Reliability is a major issue for fault-tolerant systems used in critical applications. N-modular redundancy (NMR) is one of the traditional approaches used for fault masking in fault-tolerant systems. Reconfigurable NMR architecture supported with hot or cold standby spares is a common industrial method. So far, no systematic method for creating the Markov model of reconfigurable NMR systems supported with hot standby spares has been presented. Likewise, there is no explicit parametric formula for the reliability of these systems in the literature. This paper focuses on two issues: the systematic construction of the Markov model of reconfigurable NMR system, and its evaluation through a precise and explicit formula introduces in this paper. The introduced formula gives for system designer a good view of reliability behaviour of the reconfigurable NMR systems. https://doi.org/10.1007/978-3-030-58920-2_4

1. 1
Reliability Evaluation of Reconfigurable NMR
Architecture Supported with Hot Standby Spare:
Markov Modelling and Formulation
Koorosh Aslansefat, Gholamreza Latif-Shabgahi and Mehrdad Mohammadi
Email: k.aslansefat-2018@hull.ac.uk

2. 2
Table of Content
What we are going to discuss
Introduction
Introduction, Fault Tolerance and Redundancy
Markov Modelling
Models, Assumptions, Basic Concepts
Reliability Evaluation
Reliability Evaluation of Reconfigurable NMR supported with Hot Standby Spares
Numerical Results and Conclusion
Numerical reliability evaluation of Reconfigurable NMR architectures and Conclusion

3. 3
Introduction
Redundancy
ActivePassive Hybrid
Fault tolerance is the property that enables a system to continue operating
properly in the event of the failure of (or one or more faults within) some of its
components.

4. 4
Introduction
Passive Redundancy
‫ساختار‬TMR ‫ساختار‬NMR
TMR NMR

5. 5
Introduction
Active Redundancy
‫ساختار‬TMR ‫ساختار‬NMR
Cold Standby
HOT Standby

6. 6
Introduction
Hybrid Redundancy
‫ساختار‬TMR ‫ساختار‬NMR

7. 7
Introduction
Hybrid Redundancy
‫ساختار‬TMR ‫ساختار‬NMR
Cai, B., Liu, Y., Liu, Z., Tian, X., Li,
H., & Ren, C. (2012). Reliability
analysis of subsea blowout
preventer control systems subjected
to multiple error shocks. Journal of
Loss Prevention in the Process
Industries, 25(6), 1044-1054.

8. 8
Markov Modelling

9. 9
Markov Modelling
Assumptions
In the beginning the system is healthy.
There is no common cause failure.
There is no repair or maintenance available for the system.
All modules have the same failure rate with the exponential distribution.
Upon the failure of each module the spare replaces it immediately.
All modules are performing the same computation with the same inputs.

10. 10
Markov Modelling

11. 11
Markov Modelling

12. 12
Markov Modelling

13. 13
Markov Modelling

14. 14
Markov Modelling
𝑅 𝑇𝑀𝑅_1𝑆 𝑡 =
𝑒−4𝜆𝑡
2
−6𝑐2
+ 12𝑐 + 𝑒−3𝜆𝑡
6𝑐2
− 12𝑐 − 2 +
𝑒−2𝜆𝑡
2
−6𝑐2
+ 12𝑐 + 6
𝑅5𝑀𝑅_1𝑆 𝑡
= −
𝑒−6𝜆𝑡
6
60𝑐3
− 180𝑐2
+ 180𝑐 +
𝑒−5𝜆𝑡
2
60𝑐3
− 180𝑐2
+ 180𝑐 + 12
−
𝑒−4𝜆𝑡
2
60𝑐3
− 180𝑐2
+ 180𝑐 + 30 +
𝑒−3𝜆𝑡
6
60𝑐3
− 180𝑐2
+ 180𝑐 + 60
𝑅7𝑀𝑅_1𝑆 𝑡
= −
𝑒−8𝜆𝑡
24
840𝑐4
− 3360𝑐3
+ 5040𝑐2
− 3360𝑐 +
𝑒−7𝜆𝑡
6
840𝑐4
− 3360𝑐3
+ 5040𝑐2
− 3360𝑐 − 140
−
𝑒−6𝜆𝑡
4
840𝑐4
− 3360𝑐3
+ 5040𝑐2
− 3360𝑐 − 280 +
𝑒−5𝜆𝑡
6
840𝑐4
− 3360𝑐3
+ 5040𝑐2
− 3360𝑐 − 504
−
𝑒−4𝜆𝑡
24
840𝑐4
− 3360𝑐3
+ 5040𝑐2
− 3360𝑐 − 840

15. 15
Markov Modelling
𝑅 𝑁𝑀𝑅_1𝑆 𝑡 =
𝑘= 𝑁+1 2
𝑁+1 𝛼 + 𝛽 𝑁 + 1 − 𝑘 𝑒−𝑘𝜆𝑡
𝑁 + 1 2 !
𝑁 + 1 2
𝑘 − 𝑁 + 1 2
𝛼 =
𝛥
𝑗=0
𝑁−1 2 𝑁 + 1
2
𝑗 𝑖=
𝑁+1
2
𝑁
𝑖 𝑐
𝑁+1
2
−𝑗
−1 𝑗+1
𝛽 𝑘 =
𝛥
0 𝑘 =
𝑁 + 1
2
𝑘 +
𝑁 + 1
2
𝑁 − 3
2
𝑁 + 1
2
𝑁 + 1
2
𝑂. 𝑊.

16. 16
Numerical Results

17. 17
Numerical Results
System Type c = 0.3 c = 0.5 c = 0.8 c = 1
Simple TMR 0.75 0.75 0.75 0.75
TMR + HSP 0.83 0.86 0.88 0.90
Simple 5MR 0.80 0.80 0.80 0.80
5MR + HSP 0.87 0.89 0.90 0.91
Simple 7MR 0.85 0.85 0.85 0.85
7MR + HSP 0.91 0.92 0.92 0.92
0.001  at time = 40

18. 18
Numerical Results

19. 19
Numerical Results

20. 20
Conclusion
What will be the final conclusion
A Systematic Markov Model Construction for Reconfigurable NMR supported with Hot
Standby Spares has been explained.
A parametric formula for reliability evaluation of Reconfigurable NMR architecture supported
with one hot standby spare has been obtained that can be used for system optimization.
From the parametric equation other factors like Mean Time To Failure can be easily calculated.

21. 21
References
Selected References
• Z. Liu, Y. Liu, B. Cai, X. Liu, J. Li, X. Tian and R. Ji, "RAMS Analysis of Hybrid Redundancy System of
Subsea Blowout Preventer Based on Stochastic Petri Nets," International Journal of Security & Its
Applications, vol. 7, no. 4, pp. 159-166, 2013.
• B. Cai, Y. Liu, Z. Liu, X. Tian, H. Li and C. Ren, "Reliability Analysis of Subsea Blowout Preventer Control
Systems Subjected to Multiple Error Shocks," Journal of Loss Prevention in the Process Industries, vol. 25,
no. 6, pp. 1044-1054, 2012.
• S. Distefano, F. Longo and K. S. Trivedi, "Investigating Dynamic Reliability and Availability through State–
Space Models," Computers & Mathematics with Applications, vol. 64, no. 12, pp. 3701-3716, 2012.
• F. P. Mathur and A. Avižienis, "Reliability Analysis and Architecture of a Hybrid-redundant Digital System:
Generalized Triple Modular Redundancy with Self-repair," in Spring Joint Computer Conference, ACM, 1969.
• K. Zhang, G. Bedette and R. F. DeMara, "Triple Modular Redundancy with Standby (TMRSB) Supporting
Dynamic Resource Reconfiguration," in IEEE Autotestcon, Anaheim, CA, 2006.
• Z. Zhe, L. Daxin, W. Zhengxian and S. Changsong, "Research on Triple Modular Redundancy Dynamic
Fault-Tolerant System Model," in First International Multi-Symposiums on Computer and Computational
Sciences. IMSCCS, Hanzhou, Zhejiang, 2006.

22. 22
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