## 4 Analysis
### System Dynamics Risk Modelling, Sensitivity and Uncertainty Analysis  
The assessment of risk in the SD model allows for an integrated analysis of HOE factors and their nonlinear interactions and feedback loops. The SD model also accounts for the delays and more realistic analysis of risk variation over time. When maintenance of a safety system is not performed for prolonged periods, risk will trend upwards over time and there can be a duration in which risk exceeds a critical or safe value. The SD model identifies the point in which the maintenance regime of safety systems needs to be conducted. System dynamics describe the level of uncertainty of diverse situations. This technique is specifically useful when variables are interlinked, and data is indistinct. In this model, some variables vary with time and simultaneously interact with other variables. Thus, the state of a variable is both time dependent and state dependent with respect to other variables. The time slice in a SD model is a snapshot of the BN at different instances of time (Figure 17). The conditional probability table (CPT) is the transition matrix that represents the time slice and provides insights into the transformation of the different nodes across the model and describe the causal relationships within the nodes. The mathematical model describing the state and time dependency is given by:  
\[P(X,Y)=\prod_{t=0}^{T}P(x_{t}|x_{t-1})\prod_{t=0}^{T}P(y_{t}|x_{t})P(x_{0}) \tag{3}\]  
where:  
\(X\), \(x_{t}\), \(x_{t-1}\) are state variables; \(Y\), \(y_{t}\), \(y_{t-1}\) are observable variables;  
\(P(x_{t}|x_{t-1})\) gives time dependencies between states;  
\(P(y_{t}|x_{t})\) gives state dependencies between the variables;  
\(P(x_{0})\) is initial state distribution. In the SD model, the flow variables are time-varying terms. For example, the rate of change (RoC) of the number of checks (NoC)) for perceived safety (ps) is given by:  
\[RoC(t)=5\ \frac{d}{dt}(ps) \tag{4}\]  
On the other hand, stock variables, refer to the integrated value of the flow variables. Thus, a stock variable refers to the accumulated value of the flow variable in a given time frame. For the above example, a stock variable NoC is given by:  
\[NoC=12+\ \int RoC(t)dt \tag{5}\]  
In the present analysis, random perturbations on input parameters are performed and risk is computed at each of the time instants. The nodes of the SD are mapped from the corresponding nodes in the BN model. The mapped SD model is represented in Figure A5 in Appendix A. A sample schematic showing reliability change due to time and state  
Figure 17: SD model at three different time instances showing variable dependencies. change is shown in Figure 18. The SD model, thus, brings out the effects of deficient training and inefficient emergency plan from the analysis. The results from the SD simulation are reported in Figure 19 which compare the DTS to the performance solutions for each design scenario for Case #1 (ABCB). During the 10-year life span considered, the performance solution shows a lower level of risk than the DTS solution. It can also be noted that there is no variation in risk for the DTS solution over 10 years; this is because the DTS solution has no active fire protection measures (detection or suppression system), hence no HOEs can significantly alter the level of risk. However, outside the active protection system, it is always possible that  
Figure 19: SD result- DTS vs. Performance solution. Risk profile over 10-year period (Case #1). Figure 18: Sample schematic showing time and state affecting reliability. HOEs reduce the reliability of passive protection systems (for example, obstructions of the exits and refurbishment activities) but they are not modelled in the T-H-O-Risk model. In the risk-over-time curve related to the performance solutions, the level of risk reduces after a seven year-long period of stability, because the building maintenance team has developed a thorough knowledge of the reliability of safety systems.