# On the Urban Link Fundamental Diagram Based on Velocity-Weighted Flow and Queue Length

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## Abstract

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## 1. Introduction

- (a)
- A variable named the velocity-weighted flow combining the flow and space-mean velocity of each vehicle is proposed, which contains characteristics of both velocities and flows.
- (b)
- A new kind of link fundamental diagram (LFD) based on the velocity-weighted flow and queue length is presented in this paper, which can show the relationship between traffic condition and queue length.
- (c)
- Features of the proposed LFD related to cycle times, green times, and splits are presented: The LFD shape is generally determined by the splits. Both the critical queue length and critical velocity-weighted flow increase with increasing green time. The critical queue length is more closely related to the green time than the cycle time.

## 2. Problem Formulation

#### 2.1. Traffic Flow Description

#### 2.2. Vehicle Velocity Description

## 3. Link Fundamental Diagram Based on Queue

#### 3.1. The Definition of Velocity-Weighted Flow

#### 3.2. Drawing Link Fundamental Diagram

## 4. Characters of the Link Fundamental Diagram

#### 4.1. Impact of Green Time on the Link Fundamental Diagram

#### 4.2. The Impact of Split on the Link Fundamental Diagram

#### 4.3. The Impact of Cycle Time on the Link Fundamental Diagram

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

FD | Fundamental diagram |

MFD | Macroscopic fundamental diagram |

LFD | Link fundamental diagram |

## References

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**Figure 2.**Comparison of the traditional FD and the proposed urban link FD: (

**a**) scatter diagram of traditional flow and queue length; (

**b**) scatter diagram of the velocity-weighted flow and queue length.

**Figure 4.**$\overline{q}$ versus queue length for different green times with a cycle time of 60 s: (

**a**) cycle time C is 60 s, green time g is 10 s; (

**b**) cycle time C is 60 s, green time g is 20 s; (

**c**) cycle time C is 60 s, green time g is 30 s; (

**d**) cycle time C is 60 s, green time g is 40 s; (

**e**) cycle time C is 60 s, green time g is 30 s; (

**f**) the summary figure and the fitting curve of the critical point.

**Figure 5.**$\overline{q}$ versus queue length for different green times with a cycle time of 90 s: (

**a**) cycle time C is 90 s, green time g is 20 s; (

**b**) cycle time C is 90 s, green time g is 30 s; (

**c**) cycle time C is 90 s, green time g is 40 s; (

**d**) cycle time C is 90 s, green time g is 50 s; (

**e**) cycle time C is 90 s, green time g is 60 s; (

**f**) the summary figure and the fitting curve of the critical point.

**Figure 6.**$\overline{q}$ versus queue length for different green time with a cycle time of 120 s: (

**a**) cycle time C is 120 s, green time g is 30 s; (

**b**) cycle time C is 120 s, green time g is 40 s; (

**c**) cycle time C is 120 s, green time g is 50 s; (

**d**) cycle time C is 120 s, green time g is 60 s; (

**e**) cycle time C is 120 s, green time g is 70 s; (

**f**) the summary figure and the fitting curve of the critical point.

**Figure 7.**$\overline{q}$ versus queue length for different split: (

**a**) split is 1/3; (

**b**) split is 1/2; (

**c**) split is 2/3.

**Figure 8.**$\overline{q}$ versus queue length for different cycle times: (

**a**) green time g is 20 s; (

**b**) green time g is 30 s; (

**c**) green time g is 40 s.

**Figure 9.**Scatter diagram of critical $\overline{q}$ vs. critical queue length for different splits.

Symbol | Explanation |
---|---|

${v}_{i}\left(t\right)$ | the velocity of vehicle i at time step t |

${V}_{free}$ | free-flow velocity |

${V}_{f}$ | following velocity |

${x}_{i}\left(t\right)$ | distance from vehicle i to stop line at time step t |

$l\left(t\right)$ | queue length at time step t |

$\alpha $ | Discharge shock wave |

$\beta $ | queuing shock wave |

${k}_{m}$ | density when following with vehicles |

${k}_{j}$ | jam density |

s | saturation flow |

g | green time |

r | red time |

C | cycle time |

${t}_{0}$ | beginning of the cycle |

q | traffic flow |

${q}_{C}$ | traffic flow passing an intersection in one cycle |

${q}_{g}$ | arriving flow during green time |

$\overline{q}$ | velocity-weighted traffic flow |

Symbol | Quantity | Sample |
---|---|---|

i | vehicle number | No. 549 |

${t}_{i}$ | moment of being positioned | 600 (s) |

${v}_{i}$ | instantaneous velocity | 31.27 (km/h) |

${d}_{i}$ | distance from vehicle to link entrance | 386.48 (m) |

$lin{k}_{i}$ | The number of the link where the vehicle is located | No. 30 |

${L}_{i}$ | length of the link | 826.41 (m) |

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**MDPI and ACS Style**

Yu, H.; Kong, J.; Ren, Y.; Yin, C.
On the Urban Link Fundamental Diagram Based on Velocity-Weighted Flow and Queue Length. *Symmetry* **2020**, *12*, 1852.
https://doi.org/10.3390/sym12111852

**AMA Style**

Yu H, Kong J, Ren Y, Yin C.
On the Urban Link Fundamental Diagram Based on Velocity-Weighted Flow and Queue Length. *Symmetry*. 2020; 12(11):1852.
https://doi.org/10.3390/sym12111852

**Chicago/Turabian Style**

Yu, Hansong, Junwei Kong, Ye Ren, and Chenkun Yin.
2020. "On the Urban Link Fundamental Diagram Based on Velocity-Weighted Flow and Queue Length" *Symmetry* 12, no. 11: 1852.
https://doi.org/10.3390/sym12111852