# 用户异质下铁路交通通勤者出行成本The Travel Cost of Rail Commuters under Heterogeneous Users

Abstract: In the railway transportation system, first of all, on the basis of considering the choice of departure time, commuters are divided into two groups according to the difference in the parameters of early arrival penalty and physical contact congestion. The travel behavior of the two groups of commuters when riding the subway is studied. This derives the travel costs of two groups of heterogeneous users when riding the railway. Secondly, the travel behavior of two groups of heterogeneous users under dual-sites is studied, and the travel costs of heterogeneous users under dual-sites are derived from this. Studies have shown that when considering the existence of heterogeneous differences, the travel cost of commuters is lower than the travel cost under the homogeneity assumption, and the greater the heterogeneity, the lower the cost, and as the number of commuters increases growth, the impact of heterogeneous factors on travel costs will also become more significant. Finally, the results of the calculation example also show that the heterogeneity factors need to be paid attention to when calculating the railway travel cost.

1. 引言

2. 同质性用户铁路出行模型

$r\left(t\right)$ 为铁路交通线路上的通勤者的离开率， $\left[{t}_{s},{t}_{e}\right]$ 为铁路交通线路上通勤者选择的出行峰期，显然有 $r\left(t\right)=0,\forall t\notin \left[{t}_{s},{t}_{e}\right]$。假设 $\xi$ 为在 $\left[{t}_{s},{t}_{e}\right]$ 期间调度列车的时间间隔，且在本文中，为了便于研究通勤行为，我们遵循Wu和Huang [16] 中的假设，即在一个列车调度间隔期间，铁路交通线路通勤者的离开率是连续的，并且在一个时间间隔内的出发速率基本保持不变，也就是说，搭乘在 ${t}_{0}$ 时刻开出的列车的

$C\left(t\right)=\left\{\begin{array}{l}\alpha T+\beta \left(\stackrel{˜}{t}-t\right)+\psi \xi Tr\left(t\right)+F,\text{ }t\in \left[{t}_{s},\stackrel{˜}{t}\right]\\ \alpha T+\gamma \left(\stackrel{˜}{t}-t\right)+\psi \xi Tr\left(t\right)+F,\text{ }\text{ }\text{ }t\in \left[\stackrel{˜}{t},{t}_{e}\right]\end{array}$ (1)

$C=\alpha T+\beta \left(\stackrel{˜}{t}-{t}_{s}\right)+F=\alpha T+\gamma \left({t}_{e}-\stackrel{˜}{t}\right)+F$ (2)

$r\left(t\right)=\left\{\begin{array}{l}\frac{\beta \left(t-{t}_{s}\right)}{\psi \xi T},\text{ }t\in \left[{t}_{s},\stackrel{˜}{t}\right]\\ \frac{\gamma \left({t}_{e}-t\right)}{\psi \xi T},\text{ }\text{ }\text{ }t\in \left[\stackrel{˜}{t},{t}_{e}\right]\end{array}$ (3)

$\beta {\left(t-{t}_{s}\right)}^{2}+\gamma {\left({t}_{e}-t\right)}^{2}=2\psi \xi TN$ (4)

$\gamma \left({t}_{e}-\stackrel{˜}{t}\right)=\beta \left(\stackrel{˜}{t}-{t}_{s}\right)$ (5)

${t}_{s}=\stackrel{˜}{t}-\sqrt{2\delta \psi \xi TN}/\beta$ (6)

${t}_{e}=\stackrel{˜}{t}+\sqrt{2\delta \psi \xi TN}/\gamma$ (7)

$C=\alpha T+\sqrt{2\delta \psi \xi TN}+F$ (8)

Figure 1. Homogeneous railway commuter departure rate

3. 考虑出发时间下的异质性用户出行模型

3.1. 单站点异质性用户铁路交通出行成本

Figure 2. Departure rate of railway commuters for heterogeneous users of single station

${C}_{1}\left(t\right)=\left\{\begin{array}{l}\alpha T+{\beta }_{1}\left(\stackrel{˜}{t}-t\right)+{\psi }_{1}\xi T{r}_{1}\left(t\right)+F,\text{ }t\in \left[{t}_{1},\stackrel{˜}{t}\right]\\ \alpha T+{\gamma }_{1}\left(t-\stackrel{˜}{t}\right)+{\psi }_{1}\xi T{r}_{1}\left(t\right)+F,\text{ }\text{ }\text{ }t\in \left[\stackrel{˜}{t},{t}_{2}\right]\end{array}$ (9)

${C}_{2}\left(t\right)=\left\{\begin{array}{l}\alpha T+{\beta }_{2}\left(\stackrel{˜}{t}-t\right)+{\psi }_{2}\xi T{r}_{2}\left(t\right)+F,\text{ }t\in \left[{t}_{s},{t}_{1}\right]\\ \alpha T+{\gamma }_{2}\left(t-\stackrel{˜}{t}\right)+{\psi }_{2}\xi T{r}_{2}\left(t\right)+F,\text{ }\text{ }\text{ }t\in \left[{t}_{2},{t}_{e}\right]\end{array}$ (10)

${C}_{2}=\alpha T+{\beta }_{2}\left(\stackrel{˜}{t}-{t}_{s}\right)+F=\alpha T+{\gamma }_{2}\left({t}_{e}-\stackrel{˜}{t}\right)+F$ (11)

${r}_{2}\left(t\right)=\left\{\begin{array}{l}\frac{{\mu }_{2}\left(t-{t}_{s}\right)}{\xi T},\text{ }t\in \left[{t}_{s},{t}_{1}\right]\\ \frac{\eta {\mu }_{2}\left({t}_{e}-t\right)}{\xi T},\text{ }t\in \left[{t}_{2},{t}_{e}\right]\end{array}$ (12)

${C}_{1}=\alpha T+{\beta }_{1}\left(\stackrel{˜}{t}-{t}_{1}\right)+{\psi }_{1}\xi T{r}_{1}\left({t}_{1}\right)+F=\alpha T+{\gamma }_{1}\left(\stackrel{˜}{t}-{t}_{2}\right)+{\psi }_{1}\xi T{r}_{1}\left({t}_{2}\right)+F$ (13)

${r}_{1}\left(t\right)=\left\{\begin{array}{l}{r}_{2}\left({t}_{1}\right)+\frac{{\mu }_{1}\left(t-{t}_{1}\right)}{\xi T},\text{ }t\in \left[{t}_{1},\stackrel{˜}{t}\right]\\ {r}_{2}\left({t}_{2}\right)+\frac{\eta {\mu }_{1}\left({t}_{2}-t\right)}{\xi T},\text{ }t\in \left[\stackrel{˜}{t},{t}_{2}\right]\end{array}$ (14)

${\int }_{{t}_{1}}^{\stackrel{˜}{t}}{r}_{1}\left(t\right)\text{d}t+{\int }_{\stackrel{˜}{t}}^{{t}_{2}}{r}_{1}\left(t\right)\text{d}t={N}_{1}$(15)

${\int }_{{t}_{s}}^{{t}_{1}}{r}_{2}\left(t\right)\text{d}t+{\int }_{{t}_{2}}^{{t}_{e}}{r}_{2}\left(t\right)\text{d}t={N}_{2}$(16)

${t}_{s}=\stackrel{˜}{t}-\left(\frac{\sqrt{a{N}_{1}+{N}_{2}}+\left(a-1\right)\sqrt{{N}_{2}}}{a}\right)\sqrt{2m\xi T/{u}_{2}}$(17)

${t}_{1}=\stackrel{˜}{t}-\left(\frac{\sqrt{a{N}_{1}+{N}_{2}}-\sqrt{{N}_{2}}}{a}\right)\sqrt{2m\xi T/{u}_{2}}$(18)

${t}_{2}=\stackrel{˜}{t}+\left(\frac{\sqrt{a{N}_{1}+{N}_{2}}-\sqrt{{N}_{2}}}{a}\right)\sqrt{2p\xi T/{u}_{2}}$(19)

${t}_{e}=\stackrel{˜}{t}+\left(\frac{\sqrt{a{N}_{1}+{N}_{2}}+\left(a-1\right)\sqrt{{N}_{2}}}{a}\right)\sqrt{2p\xi T/{u}_{2}}$(20)

${C}_{2}=\alpha T+{\beta }_{2}\left(\frac{\sqrt{a{N}_{1}+{N}_{2}}+\left(a-1\right)\sqrt{{N}_{2}}}{a}\right)\sqrt{2m\xi T/{u}_{2}}+F$(21)

${C}_{1}=\alpha T+{\beta }_{1}\sqrt{\frac{a{N}_{1}+{N}_{2}}{a}}\sqrt{2m\xi T/{u}_{1}}+F$(22)

Figure 3. Departure rate of railway commuters in five groups for heterogeneous users

3.2. 双站点异质性用户铁路交通出行成本

Figure 4. Dual-site transportation network

${C}_{1}\left(t\right)=\left\{\begin{array}{l}2\alpha T+{\beta }_{1}\left(\stackrel{˜}{t}-t\right)+{\psi }_{1}\xi T{r}_{1}\left(t\right)+{\psi }_{1}\xi T\left[{r}_{1}\left(t\right)+{r}_{2}\left(t+T\right)\right]+F,\text{ }t\in \left[{t}_{s},\stackrel{˜}{t}\right]\\ 2\alpha T+{\gamma }_{1}\left(t-\stackrel{˜}{t}\right)+{\psi }_{1}\xi T{r}_{1}\left(t\right)+{\psi }_{1}\xi T\left[{r}_{1}\left(t\right)+{r}_{2}\left(t+T\right)\right]+F,\text{ }\text{ }\text{ }t\in \left[\stackrel{˜}{t},{t}_{e}\right]\end{array}$ (23)

${C}_{2}\left(t\right)=\left\{\begin{array}{l}\alpha T+{\beta }_{2}\left(\stackrel{˜}{t}+T-t\right)+{\psi }_{2}\xi T\left[{r}_{1}\left(t-T\right)+{r}_{2}\left(t\right)\right]+F,\text{ }t\in \left[{{t}^{\prime }}_{s},\stackrel{˜}{t}+T\right]\\ \alpha T+{\gamma }_{2}\left(t-\stackrel{˜}{t}-T\right)+{\psi }_{2}\xi T\left[{r}_{1}\left(t-T\right)+{r}_{2}\left(t\right)\right]+F,\text{ }\text{ }\text{ }t\in \left[\stackrel{˜}{t}+T,{{t}^{\prime }}_{e}\right]\end{array}$ (24)

${r}_{1}\left(t\right)=\left\{\begin{array}{l}\frac{\left({\mu }_{1}-{\mu }_{2}\right)\left(t-{t}_{s}\right)}{\xi T},\text{ }t\in \left[{t}_{s},\stackrel{˜}{t}\right]\\ \frac{\eta \left({\mu }_{1}-{\mu }_{2}\right)\left({t}_{e}-t\right)}{\xi T},\text{ }\text{ }\text{ }t\in \left[\stackrel{˜}{t},{t}_{e}\right]\\ 0,\text{ }其他\end{array}$ (25)

${r}_{2}\left(t\right)=\left\{\begin{array}{l}\frac{{\mu }_{2}\left(t-{{t}^{\prime }}_{s}\right)}{\xi T}-{r}_{1}\left(t-T\right),\text{ }t\in \left[{{t}^{\prime }}_{s},\stackrel{˜}{t}+T\right]\\ \frac{\eta {\mu }_{2}\left({{t}^{\prime }}_{e}-t\right)}{\xi T}-{r}_{1}\left(t-T\right),\text{ }\text{ }\text{ }t\in \left[\stackrel{˜}{t}+T,{{t}^{\prime }}_{e}\right]\end{array}$ (26)

Figure 5. Dual-site transportation network

${t}_{s}=\stackrel{˜}{t}-\sqrt{2m\xi T{N}_{1}/\left({u}_{1}-{u}_{2}\right)}$(27)

${t}_{e}=\stackrel{˜}{t}+\sqrt{2p\xi T{N}_{1}/\left({u}_{1}-{u}_{2}\right)}$(28)

${{t}^{\prime }}_{s}=\stackrel{˜}{t}-\sqrt{2m\xi TN/{u}_{2}}+T$(29)

${{t}^{\prime }}_{e}=\stackrel{˜}{t}+\sqrt{2p\xi TN/{u}_{2}}+T$(30)

${C}_{1}=2\alpha T+{\beta }_{1}\sqrt{2m\xi T{N}_{1}/\left({u}_{1}-{u}_{2}\right)}+{\psi }_{1}\sqrt{2{u}_{2}m\xi TN}+F$(31)

${C}_{2}=\alpha T+{\beta }_{2}\sqrt{2m\xi TN/{u}_{2}}+F$(32)

3.3. 模型性质

${C}_{1}=\alpha T+{\beta }_{1}\left({t}_{1}-{t}_{s}\right)+F=\alpha T+{\gamma }_{1}\left({t}_{e}-{{t}^{\prime }}_{1}\right)+F$ (33)

4. 地铁出行数值分析

Figure 6. The first group of commuters travel cost change chart

Figure 7. The second group of commuters travel cost change chart

Figure 8. The graph of travel costs with the number of commuters

5. 研究结论

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