## 2 Integrated Predictive Traffic Resilience Framework
### Road Traffic Performance under Uncertainty  
A variety of traffic condition measures are available and are employed by different transportation departments [7]. In practice, the level of service (LoS) determined by the Highway Capacity Manual (HCM) has become widely adopted in measuring traffic conditions as it accounts for variabilities with peak hour factor (PHF) [21; 22; 23]. To determine the LoS of a roadway, the volume-to-capacity ratio (\(V/C\)) is defined by  
\[V/C=\frac{Hourly\ traffic\ volume}{Roadway\ capacity} \tag{1}\]  
where \(V\) is the hourly traffic volume, and \(C\) is the allowable roadway capacity. The LoS of a roadway can be described with respect to the \(V/C\) ratios [7], as shown in Table 1.  
The methods to calculate LoS vary with different constructions of roadways, such as freeways, multi-lane highways, two-lane highways, arterials, and intersections [21]. Although \(V/C\) is widely employed in measuring road traffic conditions, it is limited to measuring non-recurring congestion due to the uncertainty associated with the non-recurring congestion scenario. Since traffic conditions during an evacuation are associated with non-recurring congestion, the measurement must include uncertainty factors related to the disaster risk. Thus, the traffic volume during the evacuation under extreme weather can be redefined as  
\[V(t)=(1+\alpha)V_{0}(t) \tag{2}\]  
where \(V(t)\) is the hourly volume at time \(t\) during the evacuation period, \(V_{0}\) is the average hourly volume at time \(t\) over the past years, and \(\alpha\) is the scaling parameter. The value of \(\alpha\) can be estimated by analyzing the average change in volume from previous years at  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**LoS Class** & **Traffic State and Condition** & **\(V/C\) Ratio** \\ \hline A & Free flow & 0–0.60 \\ B & Stable flow with unaffected speed & 0.61–0.70 \\ C & Stable flow but speed is affected & 0.71–0.80 \\ D & High-density but stable flow & 0.81–0.90 \\ E & Traffic volume near or at capacity level with low speed & 0.91–1.00 \\ F & Breakdown flow & \textgreater{}1.00 \\ \hline \hline \end{tabular}
\end{table}
Table 1: LoS indicator based on the \(V/C\) ratios.  
the same time. In this case, scaling parameters can account for traffic uncertainty during evacuation scenarios. This is because traffic measurement often has a known uncertainty associated with it. Thus, the traffic volume measurement during the evacuation can be scaled by \(\alpha\), which represents the uncertainty. In this way, the uncertainty can be propagated through calculations involving the scaled variable.  
Moreover, the roadway capacity can also be affected due to lane closure, traffic accidents, or other potential damage during evacuation. The redefined road capacity can be written as  
\[C(t)=(1-\beta)C_{0}(t) \tag{3}\]  
where \(C(t)\) is the roadway capacity at time \(t\) during the evacuation period, \(C_{0}(t)\) is the maximum roadway capacity during a normal traffic period, and \(\beta\) is the scaling parameter. The value of \(\beta\) can be estimated by analyzing current and historical data. Similar to the traffic volume during the evacuation, the roadway capacity measurement is often associated with known uncertainty. Thus, it can be scaled by \(\beta\) to ensure that the uncertainty can be propagated through calculations. Introducing the scaling parameters in Equations (2) and (3) into Equation (1) yields the updated \(V\)/\(C\) ratio as  
\[\frac{V(t)}{C(t)}=\frac{(1+\alpha)V_{0}(t)}{(1-\beta)C_{0}(t)} \tag{4}\]  
The calculation of \(\alpha\) and \(\beta\) serves the purpose of capturing the relationship between \(V(t)\) and \(C(t)\) of the traffic performance. While it may initially appear that only \(V(t)\) and \(C(t)\) are sufficient for quantifying traffic resilience, the inclusion of \(\alpha\) and \(\beta\) is crucial. These parameters enable a more comprehensive understanding of how well the system can handle traffic variabilities and uncertainties in the predicted traffic demand.