# Walking Speed in a Motorbike Lane Considering the Density of Evacuees and Motorbikes
## 4 Influence of Motorbike Density on Walking Speed  
### Walking Speed in the Motorbike Section and Modeling  
In this section, we further clarify the influence of motorbike density on walking speed and discuss the modeling process. As demonstrated in Figure 4 (red section), there were 10 motorbikes set up in the motorbike section. The section lengths (\(L\)) with motorbikes were set as 10, 15, 20, and 30 m depending on the experiment cases. Considering the width (\(w\)) of the motorbike lane, the motorbike density (\(\rho_{b}\)) can be expressed as \(\rho_{b}=N/(wL)\).  
Moreover, the function \(V=V_{0}\,f(\rho_{e})\) in Section 3.1 further considers the influence of motorbike density, which can be rewritten in the following form:  
\[V=V_{0}\,f(\rho_{e})g(\rho_{b}), \tag{9}\]  
where the function \(g(\rho_{b})\) is assumed to represent the influence of motorbike density, which can also be rewritten as \(V/(V_{0}\,f(\rho e))\). Since the \(f(\rho_{e})\) was calculated in Sections 3.1 and 3.2, and the evacuee density in the motorbike section was around 0.06-0.79 person/m\({}^{2}\), we could further apply Equations (5) and (7) to Equation (9). In addition, to establish a criterion of walking speed not affected by \"motorbike density (\(\rho_{b}\))\", we also considered \(V_{0}\) as the average walking speed (1.45 m/s), as mentioned in Section 3.1. The relationship between function \(g(\rho_{b})\) and the reciprocal of \(\rho_{b}\) is shown in Figure 12.  
Figure 12 indicates that \(g(\rho_{b})\) could be mainly divided into four groups, as four motorbike densities were considered in experiments. Moreover, the distribution of data points in Figure 12 reveals that \(g(\rho_{b})\) gradually decreased with the increase in motorbike density (i.e., \(1/\rho_{b}\) decreased), and the tendency seemingly showed an exponential relationship. Thus, we further assumed function \(g(\rho_{b})\) to be in the same form as in Equation (4) and conducted a nonlinear regression by applying the least square method to estimate parameters in the  
function \(g(\rho_{b})\). The regression curve is shown as a black line in Figure 12, and the regression function was generated as follows:  
\[g(\rho_{b})=\left(1-1.14e^{-\frac{0.55}{\rho_{b}}}\right). \tag{10}\]  
Apparently, the regression function was well fitted by the mean of function \(g(\rho_{b})\) for each motorbike density. However, for the same motorbike density, the difference in individual walking speed persisted.  
Moreover, we mainly focused on actual motorbike density in an accident and a relatively lower density than an actual accident, according to the investigation in Section 2.2. When a tunnel fire accident occurs, an extreme situation is that motorbikes are probably congested in the lane such that people cannot walk in the motorbike lane; however, such motorbike density conditions were not explored in the present experiments. Whether the current regression function can explain the walking speed under the condition of high motorbike density is unclear. Therefore, the regression function in Figure 12 is only expressed until the reciprocal of \(\rho_{b}\) is 2 (i.e., the motorbike density is equal to 0.50 motorbikes/m\({}^{2}\)).