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## Bus delay model considering influence of stop at upstream of intersection

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Abstract

The bus delay under the influence of upstream stop at intersection was analyzed. The modeling scenario was divided into three categories according to whether the system set the bus lane and the bus station position was father than the farthest queue point of the intersection by analyzing the location of the stop upstream at the intersection, the signal timing, the stop time, the input flow and other factors. The bus delay model under the influence of the stop was established based on the traffic wave theory. All models were analyzed and calculated by the numerical simulation. Results show that the expected bus delay is only related to the cycle and the green signal ratio in the BL scenario. In the NBL1 scenario, the expected bus delay increases with the increase of traffic flow, cycle and red light duration, and has nothing to do with the bus stop time. In the NBL2 scenario, the expected bus delay first decreases and then increases as the distance of the station increases. The expected bus delay under NBL1 is the largest with the increase of traffic flow under the same conditions.

Keywords： bus delay ; bus stop ; farthest queuing point ; bus lane ; traffic wave

ZHU Wen-tao, QIAN Guo-min, MA Dong-fang, WANG Dian-hai. Bus delay model considering influence of stop at upstream of intersection. Journal of Zhejiang University(Engineering Science)[J], 2020, 54(4): 796-803 doi:10.3785/j.issn.1008-973X.2020.04.019

### 图 1

Fig.1   Traffic wave propagation diagram

$N = \left( {{u_A} - {u_S}} \right){k_A}t = \left( {{u_B} - {u_S}} \right){k_B}t.$

${u_{{S}}} = \frac{{{q_B} - {q_A}}}{{{k_B} - {k_A}}} = \frac{{\Delta q}}{{\Delta k}}.$

${d_{{\rm{far}}}} = {Q_0} + {v_1}({t_{\rm{d}}} + {t_{\rm{r}}}) = {v_2}{t_{\rm{d}}}.$

$D_{{\rm{AM}}}^3 = t_{{\rm{AM}}}^3 - {t_{{\rm{vf}}}} = {t_{\rm{r}}} - \frac{{{t_{\rm{r}}}\left[ {{d_{{\rm{far}}}} + {t_0}{v_{\rm{f}}}} \right]}}{{{t_{{\rm{far}}}}({v_1} + {v_{\rm{f}}})}},$

$\frac{{{d_{\rm{s}}}}}{{{v_1}}} - \frac{{{d_{{\rm{far}}}} - {d_{\rm{s}}}}}{{{v_{\rm{f}}}}} < {t_0} < {t_{{\rm{far}}}}.$

### 图 8

Fig.8   Bus operation trajectory of scenario 4 when ${d_{\rm{A}}} \leqslant {d_{\rm{s}}} < {d_{\rm{M}}}$

### 图 9

Fig.9   Bus operation trajectory map under different scenarios when ${d_{\rm{M}}} \leqslant {d_{\rm{s}}} < {d_{\rm{N}}}$

\left. \begin{aligned} & D\left( {{t_{\rm{D}}},{d_{\rm{D}}}} \right) = D\left( {\frac{{{d_{\rm{s}}}}}{{{v_1}}},{d_{\rm{s}}}} \right) , \\ & E\left( {{t_{\rm{E}}},{d_{\rm{E}}}} \right) = E\left( {\frac{{{d_{\rm{s}}} - {d_{\rm{M}}}}}{{{v_1}}} + {t_{\rm{M}}},{d_{\rm{s}}}} \right) , \\ & F\left( {{t_{\rm{F}}},{d_{\rm{F}}}} \right) = F\left( {\frac{{{d_{\rm{s}}}}}{{{v_2}}} + {t_{\rm{r}}},{d_{\rm{s}}}} \right) . \\ \end{aligned} \right\}

\begin{aligned} & D_{{\rm{MN}}}^1 = t_{{\rm{MN}}}^1 - {t_{{\rm{vf}}}} = \\ &\quad \left\{ \begin{aligned} & {t_{\rm{r}}} - {t_{\rm{G}}} + \frac{{{d_{\rm{M}}}{t_{\rm{G}}}}}{{{v_2}{t_{\rm{M}}}}}\;\;{\rm{ ,}}\quad{t_{\rm{A}}} \leqslant {t_{\rm{G}}} < {t_{\rm{M}}} ;\\ &\frac{{\left( {{t_{\rm{E}}} - {t_{{{\rm{G}}^{\rm{'}}}}}} \right)\left( {{t_{\rm{R}}} - {t_{\rm{M}}}} \right)}}{{{t_{\rm{E}}} - {t_{\rm{M}}}}}{\rm{ , }}\quad{t_{\rm{M}}} \leqslant {t_{{{\rm{G}}^{\rm{'}}}}} < {t_{\rm{E}}} . \\ \end{aligned} \right. \\ \end{aligned}

$\max \left({t_{{\rm{far}}}} - C, - \frac{{{d_{{\rm{far}}}}}}{{{v_{\rm{f}}}}} - {t_{\rm{s}}}\right) < {t_0} < {t_{\rm{E}}} - {t_{\rm{s}}} - \frac{{{d_{{\rm{far}}}} - {d_{\rm{s}}}}}{{{v_{\rm{f}}}}}.$

${t_{\rm{E}}} - {t_{\rm{s}}} - \frac{{{d_{{\rm{far}}}} - {d_{\rm{s}}}}}{{{v_{\rm{f}}}}} < {t_0} < {t_{\rm{D}}} - \frac{{{d_{{\rm{far}}}} - {d_{\rm{s}}}}}{{{v_{\rm{f}}}}}.$

${t_{\rm{D}}} - \frac{{{d_{{\rm{far}}}} - {d_{\rm{s}}}}}{{{v_{\rm{f}}}}} < {t_0} < {t_{{\rm{far}}}}.$

## 4. 数值分析

1）BL场景. 由式（9）可知，该场景下的延误 $D$仅与 $C$、绿信比 $\lambda$等相关配时参数有关. $D$随着 $\lambda$的增加而减小，随着 $C$的增加呈线性增加，具体的 $C$- $\lambda$- $D$三者关系如图11所示.

### 图 11

Fig.11   C-λ-D relationship diagram in BL scenario

2）NBL1场景. 图12（a）中假设 ${t_{\rm{s}}}$=15 s， $\lambda$=0.5，考虑 ${Q_{\rm{in}}}$$C$$D$的影响关系，其中 $C$的取值为0~150 s， ${Q_{\rm{in}}}$的取值为0~1 500 veh/h. 从图12可知：a）当 $C$固定时， ${Q_{\rm{in}}}$越大， $D$越大；b）当 ${Q_{\rm{in}}}$保持一定时， $C$越大， $D$越大；c） ${Q_{\rm{in}}}$越大，增加相同幅度的 $C$所带来的 $D$的增幅越大.

### 图 12

Fig.12   Analysis of bus delay in NBL1 scenario

4） ${Q_{\rm{in}}}$对3种场景下 $D$的影响. 假设各场景中的 $C$=120 s，λ=0.5， ${t_{\rm{s}}}$=15 s， ${Q_{\rm{in}}}$为0~1 500 veh/h，其中NBL2场景下的 ${d_{\rm{s}}}$取为 ${{{d_{{\rm{far}}}}} / 2}$. 通过数值模拟发现：a）BL场景下由于公交专用道的设置，公交运行环境相对独立， $D$随着 ${Q_{\rm{in}}}$的增大保持固定不变，且期望延误较小；b）NBL1场景中的 $D$随着 ${Q_{\rm{in}}}$的增加而增加，且增加趋势越来越大，期望延误在3种场景中最大；c）由于本文所定义的 $D$不包含 ${t_{\rm{s}}}$，在NBL2场景下部分公交车可以利用交叉口排队时间进行停靠作业，该场景下的公交期望延误 $D$相对于NBL1场景较小.3种情景下的模拟结果如图14所示.

### 图 14

Fig.14   Impact of input traffic on delays in three scenarios

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