# Fire Risk Analysis in Large Multi-Compartment Structures Using a Hybrid Multiscale Approach
## 2 The Hybrid Multiscale Approach
### The Network Model
#### 2.2.2 Model Probabilities  
_Probability of barrier failure_  
The pdf of the barrier failure between a compartment \(C1\), where the fire is fully developed, and another compartment \(C2\) at time \(t\) can be written as  
\[p_{bf}(C1\to C2)=\frac{1}{\sigma_{bf}\sqrt{2\pi}}exp\left(-\frac{ \left[\left(t-t_{fo}\right)-\mu_{bf}\right]^{2}}{2\sigma_{bf}^{2}}\right) \tag{8}\]  
where \(\mu_{bf}\) and \(\sigma_{bf}\) are the mean and standard deviation of duration of barrier failure, respectively.  
The corresponding cumulative probability at time \(t\) is thus given by  
\[P_{bf}(C1\to C2)=\int_{t_{fo}}^{t}p_{bf}(C1\to C2)dt\text{ for }t_{fo}\leq t\leq t_{fo}+\tau_{fd} \tag{9}\]  
The above equation can be integrated analytically using the error function as follows:  
\[P_{bf}(C1\to C2)=\frac{1}{2}erf\left(\frac{t-t_{fo}-\mu_{bf}}{\sigma_{bf} \sqrt{2}}\right)+\frac{1}{2}erf\left(\frac{\mu_{bf}}{\sigma_{bf}\sqrt{2}} \right)\text{ for }t_{fo}+\tau_{fd} \tag{10}\]  
_Probability of a fully developed fire_  
If ignition occurred in compartment \(C1\), the fire may grow to a fully developed fire. If the hot layer temperature is less than the critical value for flashover (i.e., 550 \({}^{\circ}\)C), flashover would not be expected to occur in the compartment, and the probability of fire growth to a fully developed fire is zero. Otherwise, flashover can occur in the compartment.  
The pdf of fire growth to a fully developed fire at time \(t\) is  
\[p_{fd}(C1)=\frac{1}{\sigma_{fo}\sqrt{2\pi}}exp\left(-\frac{\left[\left(t-t_{ ig}\right)-\mu_{fo}\right]^{2}}{2\sigma_{fo}^{2}}\right) \tag{11}\]  
When flashover occurred in the compartment, the cumulative probability can be calculated as:  
\[P_{fd}(C1)=\left\{\begin{array}{ll}\int_{t_{ig}}^{t}p_{fd}(C1)dt&for\ t_{ig} \leq t\leq t_{ig}+\tau_{fo}+\tau_{fd}\\ &0\text{ }otherwise\end{array}\right. \tag{12}\]  
where \(\mu_{fo}\) and \(\sigma_{fo}\) are the mean and standard deviation of the flashover time in the compartment, respectively.  
When flashover does not occur in the compartment, the cumulative probability is deduced from Equation (12). Then, a random number \(R\) between 0 and 1 is generated and compared to the computed cumulative probability. If \(R\leq P_{fd}\) (\(C1\)), flashover occur in \(C1\) and the flashover time \(t_{fo}\) is equal to the current time \(t\). Otherwise, flashover does not occur at time \(t\) in \(C1\).  
_Probability of ignition_  
Given the A3 assumption, the probability of ignition of \(C2\) by \(C1\) is expressed as  
\[P_{ig}(C1\to C2)=P_{bf}(C1\to C2)\times P_{fd}(C1) \tag{13}\]  
The statistical approach consists in drawing a random number \(0<R<1\) and comparing \(R\) to the probability of ignition. If \(R\leq P_{ig}(C1\to C2)\), then ignition occurs in \(C2\) and the ignition time \(t_{ig}\) is equal to the current time \(t\). If a target compartment is connectedto several fully developed fire compartments, the probability of ignition increases due to the heat transferred from all adjacent fire compartments. For example, if C2 has two adjacent fire compartments C1 and C3, the probability of C2 ignition becomes  
\[P_{ig}(C1\to C2\;or\;C3\to C2)=P_{ig}(C1\to C2)+P_{ig}(C3\to C2)-P_{ig}(C1\to C2) \times P_{ig}(C3\to C2) \tag{14}\]  
_Fire spread probability_  
According to the assumption A5, the probability of fire spread from C1 to C2 is equal to  
\[P_{spread}(C1\to C2)=P_{ig}(C1\to C2)\times P_{fd}(C2) \tag{15}\]  
where \(P_{ig}(C1\to C2)\) is the probability of ignition of C2 by C1, and \(P_{fd}(C2)\) is the probability of fully developed fire in C2.