# Coordination in Emergency Response System Design:
## 3 Modeling the Response Times  
In this section, we focus on emergency response planning with two resources, i.e., fire stations and ambulance stations, for the ease of exposition. Our aim is to lay the groundwork for the mathematical formulation of the problem in the next Section. In the event of an emergency, at any node on the network, the closest fire and ambulance stations will be informed simultaneously, and they _will_ dispatch response teams via the shortest routes to the scene. At any node \(i\), we assume that, if the response time of the fire station (\(RTFS_{i}\)) is greater than the response time of the ambulance station (\(RTAS_{i}\)), then the EMTs will have to wait for the firemen, and vice versa. The amount of time one of the teams would be waiting is \(WT_{i}=|RTFS_{i}-RTAS_{i}|\). Let \(RTFS\), \(RTAS\), and \(\overline{WT}\) denote the prespecified targets for the response and wait times across the road network.  
Figure 1a, where each node is depicted in terms of its (\(RTFS_{i}\), \(RTAS_{i}\)), enables us to categorize the level of collaboration among the ambulances and firetucks. Clearly, the highest level of collaboration is when the node is on the \(45^{\circ}\) line, e.g., the star on line 2 in Figure 1a, where \(WT_{i}=0\). The figure is divided into four quadrants using \(\overline{RTFS}\) and \(\overline{RTAS}\)  
\begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline \multirow{2}{*}{**Paper**} & \multicolumn{4}{c}{**Network**} & \multicolumn{4}{c}{**Sub-Problem**} \\  & **Rural** & **Highway** & **Rail** & **Urban** & **Covering** & **Location** & **Evacuation** & **Routing** \\ \hline [PERSON] and [PERSON] (1988) [24] & \(\times\) & & & & & \(\times\) & & \\ [PERSON] et al. (2007) [26] & \(\times\) & & & & \(\times\) & & \(\times\) \\ [PERSON] and Androutsopoulos (2008) [27] & \(\times\) & & & & \(\times\) & \(\times\) & \(\times\) \\ [PERSON] et al. (2017) [28] & & \(\times\) & & & & \(\times\) & \(\times\) & \(\times\) \\ [PERSON] and [PERSON] (2019) [29] & & & & \(\times\) & \(\times\) & \(\times\) & & \\ Mohabbati-Kalejahi and Vinel (2021) [30] & & \(\times\) & & \(\times\) & \(\times\) & & \\ [PERSON] (2022) [31] & & & & \(\times\) & & \(\times\) & \(\times\) & \\ [PERSON] et al. (2023) [32] & & & \(\times\) & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 2: Emergency response for hazmat transportation.  
Line 1 and line 3 have \(\overline{WT}\) as intercept and a 45\({}^{\circ}\) slope. Thus, \(WT_{i}>\overline{WT}\) at both the circle and the square in Figure 1a, where the firemen wait in the former and the EMTs wait in the latter. The shaded area in Figure 1b depicts the poor, adequate, and preferred response time areas. If a node is in a preferred response time area (dark shaded), then its \(RTFS_{i}\), \(RTAS_{i}\), and \(WT_{i}\) are all below their respective targets. By contrast, the light shaded area in Figure 1b represents the poor response time area, where both response times are above their respective targets. We deem the dotted area in Figure 1b to be an adequate response: although the wait time is above its target, the two response times are in the preferable range. The aim is to maximize the number of nodes moved to the preferred response time area by the construction of new ambulance and fire stations.