Studying the Efficiency of the Apache Kafka System Using the Reduction Method, and Its Effectiveness in Terms of Reliability Metrics Subject to a Copula Approach
Abstract
:1. Introduction
2. Model Description and Notations
2.1. State Description
2.2. Assumptions
- At the beginning, all subsystems operate flawlessly.
- The system functions efficiently as long as at least one unit within each subsystem is in good working condition following the 1-out-of-3: G policy.
- The system will be inoperative if three units fail from each subsystem. Additionally, if the ZooKeeper unit within subsystem 2 fails, it will also lead to system inoperability.
- The rate at which failures occur remains consistent over time, and these rates are distributed according to an exponential distribution.
- The process of repairing states that have partially failed is achieved through general distribution, while the complete restoration of failed states is accomplished through the application of the Gumbel–Hougard family copula distribution.
- A repairman is employed on a full-time basis and has the capability to fix units that have experienced partial or complete failure.
- The repaired system functions with the same reliability and efficiency as a new one.
2.3. Notations
3. Mathematical Model and Solution
4. Model Analysis
4.1. System Availability
4.2. System Reliability Analysis
4.3. Mean Time to Failure (MTTF) Analysis
4.4. Sensitivity Analysis of the MTTF
4.5. Cost Analysis
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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State | Description |
---|---|
At this stage, it is expected that all components of the subsystem are operating correctly and without any issues. | |
, , | The mentioned state has worsened and is considered to be a minor failure within a system. However, despite the failure of any single unit within subsystems 1, 2, and 3, the overall system remains operational as two units are still functioning properly. Efforts are underway to restore the system through comprehensive repairs. |
, , | After experiencing the malfunction of at least two units within subsystems 1, 2, and 3, the system has reached a significantly impaired operational state, requiring ongoing repairs to restore full functionality. |
, , | These states have experienced significant breakdowns due to the failure of more than two units within subsystems 1, 2, and 3. To address this issue, the Gumbel–Hougard copula distribution is being employed to rectify the system. |
Due to a malfunction in the ZooKeeper unit, subsystem 2 has completely stopped working. The issue was resolved using copula distribution. |
Time | Case 1 | Case 2 | Case 3 |
---|---|---|---|
0 | 1 | 1 | 1 |
1 | 0.985207 | 0.972606 | 0.997056 |
2 | 0.985002 | 0.964517 | 0.997001 |
3 | 0.985202 | 0.962104 | 0.997037 |
4 | 0.985277 | 0.961376 | 0.997054 |
5 | 0.9853 | 0.961153 | 0.99706 |
6 | 0.985307 | 0.961084 | 0.997063 |
7 | 0.985309 | 0.961062 | 0.997063 |
8 | 0.98531 | 0.961055 | 0.997064 |
9 | 0.98531 | 0.961053 | 0.997064 |
10 | 0.98531 | 0.961052 | 0.997064 |
Time | Case 1 | Case 2 |
---|---|---|
0 | 1 | 1 |
1 | 0.96434 | 0.992194 |
2 | 0.922462 | 0.984121 |
3 | 0.876713 | 0.975801 |
4 | 0.828879 | 0.967258 |
5 | 0.780305 | 0.958509 |
6 | 0.731992 | 0.949574 |
7 | 0.684668 | 0.940472 |
8 | 0.63885 | 0.931218 |
9 | 0.594894 | 0.92183 |
10 | 0.553027 | 0.912322 |
Failure Rates | MTTF () | MTTF () | MTTF () | MTTF () |
---|---|---|---|---|
0.01 | 14.9052 | 15.7955 | 17.5875 | 14.9052 |
0.02 | 14.2286 | 14.9052 | 16.2737 | 14.9052 |
0.03 | 13.3724 | 13.8651 | 14.9052 | 14.9052 |
0.04 | 12.5392 | 12.8898 | 13.688 | 14.9052 |
0.05 | 11.7801 | 12.0233 | 12.6432 | 14.9052 |
0.06 | 11.1031 | 11.2651 | 11.7516 | 14.9052 |
0.07 | 10.5027 | 10.6029 | 10.9881 | 14.9052 |
0.08 | 9.9701 | 10.0229 | 10.3297 | 14.9052 |
0.09 | 9.49622 | 9.51224 | 9.7578 | 14.9052 |
0.1 | 9.07289 | 9.06025 | 9.25714 | 14.9052 |
Failure Rates | ||||
---|---|---|---|---|
0.01 | −41.9193 | −62.5954 | −107.932 | −569.162 |
0.02 | −82.4095 | −102.874 | −140.725 | −370.604 |
0.03 | −85.9876 | −102.218 | −130.34 | −263.725 |
0.04 | −79.949 | −92.2716 | −112.923 | −198.283 |
0.05 | −71.7737 | −81.0735 | −96.4012 | −154.875 |
0.06 | −63.7345 | −70.7879 | −82.3463 | −124.444 |
0.07 | −56.4945 | −61.8877 | −70.7466 | −69.3555 |
0.08 | −50.179 | −28.6126 | −61.232 | −51.685 |
0.09 | −44.7334 | −24.9794 | −53.4053 | −45.353 |
Time | = 0.5 | = 0.4 | = 0.3 | = 0.2 | = 0.1 |
---|---|---|---|---|---|
0 | 0 | 0 | 0 | 0 | 0 |
1 | 0.489226 | 0.589226 | 0.689226 | 0.789226 | 0.889226 |
2 | 0.974191 | 1.17419 | 1.37419 | 1.57419 | 1.77419 |
3 | 1.4593 | 1.7593 | 2.0593 | 2.3593 | 2.6593 |
4 | 1.94455 | 2.34455 | 2.74455 | 3.14455 | 3.54455 |
5 | 2.42984 | 2.92984 | 3.42984 | 3.92984 | 4.42984 |
6 | 2.91515 | 3.51515 | 4.11515 | 4.71515 | 5.31515 |
7 | 3.40045 | 4.10045 | 4.80045 | 5.50045 | 6.20045 |
8 | 3.88576 | 4.68576 | 5.48576 | 6.28576 | 7.08576 |
9 | 4.37107 | 5.27107 | 6.17107 | 7.07107 | 7.97107 |
10 | 4.85638 | 5.85638 | 6.85638 | 7.85638 | 8.85638 |
Time | = 0.5 | = 0.4 | = 0.3 | = 0.2 | = 0.1 |
---|---|---|---|---|---|
0 | 0 | 0 | 0 | 0 | 0 |
1 | 0.483578 | 0.583578 | 0.683578 | 0.783578 | 0.883578 |
2 | 0.951341 | 1.15134 | 1.35134 | 1.55134 | 1.75134 |
3 | 1.41442 | 1.71442 | 2.01442 | 2.31442 | 2.61442 |
4 | 1.87608 | 2.27608 | 2.67608 | 3.07608 | 3.47608 |
5 | 2.33733 | 2.83733 | 3.33733 | 3.83733 | 4.33733 |
6 | 2.79844 | 3.39844 | 3.99844 | 4.59844 | 5.19844 |
7 | 3.25951 | 3.95951 | 4.65951 | 5.35951 | 6.05951 |
8 | 3.72057 | 4.52057 | 5.32057 | 6.12057 | 6.92057 |
9 | 4.18162 | 5.08162 | 5.98162 | 6.88162 | 7.78162 |
10 | 4.64267 | 5.64267 | 6.64267 | 7.64267 | 8.64267 |
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Elshoubary, E.E.; Radwan, T. Studying the Efficiency of the Apache Kafka System Using the Reduction Method, and Its Effectiveness in Terms of Reliability Metrics Subject to a Copula Approach. Appl. Sci. 2024, 14, 6758. https://doi.org/10.3390/app14156758
Elshoubary EE, Radwan T. Studying the Efficiency of the Apache Kafka System Using the Reduction Method, and Its Effectiveness in Terms of Reliability Metrics Subject to a Copula Approach. Applied Sciences. 2024; 14(15):6758. https://doi.org/10.3390/app14156758
Chicago/Turabian StyleElshoubary, Elsayed E., and Taha Radwan. 2024. "Studying the Efficiency of the Apache Kafka System Using the Reduction Method, and Its Effectiveness in Terms of Reliability Metrics Subject to a Copula Approach" Applied Sciences 14, no. 15: 6758. https://doi.org/10.3390/app14156758