# Applying Real-Time Travel Times to Estimate Fire Service Coverage Rate for High-Rise Buildings
## 2 Methods
### Dynamic Estimation Model  
Based on the minimum fire apparatus requirement for high-rise buildings, the dynamic model for estimating the effective coverage rate of the fire service for high-rise buildings can be established. Due to the economic constraints, some fire stations are not equipped with both water fire trucks and aerial fire trucks. Therefore, the travel times and distances of water fire trucks and aerial fire trucks arriving at the fire scene may be different. It is assumed that there are \(k\) water fire trucks, \(m\) aerial fire trucks, and \(n\) high-rise buildings in the region. Indeed, high-rise buildings can be regarded as fire service demand points. \(P_{i}\) represents the \(i\)th fire service demand point (high-rise building), and \(T_{\min}(P_{i}\), \(WT)\) represents the travel time of the first arrived water fire truck to the \(P_{i}\). \(T_{\min}(P_{i}\), \(WT)\) can be obtained by searching for the minimum value of travel times of water fire trucks, as Equation (1). Once \(T_{\min}(P_{i}\), \(WT)\) is searched, the corresponding travel distance \(D(P_{i}\), \(WT)\) is also determined. Then the corresponding travel velocity \(V(P_{i}\), \(WT)\) is equal to \(D(P_{i}\), \(WT)\) divided by \(T_{\min}(P_{i}\), \(WT)\) as Equation (2). Similarly, the travel time of the first arrived aerial fire truck to \(P_{i}\) (\(T_{\min}(P_{i}\), \(AT)\)) can be obtained as Equation (3),and its corresponding travel distance is represented by \(D(P_{i},AT)\). Subsequently, \(V(P_{i},AT)\) is calculated as Equation (4). \[T_{\min}(P_{i},WT)=\min\{T(P_{i},WT_{1}),T(P_{i},WT_{2}),T(P_{i},WT_{3})...T(P_{i },WT_{k})\} \tag{1}\]  
\[V(P_{i},WT)=\frac{D(P_{i},WT)}{T_{\min}(P_{i},WT)} \tag{2}\]  
\[T_{\min}(P_{i},AT)=\min\{T(P_{i},AT_{1}),T(P_{i},AT_{2}),T(P_{i},AT_{3})...T(P _{i},AT_{m})\} \tag{3}\]  
\[V(P_{i},AT)=\frac{D(P_{i},AT)}{T_{\min}(P_{i},AT)} \tag{4}\]  
The earlier the fire truck arrives at the fire scene, the more conducive it is to put out the high-rise building fire. In general, the fire truck arriving first at the fire scene plays a more important role than the fire truck arriving later. Therefore, the minimum value of \(T_{\min}(P_{i},WT)\) and \(T_{\min}(P_{i},AT)\) should contribute more to the complex travel time of a water fire truck and aerial fire truck (\(T(P_{i},C)\)). In Equation (5), \(C_{1}\) and \(C_{2}\) represent the weight coefficients of \(T_{\min}(P_{i},WT)\) and \(T_{\min}(P_{i},AT)\), respectively. Both water fire trucks and aerial fire trucks are indispensable in fire-fighting and the rescue of high-rise buildings. Therefore, the difference between \(C_{1}\) and \(C_{2}\) should be small, and we assumed that the values of \(C_{1}\) and \(C_{2}\) are 0.4 or 0.6 in this study. Indeed, when the number of fire stations equipped with water fire trucks is similar to the number of fire stations equipped with aerial fire trucks, the difference between the values of \(C_{1}\) and \(C_{2}\) will have little effect on the final result. If \(T_{\min}(P_{i},WT)\leq T_{\min}(P_{i},AT)\), \(C_{1}=0.6\) and \(C_{2}=0.4\). If \(T_{\min}(P_{i},WT)>T_{\min}(P_{i},AT)\), \(C_{1}=0.4\) and \(C_{2}=0.6\). Similarly, the complex travel distance of a water fire truck and aerial fire truck (\(D(P_{i},C)\)) can be calculated as Equation (6), and the complex travel velocity of them (\(V(P_{i},C)\)) equals \(D(P_{i},C)\) divided by \(T(P_{i},C)\) as Equation (7). Obviously, the average travel time (\(AT(C)\)), the average travel distance (\(AD(C)\)) and the average travel velocity (\(AV(C)\)) for all points can be calculated as Equations (8)-(10).