## 3 CPS Resilience Quantitative Assessment Framework
### Cyber Layer: Security Architecture and Modeling
#### 3.3.2 Markov Process Modeling
\begin{split} G_{1}&=(2\lambda\mu_{3}{{\mu_{2}}^{2}}{{ \mu_{4}}^{2}}+{\mu_{3}}{{\mu_{4}}{\mu_{2}}^{2}}\lambda\ ^{2}+2\mu_{3}{{\mu_{4}}{\mu_{2}}^{2}}\lambda^{2}+\mu_{1}{{\mu_{3}}}{{\mu_{4}}{ \mu_{2}}^{2}}\lambda+2\mu_{3}\lambda^{2}{{\mu_{2}}{\mu_{4}}^{2}}\\ &\quad+\mu_{1}{{\mu_{3}}}{{\mu_{2}}}{{\mu_{4}}}\lambda+2\mu_{2}{{ \mu_{3}}}{{\mu_{4}}}\lambda^{3}+\mu_{1}{{\mu_{2}}}{{\mu_{3}}}{{\mu_{4}}}\lambda^ {2})\\ G_{2}&=(\lambda\mu_{3}{{\mu_{2}}^{2}}{{\mu_{4}}^{2}}+ \mu_{3}\mu_{4}{{\mu_{2}}^{2}}\lambda\ ^{2}+\mu_{3}\lambda^{2}{{\mu_{2}}}{{\mu_{4}}^{2}}+\mu_{2}{{\mu_{3}}}{{\mu_{4}} }\lambda^{3})\\ G_{3}&=2(\sigma\mu_{3}\lambda^{2}\mu_{2}{{\mu_{4}}^{2 }}+\sigma\mu_{2}\mu_{3}\mu_{4}\lambda^{3})\\ G_{4}&=2(\mu_{2}\mu_{3}\mu_{4}\lambda^{3}+\mu_{3}\mu_ {4}\mu_{2}\lambda^{2}-\sigma\mu_{3}\mu_{4}{{\mu_{2}}^{2}}\lambda^{2}-\sigma \mu_{2}\mu_{3}\mu_{4}\lambda^{3})\\ G_{5}&=6(\sigma^{2}{{\mu_{4}}^{2}}{{\mu_{2}}^{3}} \lambda^{5}+\sigma^{2}\mu_{4}\mu_{2}\lambda^{4})\\ G_{6}&=6(\sigma{{\mu_{3}}}{{\mu_{4}}^{2}}\lambda^{3 }+3\sigma\mu_{3}\mu_{4}\lambda^{4}+2\sigma\mu_{2}\mu_{3}\mu_{4}\lambda^{3}- \sigma^{2}\mu_{3}{{\mu_{4}}^{2}}\lambda^{3}\\ &\quad-3\sigma^{2}\mu_{3}\mu_{4}\lambda^{4}-2\sigma^{2}\mu_{2}\mu _{3}\mu_{4}\lambda^{3})\\ G_{7}&=6(2\sigma^{2}\mu_{3}\mu_{2}\lambda^{3}+2 \sigma^{2}\mu_{3}\mu_{2}\lambda^{4}-3\sigma\mu_{3}{{\mu_{2}}^{2}}\lambda^{3}- 3\sigma\mu_{3}\mu_{2}\lambda^{4}\\ &\quad+\mu_{3}{{\mu_{2}}^{2}}\lambda^{3}+\mu_{3}\mu_{2}\lambda^{ 4})\\ G&=G_{1}+3G_{2}+3G_{3}+3G_{4}+G_{5}+G_{6}+G_{7}\end{split}  \tag{10}\]  
From the resulting steady-state probabilities, several resilience-related evaluation metrics can be defined:  
(1) Steady-State Availability Probability (_AP_): This metric reflects the long-term stability of the system's operational capacity under cyber attacks and other adverse conditions. It is defined as the total probability of the system being in any of the available states:  
\[AP=\sum\limits_{i\in S_{\text{avail}}}p_{i} \tag{11}\]  
A higher availability probability indicates stronger system continuity and operational robustness. (2) Steady-State Degradation Probability (DP) and Steady-State Escape (Failure) Probability (EP): These two metrics measure the system's ability to mitigate threat accumulation and limit the extent of damage. Specifically, DP represents the probability of entering a degraded but observable state, while EP corresponds to stealthy or undetected failure states. Since the system can only be in one of three mutually exclusive states--available, degraded, or failed--the following holds: \(AP+EP+DP=1\). Lower values of EP and DP indicate better resilience in terms of failure prevention and threat containment. (3) Steady-State Abnormal Behavior Perception Probability (ABP): This metric captures the system's ability to detect anomalies during abnormal or failed conditions. It is defined as the proportion of detectable failure and degradation states among all non-available states:  
\[ABP=\frac{\sum\limits_{i\in S_{\text{aware}}}p_{i}}{1-AP} \tag{12}\]  
A higher ABP reflects stronger perception and detection capabilities, which are crucial for timely response and recovery in adversarial environments.