensity of the generalized Pólya process | |
Intensity function, | |
Parameters of the generalized Pólya process | |
The arrival of external shocks | |
M | The number of shocks until the system fails |
The inter-arrival time between the ith and the th shocks | |
The failure threshold of the censored shock model | |
The failure rate of the system | |
The density function of system lifetime T | |
The distribution function of system lifetime T | |
The reliability function of system lifetime T | |
The mean lifetime of the system | |
J | The Jacobian determinant |
The conditional density function of | |
The repair time of the system after the nth failure, | |
System repair cost per unit time | |
Operating reward rate per unit time | |
Replacement cost | |
N | Replacement policy |
W | Random length of a cycle under the replacement policy N |
Expected length of the renewal cycle | |
Long-run average cost per time | |
m | The real value of |
, | Parameters of the generalized Pólya process |
Acronym | |
Customer relationship management | |
Customer lifetime value | |
Generalized Pólya process | |
Homogeneous Poisson process | |
Non-homogeneous Poisson process | |
Increasing failure rate average | |
Decreasing failure rate average | |
Censored shock model | |
Generalized Pólya censored shock model | |
Homogeneous Poisson censored shock model | |
Non-homogeneous Poisson censored shock model |
Appendix A. Proof of Theorem 1
Appendix B. Proof of Theorem 2
Appendix C. Proof of Theorem 4
Appendix D. Proof of Theorem 5
- (1)
- When , i.e., , is increasing, such that ;
- (2)
- When , i.e., , is decreasing, such that .
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Model | Parameter | Failure Mechanism | System Lifetime |
---|---|---|---|
shock model | |||
censored shock model |
N | N | N | N | N | |||||
---|---|---|---|---|---|---|---|---|---|
1 | 13 | 30 | 54 | 78 | 0.1974 | ||||
2 | 14 | 32 | 56 | 80 | 0.3798 | ||||
3 | 15 | 34 | 58 | 82 | 0.5038 | ||||
4 | 16 | 36 | 60 | 84 | 0.5941 | ||||
5 | 17 | 38 | 62 | 86 | 0.7549 | ||||
6 | 18 | 40 | 64 | 88 | 1.2810 | ||||
7 | 20 | 42 | 66 | 90 | 1.3709 | ||||
8 | 22 | 44 | 68 | 92 | 1.6431 | ||||
9 | 24 | 46 | 70 | 0.1368 | 94 | 2.0699 | |||
10 | 26 | 48 | 72 | 0.1567 | 96 | 2.4149 | |||
11 | 28 | 50 | 74 | 0.1693 | 98 | 2.5950 | |||
12 | 29 | −0.9944 | 52 | 76 | 0.1859 | 100 | 3.0909 |
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Bian, L.; Peng, B.; Ye, Y. Reliability Analysis and Optimal Replacement Policy for Systems with Generalized Pólya Censored δ Shock Model. Mathematics 2023, 11, 4560. https://doi.org/10.3390/math11214560
Bian L, Peng B, Ye Y. Reliability Analysis and Optimal Replacement Policy for Systems with Generalized Pólya Censored δ Shock Model. Mathematics. 2023; 11(21):4560. https://doi.org/10.3390/math11214560
Chicago/Turabian StyleBian, Lina, Bo Peng, and Yong Ye. 2023. "Reliability Analysis and Optimal Replacement Policy for Systems with Generalized Pólya Censored δ Shock Model" Mathematics 11, no. 21: 4560. https://doi.org/10.3390/math11214560