# A Statistically Based Model for the Characterization of Vehicle Interactions and Vehicle Platoons Formation on Two-Lane Roads

^{*}

## Abstract

**:**

## 1. Introduction

- To overcome the approach that we find in a part of the technical literature, which sets a deterministic threshold for the distinction between free and non-free vehicles;
- To decouple analyses from the specification of certain probabilistic distributions for the headways and the estimation of the related parameters that we can find in research works more focused on the probabilistic modelling aspects;
- To have an agile but not simplistic method for evaluations in technical applications related to traffic quality assessment on two-lane roads.

## 2. Performance Measures and Transport Sustainability on Two-Lane Roads

- Reflect road users’ perception of the quality of traffic flow;
- Easy to be measured and estimated using field data;
- Correlate to traffic and roadway conditions in a meaningful way;
- Compatible with performance measures used on other facilities;
- Able to describe both uncongested and congested conditions;
- Useful in analyses concerning traffic safety, transport economics, and environmental impacts.

## 3. Literary Review

## 4. The Two-Lane Road Statistical Platooning Model

- ${V}_{A}\left({v}_{A},\tau \right)$, which is the random process of the speeds implemented by the type A vehicles, with a time headway $\tau $ with respect to the following vehicle;
- ${V}_{B}\left({v}_{B},\tau \right)$, which is the random process of the speeds implemented by the type B vehicles, with the time headway $\tau $ with respect to the previous vehicle;
- ${V}_{AB}\left({v}_{A},{v}_{B},\tau \right)$, which is the random process of the speeds difference ${v}_{AB}={v}_{A}-{v}_{B}$ for the pairs of vehicles A and B, separated by the headway $\tau $.

## 5. Data Processing and Results

#### 5.1. The Case Study

#### 5.2. Free-Vehicle Percentage as a Headway Function

^{2}, $K\left(\overline{\tau}\right)=80.86$ (km/h)

^{2}, and $VAR\left[{V}_{AB}\left(\overline{\tau}\right)\right]=63.42$ (km/h)

^{2}. Furthermore, ${D}_{L}=\underset{\tau \to \infty}{lim}VAR\left[{V}_{B}\left(\tau \right)\right]=177.69$ (km/h)

^{2}, and then $VAR\left[{V}_{AB}\left(\infty \right)\right]=355.38$ (km/h)

^{2}. In those circumstances, we obtain:

#### 5.3. Free Vehicle Percentage as a Traffic Flow Function

- 4.
- For each value ${\tau}_{j}^{\left(q\right)}$, the amount of free vehicles ${\alpha}_{j}^{\left(q\right)}\xb7{n}_{j}^{\left(q\right)}$ and non-free vehicles $(1-{\alpha}_{j}^{\left(q\right)})\xb7{n}_{j}^{\left(q\right)}={\beta}_{j}^{\left(q\right)}\xb7{n}_{j}^{\left(q\right)}$ with ${\alpha}_{j}^{\left(q\right)}$ according to Equation (27) for $\tau ={\tau}_{j}^{\left(q\right)}$;
- 5.
- The estimated value ${T}_{q}$ of the mean ${M}_{q}\left[\tau \right]$ for all the headway values ${\tau}_{j}^{\left(q\right)}$, which is ${T}_{q}=\sum {\tau}_{j}^{\left(q\right)}\xb7{n}_{j}^{\left(q\right)}/\sum {n}_{j}^{\left(q\right)}$;
- 6.
- The estimated value ${T}_{lq}$ of the mean ${M}_{lq}\left[\tau \right]$ for the headway values regarding only free vehicles, which is ${T}_{lq}=\sum \text{}{\alpha}_{j}^{\left(q\right)}\xb7{\tau}_{j}^{\left(q\right)}\xb7{n}_{j}^{\left(q\right)}/\sum {\alpha}_{j}^{\left(q\right)}\xb7{n}_{j}^{\left(q\right)}$;
- 7.
- The estimated value ${T}_{cq}$ of the mean ${M}_{cq}\left[\tau \right]$ or the headway values regarding only non-free vehicles, which is ${T}_{cq}=\sum \text{}{\beta}_{j}^{\left(q\right)}\xb7{\tau}_{j}^{\left(q\right)}\xb7{n}_{j}^{\left(q\right)}/\sum {\beta}_{j}^{\left(q\right)}\xb7{n}_{j}^{\left(q\right)}$.

## 6. Discussion

#### The Effects of the Elimination of the No-Overtaking Rule in a Downstream Section

- 8.
- A negative exponential distribution for flows up to 300 veh/h;
- 9.
- An Erlang distribution for flows greater than 300 veh/h;

## 7. Conclusions

- Allows us to overcome the approach that we find in a part of the technical literature, which sets a deterministic headway threshold for the distinction between free and non-free vehicles;
- Decouples analyses from the specification of certain probabilistic distributions for the headways and the estimation of the related parameters that we can find in research works more focused on the probabilistic modeling aspects;
- Achieves greater adaptability to experimental data due to its data-driven approach based on behaviors that emerge statistically from real-life, compared to other models in the literature;
- Provides an agile but not simplistic method for evaluations in technical applications related to traffic quality assessment on two-lane roads;
- Offers helpful support for two-lane road operating conditions improvement to promote their safety and efficiency as part of a sustainable transport system.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Guerrieri, M.; Mauro, R. A Concise Introduction to Traffic Engineering: Theoretical Fundamentals and Case Studies; Springer Nature: Cham, Switzerland, 2020. [Google Scholar]
- Brackstone, M.; Waterson, B.; McDonald, M. Determinants of Following Headway in Congested Traffic. Transp. Res. Part F Traffic Psychol. Behav.
**2009**, 12, 131–142. [Google Scholar] [CrossRef] [Green Version] - Mauro, R.; Branco, F.; Guerrieri, M. Contribution to the Platoon Distribution Analysis in Steady-State Traffic Conditions. Period. Polytech. Civ. Eng.
**2014**, 58, 217–227. [Google Scholar] [CrossRef] [Green Version] - Al-Kaisy, A.; Jafari, A.; Washburn, S. Following Status and Percent Followers on Two-Lane Highways: Empirical Investigation. Civ. Eng. Res. J.
**2019**, 7, 61–70. [Google Scholar] [CrossRef] [Green Version] - Čutura, B.; Cvitanić, D.; Lovrić, I. Estimating Percent-Time-Spent-Following on Two-Lane Rural Roads. Građevinar
**2018**, 70, 563–570. [Google Scholar] - Washburn, S.; Al-Kaisy, A.; Luttinen, T.; Dowling, R.; Watson, D.; Jafari, A.; Bian, Z.; Elias, A. Improved Analysis of Two-Lane Highway Capacity and Operational Performance; The National Academies Press: Washington, DC, USA, 2018. [Google Scholar] [CrossRef]
- Transportation Research Board. Highway Capacity Manual: A Guide for Multimodal Mobility Analysis, 6th ed.; The National Academies Press: Washington, DC, USA, 2016. [Google Scholar] [CrossRef]
- Broughton, K.L.; Switzer, F.; Scott, D. Car Following Decisions under Three Visibility Conditions and Two Speeds Tested with a Driving Simulator. Accid. Anal. Prev.
**2007**, 39, 106–116. [Google Scholar] [CrossRef] - Singh, S.; Santhakumar, S.M. Empirical Analysis of Impact of Multi-Class Commercial Vehicles on Multi-Lane Highway Traffic Characteristics under Mixed Traffic Conditions. Int. J. Transp. Sci. Technol.
**2021**. [Google Scholar] [CrossRef] - Singh, S.; Santhakumar, S.M. Platoon-Based Impact Assessment of Heavy-Duty Vehicles on Traffic Stream Characteristics of Highway Lanes under Mixed Traffic Environment. Int. J. Intell. Transp. Syst. Res.
**2021**, 20, 29–45. [Google Scholar] [CrossRef] - Jiang, Y.; Wang, S.; Yao, Z.; Zhao, B.; Wang, Y. A Cellular Automata Model for Mixed Traffic Flow Considering the Driving Behavior of Connected Automated Vehicle Platoons. Phys. A Stat. Mech. Appl.
**2021**, 582, 126262. [Google Scholar] [CrossRef] - Pompigna, A.; Mauro, R. Long-Term Relationship Estimation and Coupling/Decoupling Analysis between Motorway Traffic and Gross Value Added. Specification of an ARDL Cointegration Approach and Application to the Italian Case Study. Arch. Transp.
**2021**, 60, 39–56. [Google Scholar] [CrossRef] - Pompigna, A.; Mauro, R. Input/Output Models for Freight Transport Demand: A Macro Approach to Traffic Analysis for a Freight Corridor. Arch. Transp.
**2020**, 54, 21–42. [Google Scholar] [CrossRef] - Sénquiz-Díaz, C. Transport Infrastructure Quality and Logistics Performance in Exports. ECONOMICS-Innov. Econ. Res.
**2021**, 9, 107–124. [Google Scholar] [CrossRef] - Raicu, S.; Costescu, D.; Popa, M.; Dragu, V. Dynamic Intercorrelations between Transport/Traffic Infrastructures and Territorial Systems: From Economic Growth to Sustainable Development. Sustainability
**2021**, 13, 11951. [Google Scholar] [CrossRef] - Sustainable Transport, Sustainable Development: Interagency Report Second Global Sustainable Transport Conference; United Nations Department for Economic and Social Affairs: New York, NY, USA, 2021.
- Afrin, T.; Yodo, N. A Survey of Road Traffic Congestion Measures towards a Sustainable and Resilient Transportation System. Sustainability
**2020**, 12, 4660. [Google Scholar] [CrossRef] - Al-Kaisy, A.; Amirhossein, J.; Washburn, S.; Luttinen, T.; Dowling, R. Traffic Operations on Rural Two-Lane Highways: A Review on Performance Measures and Indicators. Transp. Res. Rec.
**2018**, 2672, 66–74. [Google Scholar] [CrossRef] [Green Version] - Guerrieri, M. Two-Lane Highways Crest Curve Design. The Case Study of Italian Guidelines. Appl. Sci.
**2020**, 10, 8182. [Google Scholar] - Transportation Research Board. Highway Capacity Manual; The National Academies Press: Washington, DC, USA, 1985. [Google Scholar]
- Luttinen, R.T. Percent Time-Spent-Following as Performance Measure for Two-Lane Highways. Transp. Res. Rec.
**2001**, 1776, 52–59. [Google Scholar] [CrossRef] - Al-Kaisy, A.; Durbin, C. Evaluating New Methodologies for Estimating Performance on Two-Lane Highways. Can. J. Civ. Eng.
**2008**, 35, 777–785. [Google Scholar] [CrossRef] - Catbagan, J.; Nakamura, H. An Improved Follower Recognition Procedure to Estimate Follower Flow-Based Quality of Service Measures. J. East. Asia Soc. Transp. Stud.
**2010**, 8, 1822–1837. [Google Scholar] - Pompigna, A.; Rupi, F. Differences between HCM Procedures and Fundamental Diagram Calibration for Operational LOS Assessment on Italian Freeways. Transp. Res. Procedia
**2015**, 5, 103–118. [Google Scholar] [CrossRef] [Green Version] - Pompigna, A.; Rupi, F. Lane-Distribution Models and Related Effects on the Capacity for a Three-Lane Freeway Section: Case Study in Italy. J. Transp. Eng. Part A Syst.
**2017**, 143, 05017010. [Google Scholar] [CrossRef] - Al-Kaisy, A.; Freedman, Z. Estimating Performance on Two-Lane Highways: Case Study Validation of a New Methodology. Transp. Res. Rec.
**2010**, 2173, 72–79. [Google Scholar] [CrossRef] - Chishaki, T.; Tamura, Y. Headway Distribution Model Based on the Distinction between Leaders and Followers. Transp. Traffic Theory
**1984**, 9, 43–63. [Google Scholar] - Mauro, R. Traffic and Random Processes; Springer International Publishing: Cham, Switzerland, 2015. [Google Scholar]
- Buckley, D. A semi-Poisson model of traffic flow. Transp. Sci.
**1968**, 2, 107–132. [Google Scholar] [CrossRef] - Cowan, R.J. Useful Headway Models. Transp. Res.
**1975**, 9, 371–375. [Google Scholar] [CrossRef] - Branston, D. Models of Single Lane Time Headway Distributions. Transp. Sci.
**1976**, 10, 125–148. [Google Scholar] [CrossRef] - Tamura, Y.; Chishaki, T.; Mino, S. Modeling of Elementary and Actual Speed Distributions for Traffic Flow. Doboku Gakkai Ronbunshu
**1987**, 1987, 127–135. [Google Scholar] [CrossRef] [Green Version] - Luttinen, R.T. Level of Service on Finnish Two-Lane Highways. In Proceedings of the Transportation Research Circular E-C018: Fourth International Symposium on Highway Capacity, Maui, HI, USA, 27 June–1 July 2000; TRB, National Research Council: Washington, DC, USA, 2000; pp. 175–187. [Google Scholar]
- Brilon, W.; Weiser, F. Two-Lane Rural Highways: The German Experience. Transp. Res. Rec.
**2006**, 1988, 38–47. [Google Scholar] [CrossRef] - Forschungsgesellschaft für Straßen-und Verkehrswesen. Handbuch fuer die Bemessung von Strassenverkehrsanlagen. In HBS 2015; FGSV-Verlag: Cologne, Germany, 2015. [Google Scholar]
- Drew, D.R. Traffic Flow Theory and Control; McGraw-Hill: New York, NY, USA, 1968. [Google Scholar]
- Topolnik, D.; Zebec, Z.; Horvat, R. Overtaking as Indicator of Road Traffic Conditions. Promet-TrafficTransp.
**2000**, 12, 213–216. [Google Scholar] - Vlahogianni, E.I. Modeling Duration of Overtaking in Two Lane Highways. Transp. Res. Part F Traffic Psychol. Behav.
**2013**, 20, 135–146. [Google Scholar] [CrossRef] - Esposito, T.; Mauro, R. Fondamenti di Infrastrutture Viarie; Hevelius: Benevento, Italy, 2003. [Google Scholar]
- Italian Guidelines for the Design of Road Infrastructures (D.M. 5/11/2001): Italy. Roma, Italy, 2002. Available online: https://www.mit.gov.it/mit/mop_all.php?p_id=1983 (accessed on 10 March 2022).
- Pompigna, A.; Mauro, R. Smart Roads: A State of the Art of Highways Innovations in the Smart Age. Eng. Sci. Technol. Int. J.
**2022**, 25, 100986. [Google Scholar] [CrossRef]

**Figure 6.**Means (

**a**) and standard deviations (

**b**) of the speeds ${V}_{A}$ and ${V}_{B}$ for the unit amplitude classes of $\tau $.

**Figure 7.**Covariance of the velocities ${V}_{A}$ and ${V}_{B}$ for the unit amplitude classes of $\tau $.

**Figure 9.**${T}_{q}$, ${T}_{lq}$, and ${T}_{cq}$ values and trend estimations. for ${M}_{q}\left[\tau \right]$, ${M}_{lq}\left[\tau \right]$, and ${M}_{cq}\left[\tau \right]$.

**Figure 10.**Values for ${G}_{L}$ and ${G}_{C}$ and trend estimations for ${\Gamma}_{L}\left(q\right)$ and ${\Gamma}_{C}\left(q\right)$.

**Figure 11.**Non-free vehicles percentage in [23] data set and comparison with result from Equation (33).

**Figure 12.**Comparison between ${G}_{C}$ points distribution, ${\Gamma}_{C}\left(q\right)$ trend by Equation (33), and PF estimations with a 3 s threshold.

**Figure 13.**FD estimation in 5 min int.—comparison between Equation (33) and the 3 s headway threshold.

**Figure 14.**Traffic flows (veh/h) and non-free vehicles perc. (%) for 5 min intervals throughout the day in the two directions of travel.

**Figure 16.**Third degree polynomial interpolation for ${S}_{s}\left(q\right)$ and ${S}_{r}\left(q\right)$.

$\mathit{Q}$ (veh/h) | $\mathit{Q}$/3600 (veh/s) | $\mathit{k}$ | ${\mathit{S}}_{\mathit{s}}\left(\mathit{Q}\right)$ |
---|---|---|---|

1 | 0.000 | 1 | 99.67% |

10 | 0.003 | 1 | 96.72% |

50 | 0.014 | 1 | 84.65% |

100 | 0.028 | 1 | 71.65% |

200 | 0.056 | 1 | 51.34% |

300 | 0.083 | 1 | 36.79% |

800 | 0.222 | 2 | 3.06% |

1200 | 0.333 | 3 | 0.05% |

1500 | 0.417 | 4 | 0% |

1700 | 0.472 | 5 | 0% |

1800 | 0.500 | 6 | 0% |

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**MDPI and ACS Style**

Mauro, R.; Pompigna, A.
A Statistically Based Model for the Characterization of Vehicle Interactions and Vehicle Platoons Formation on Two-Lane Roads. *Sustainability* **2022**, *14*, 4714.
https://doi.org/10.3390/su14084714

**AMA Style**

Mauro R, Pompigna A.
A Statistically Based Model for the Characterization of Vehicle Interactions and Vehicle Platoons Formation on Two-Lane Roads. *Sustainability*. 2022; 14(8):4714.
https://doi.org/10.3390/su14084714

**Chicago/Turabian Style**

Mauro, Raffaele, and Andrea Pompigna.
2022. "A Statistically Based Model for the Characterization of Vehicle Interactions and Vehicle Platoons Formation on Two-Lane Roads" *Sustainability* 14, no. 8: 4714.
https://doi.org/10.3390/su14084714