# Comparison of Traffic Flow Models with Real Traffic Data Based on a Quantitative Assessment
## 2 Theoretical Background
### Modeling Equilibrium Traffic Flow Relationships
#### 2.1.1 Single-Regime Traffic Flow Models
Similarly to [PERSON] or [PERSON], the exponential function was also applied by [PERSON], [PERSON]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Author (Year of Publication)** & **Basic Function** & **Parameters** \\ \hline [PERSON] (1935) & \(\mathrm{v}=\mathrm{v}_{\mathrm{sw}}\left(1-\frac{\mathrm{k}}{\mathrm{k}_{ \mathrm{max}}}\right)\) & \(\mathrm{v}_{\mathrm{sw}},\mathrm{k}_{\mathrm{max}}\) \\ [PERSON] (1959) & \(\mathrm{v}=\mathrm{v}_{\mathrm{opt}}\ln\left(\frac{\mathrm{k}_{\mathrm{max}}} {\mathrm{k}_{\mathrm{opt}}}\right)\) & \(\mathrm{v}_{\mathrm{opt}},\mathrm{k}_{\mathrm{max}}\) \\ [PERSON] (1960) & \(\mathrm{v}=\mathrm{v}_{\mathrm{sw}}\exp\left(-\frac{\mathrm{k}}{\mathrm{k}_{ \mathrm{opt}}}\right)\) & \(\mathrm{v}_{\mathrm{sw}},\mathrm{k}_{\mathrm{opt}}\) \\ [PERSON] (1961) & \(\mathrm{v}=\mathrm{v}_{\mathrm{sw}}\left(1-\exp\left(-\frac{\mathrm{a}}{\mathrm{ i}_{\mathrm{max}}}\left(\frac{1}{\mathrm{k}_{\mathrm{max}}}\right)\right)\right)\) & \(\mathrm{v}_{\mathrm{sw}},\mathrm{k}_{\mathrm{max}},\mathrm{a}\) \\ [PERSON] (1967) & \(\mathrm{v}=\mathrm{v}_{\mathrm{sw}}\left(1-\left(\frac{\mathrm{a}}{\mathrm{ k}_{\mathrm{max}}}\right)^{n}\right)\) & \(\mathrm{v}_{\mathrm{sw}},\mathrm{k}_{\mathrm{max}},\mathrm{n}\) \\ [PERSON](1967) & \(\mathrm{v}=\mathrm{v}_{\mathrm{sw}}\exp\left(-\frac{1}{2}\left(\frac{\mathrm{k}}{ \mathrm{k}_{\mathrm{opt}}}\right)^{2}\right)\) & \(\mathrm{v}_{\mathrm{sw}},\mathrm{k}_{\mathrm{opt}}\) \\ [PERSON] (1965) & \(\mathrm{v}=\mathrm{v}_{\mathrm{sw}}\left(1-\left(\frac{\mathrm{k}}{\mathrm{k}_{ \mathrm{max}}}\right)^{n+1}\right)\) & \(\mathrm{v}_{\mathrm{sw}},\mathrm{k}_{\mathrm{max}},\mathrm{n}\) \\ [PERSON] (1980) & \(\mathrm{v}=\mathrm{v}_{\mathrm{sw}}\left(1-\frac{\mathrm{k}}{\mathrm{k}_{ \mathrm{max}}}\right)^{4}\) & \(\mathrm{v}_{\mathrm{sw}},\mathrm{k}_{\mathrm{max}}\) \\ [PERSON] and [PERSON] (1995) & \(\mathrm{v}=\mathrm{v}_{\mathrm{sw}}\left(1+\exp\left(\frac{\frac{\mathrm{a}}{ \mathrm{i}_{\mathrm{max}}}-\mathrm{a}\mathrm{b}}{\mathrm{b}\mathrm{b}\mathrm{b} \mathrm{b}}\right)-3.72\times 10^{-6}\right)\) & \(\mathrm{v}_{\mathrm{sw}},\mathrm{k}_{\mathrm{max}}\) \\ [PERSON] (1995) & \(\mathrm{v}=\mathrm{v}_{\mathrm{sw}}(1-\exp\left(\frac{\mathrm{i}_{\mathrm{i}_{ \mathrm{min}}}}{\mathrm{v}_{\mathrm{sw}}}\left(1-\frac{\mathrm{b}}{\mathrm{k}_{ \mathrm{i}_{\mathrm{max}}}}\right)\right)\) & \(\mathrm{v}_{\mathrm{sw}},\mathrm{k}_{\mathrm{max}},\mathrm{C}_{\mathrm{i}}\) \\ [PERSON] (1995) & \(\mathrm{v}=\mathrm{v}_{\mathrm{sw}}\left(\frac{\mathrm{k}_{\mathrm{min}}-\mathrm{ k}^{\mathrm{a}}}{\mathrm{b}\mathrm{b}\mathrm{b}\mathrm{b}\mathrm{b}}\right)\) & \(\mathrm{v}_{\mathrm{sw}},\mathrm{k}_{\mathrm{max}},\mathrm{n},\mathrm{m}\) \\ [PERSON] (2008) & \(\mathrm{v}=\mathrm{v}_{\mathrm{min}}+\left(\frac{\mathrm{v}_{\mathrm{sw}}-\mathrm{ v}_{\mathrm{min}}}{\left(\frac{\mathrm{k}_{\mathrm{i}_{\mathrm{max}}}}{1+\mathrm{v}_{ \mathrm{sw}}}\right)^{5}}\right)\) & \(\mathrm{v}_{\mathrm{sw}},\mathrm{v}_{\mathrm{min}},\mathrm{k}_{\mathrm{opt}},\mathrm{a}, \mathrm{b}\) \\ [PERSON] (2011) & \(\mathrm{v}=\frac{\mathrm{vand [PERSON] [2], and [PERSON] and [PERSON] [15]. [PERSON]'s exponential model, similarly to [PERSON] model, was deduced from the car-following theory. The model satisfies both BC1 and BC2. Speed in the model falls rapidly as density increases, which is considered a limitation [4]. [PERSON] and [PERSON] [2] assumed that vehicle flow is strongly affected by the parameter of kinematic wave speed at jam density C\({}_{\text{j}}\)--the parameter was included in the model. [PERSON] and [PERSON] [15] proposed a model based on two parameters extended by three constant numerical values that cannot be interpreted in terms of physical parameters and may be seen as a downside. [PERSON]'s model [16] was derived from a microscopic model of following the leader and combines [PERSON] and [PERSON] model into a single-regime model [17]. The advantage of the model is that it ensures a good match to data for a wide variety of facility types and for entire ranges of densities.