System-of-Systems Resilience Analysis and Design Using Bayesian and Dynamic Bayesian Networks
Abstract
:1. Introduction
2. Hierarchy Structure of an SoS
2.1. Hierarchical Representation of an SoS
2.2. Challenges of Inter-Dependency Analysis of the SoS under Uncertainty
- It allows an intuitive and visual representation of inter-dependency in the SoS and provides a way for fusing field data, statistical analysis results, simulation outputs, analytical equations, and expert opinions in one summary model;
- It takes into account the uncertainty state of the constituent system, which also influences the completion of activities and achievement of SoS capabilities;
- It can fuse new observations with prior information to update the state of the system in real time;
- Uncertainty of the SoS can be explicitly calculated through the system state values and by propagating the uncertainty through the network.
3. Bayesian and Dynamic Bayesian Networks
3.1. Bayesian Network
3.2. Dynamic Bayesian Network
- Anchor node (A): a node outside the temporal plate of a DBN that only affects the nodes in the first time step;
- Terminal nodes (T): a node outside the temporal plate of a DBN that only affects the nodes in the last time step.
4. Resilience Metric and Resilience-Based System Importance
4.1. Resilience Analysis Metric
4.2. Resilience-Based System Importance
Algorithm 1 Resilience-based System Importance Algorithm. |
Input: the system of SoS Output: 1: Calculate the resilience of the system () 2: Calculate the resilience of the system () 3: Calculate the resilience-based system importance ( = − ) 4: Return |
5. Design Resilience under Budgetary Constraint
5.1. Single System Functionality Curve
- : The maximum of loss in the functionality of the system;
- : Recovery time of the system;
- : Improvement in the absorption of the system (in percent);
- : Improvement in the recovery time of the system (in percent);
- : The total budget limit for the system.
5.2. System Functionality Optimization under Budgetary Constraint
6. Numerical Example and Results
6.1. Hierarchical Representation of the NextGen
6.2. Using BN to Evaluate the Capability of NextGen
6.3. Using DBN to Analyze the Resilience of NextGen
- (1)
- The development failures of system , , lead to the reduction in the functionalities at the same time step T in the temporal plate;
- (2)
- The functionalities of the system , , decline to the minimum and then recover slowly; the functions of their functionality have been known.
6.4. Resilience Design under Budget Constraint
7. Discussion and Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
SoS | System-of-System |
BN | Bayesian Network |
DBN | Dynamic Bayesian Network |
LCS | littoral combat ship |
RSI | resilience system importance |
NextGen | Next Generation Air Transportation |
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Symbols of Systems | System Name | Supported Technologies |
---|---|---|
Free flight traffic management | ||
Airport surface detection equipment | , | |
En route automation modernization | ,, | |
Next Generation air-to-ground communication | – | |
Standard terminal automation replacement | , | |
Airport surveillance radar | , | |
Aviation surface weather observation network | ||
Integrated terminal weather system | , | |
Instrument flight procedures automation | ,, | |
Terminal automation modernization replacement | , | |
Automatic dependent surveillance broadcast | ,– | |
Traffic flow management infrastructure | – | |
System-wide information management | ,, | |
En route control center system modernization | , | |
Airport movement area safety system | ||
Traffic management advisor | – | |
Precision runway monitor | , | |
En route communication gateway | , |
Failure Degree | Capability Loss () | Recovery Time () | Resilience |
---|---|---|---|
Nonserious failure | 8.75 | 12 | 0.729 |
Moderate failure | 9.74 | 14 | 0.696 |
Severe failure | 10.16 | 16 | 0.635 |
System | ||||||
---|---|---|---|---|---|---|
40% | 10 | 4.23 | 3.52 | 40% | 7 | |
60% | 10 | 7.71 | 6.53 | 40% | 7 | |
50% | 10 | 6.25 | 7.45 | 40% | 7 | |
70% | 10 | 6.23 | 6.54 | 40% | 7 | |
60% | 10 | 5.22 | 5.58 | 40% | 7 | |
60% | 10 | 7.23 | 9.37 | 40% | 7 | |
50% | 10 | 6.65 | 7.16 | 40% | 7 | |
50% | 10 | 6.17 | 5.11 | 40% | 7 | |
50% | 10 | 5.64 | 6.22 | 40% | 7 | |
50% | 10 | 6.20 | 5.43 | 40% | 7 | |
70% | 10 | 7.25 | 9.41 | 40% | 7 | |
70% | 10 | 9.71 | 7.28 | 40% | 7 | |
60% | 10 | 8.28 | 7.50 | 40% | 7 | |
40% | 10 | 5.79 | 6.37 | 40% | 7 | |
40% | 10 | 4.24 | 4.44 | 40% | 7 | |
70% | 10 | 3.23 | 5.62 | 40% | 7 | |
60% | 10 | 4.82 | 6.97 | 40% | 7 | |
40% | 10 | 6.29 | 5.25 | 40% | 7 |
0.020 | 0.087 | 0.067 | 0.110 | 0.083 | 0.103 | 0.053 | 0.083 | 0.080 | 0.083 | 0.127 | 0.140 | 0.090 | 0.040 | 0.027 | 0.130 | 0.103 | 0.030 | |
10.98 | 47.60 | 36.61 | 60.41 | 45.77 | 56.56 | 29.30 | 45.77 | 43.94 | 45.77 | 69.57 | 76.89 | 49.43 | 21.97 | 14.65 | 71.40 | 56.75 | 16.48 | |
0 | 20% | 20% | 60% | 50% | 30% | 10% | 10% | 40% | 10% | 50% | 30% | 20% | 0 | 0 | 60% | 50% | 0 | |
30% | 40% | 30% | 30% | 30% | 30% | 30% | 70% | 30% | 70% | 30% | 60% | 40% | 30% | 30% | 30% | 30% | 30% | |
10.56 | 41.54 | 34.85 | 57.00 | 42.84 | 49.80 | 28.13 | 41.94 | 41.22 | 44.21 | 64.48 | 72.81 | 46.56 | 19.11 | 13.32 | 66.24 | 53.01 | 15.75 |
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Jiao, T.; Yuan, H.; Wang, J.; Ma, J.; Li, X.; Luo, A. System-of-Systems Resilience Analysis and Design Using Bayesian and Dynamic Bayesian Networks. Mathematics 2024, 12, 2510. https://doi.org/10.3390/math12162510
Jiao T, Yuan H, Wang J, Ma J, Li X, Luo A. System-of-Systems Resilience Analysis and Design Using Bayesian and Dynamic Bayesian Networks. Mathematics. 2024; 12(16):2510. https://doi.org/10.3390/math12162510
Chicago/Turabian StyleJiao, Tianci, Hao Yuan, Jing Wang, Jun Ma, Xiaoling Li, and Aimin Luo. 2024. "System-of-Systems Resilience Analysis and Design Using Bayesian and Dynamic Bayesian Networks" Mathematics 12, no. 16: 2510. https://doi.org/10.3390/math12162510