## 3 Methodology
### Congestion Assessment Methodology  
A method was developed to determine the level of congestion that was extant at the time of each crash. As mentioned earlier, the recurrent bottleneck identification approach developed by [PERSON] et al. [11] was employed. It makes use of a Congestion Index (CI), an Average Historic Congestion Index (AHCI), and a Recurrent Bottleneck Location Identification (RBLI). The congestion index, \(Cl(i,\,t,\,m)\), labels each segment \(i\) as being congested or uncongested at time \(t\), as illustrated in Figure 2b. The congestion value for each segment is determined. If the congestion value is below a threshold \(\alpha\), segment (\(i\)) is classified as being congested at time \(t\). Here, a value of 80% was used as the threshold. This value is consistent with the ratio of (the speed at capacity) to (the speed at free flow) presented in the US Highway Capacity Manual [34]. The Average Historic Congestion Index, \(AHCl(i,\,t)\), is defined as the faction of (week or all) days in the study period \(T\) (typically one or two years) where segment \(i\) was congested at time \(t\), based \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \multirow{2}{*}{**Crash ID**} & **Vehicle** & \multirow{2}{*}{**Severity**} & \multirow{2}{*}{**Direction**} & \multirow{2}{*}{**Crash Day**} & \multirow{2}{*}{...} & \multicolumn{2}{c}{**Crash**} & \multirow{2}{*}{**Locality**} & \multirow{2}{*}{**Weather**} \\  & & & & & & & & & & & \\
1034169 & VEH & O (No injury) & Westbound & Tuesday &... & Rear-end & Rural & Clear \\
4891526 & CMV & K (death) & Northbound & Thursday &... & Animal & Urban & Wet \\ : & : & : & : & : & : & : & : & : & : \\ \hline \end{tabular}
\end{table} 
Table 1: The sample of Traffic Engineering Accident Analysis System (TEAAS) crash data. Figure 2: Traffic flow contours: (**a**) congestion value; (**b**) congestion index contour. on the specified congestion index ((_CI_(_i_, \(t\), _m_))). _AHCI_ is the key parameter for identifying recurrent congestion. It is used to denote the probability that segment \(i\) was congested at time \(t\). AHCI contour maps are used to define the recurrent bottlenecks as well as their influence areas. To illustrate, Figure 3 shows three different patterns that were observed in Spring and Fall seasons of 2013 in North Carolina. Each block corresponds to an AHCI value at a TMC segment at a certain time of day. A TMC with the highest AHCI (more than 50%) value a given time of day is defined as a recurrent bottleneck segment, \(\beta\), which was proposed from [PERSON] et al. [11]. Figure 3: Spatial patterns of Average Historic Congestion Index (AHCI): (**a**) Spatial Pattern 1 of AHCI (Westbound on I-40, 2013); (**b**) Spatial Pattern 2 of AHCI (Northbound on I-85, 2013); (**c**) Spatial Pattern 3 of AHCI (Eastbound on I-40, 2013). The question then is what other nearby TMC segments are also in a recurrent bottleneck area? In reference to Figure 3a, the TMC with the highest AHCI above 50% (TMC A) is located just upstream of a significant drop in AHCI value. Therefore, this TMC segment is clearly in a recurrent bottleneck and this pattern is defined as Pattern 1. In Figure 3b, the TMC segment with the highest AHCI value (TMC B) is not located immediately upstream of a significant drop in the AHCI value. However, TMC B is also part of the bottleneck area and this pattern is defined as Pattern 2. In Figure 3c, the TMC segment with the highest AHCI value (greater than 50%) is in the bottleneck area (TMC C); however, a significant drop is not observed in the AHCI value for this TMC segment or the one downstream of it. In addition, the downstream TMC segment has an AHCI value below 50%. Therefore, the bottleneck area is defined by only TMC C and this pattern is defined as Pattern 3. In this study, the bottleneck influence area includes all the TMC segments upstream of a bottleneck area with AHCI values greater than 20%, with the assumption that congestion occurs on at least one of every five weekdays.