# Congestion-Free Ant Traffic: Jam Absorption Mechanism in Multiple Platoons

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model and Simulation Scenario

#### 2.1. Model

- The binary variable ${s}_{i}\left(t\right)$ is either zero or one depending on whether the cell is empty (zero) or occupied (one) by an ant at time step t.
- The pheromone concentration ${\sigma}_{i}\left(t\right)$ is a numerical variable ranging from zero to ${\sigma}_{\mathrm{sat}}$. ${\sigma}_{i}\left(t\right)=0$ means that there is no pheromone in cell i at time step t, whereas ${\sigma}_{i}\left(t\right)={\sigma}_{\mathrm{sat}}$ means that the cell is saturated with pheromone at that time step.
- ${\mathsf{\Omega}}_{i}$ represents a resistance or drag that the cells present against the ant motion. ${\mathsf{\Omega}}_{i}$ represents factors related to trail conditions such as obstacle, rough trail, or uphill, which have a negative impact on the motion of ants. In short, ${\mathsf{\Omega}}_{i}$ provides opposition to the motion in the preferred direction, which is represented in ATM by a negative velocity (velocity in the opposite direction). In a given simulation, ${\mathsf{\Omega}}_{i}$ is constant for a given cell. To avoid ants moving backwards, ${\mathsf{\Omega}}_{i}$ is designed to always be positive, but less than the non-zero minimum ant velocity $\left({v}_{min}\right)$ (as explained later, ${\mathsf{\Omega}}_{i}$ is used while defining the heterogeneous trail.).

- ${v}_{j}\left(t\right)$ is the instantaneous velocity of ant j at time step t. ${v}_{j}\left(t\right)$ is continuous and ranges from zero to one.
- ${p}_{j}\left(t\right)$ is the position of ant j on the trail at time step t and ranges from zero to L. Similar to ${v}_{j}\left(t\right)$, ${p}_{j}\left(t\right)$ is also continuous.

#### 2.1.1. Stage I: Ant Motion

#### 2.1.2. Stage II: Pheromone Updating

- Evaporation:$${\sigma}_{i}^{\prime}(t+1)={\sigma}_{i}\left(t\right)-({\sigma}_{i}\left(t\right)\times er),\phantom{\rule{1.em}{0ex}}if\phantom{\rule{1.em}{0ex}}{\sigma}_{i}\left(t\right)>0.$$
- Accumulation:$${\sigma}_{i}(t+1)=\left(\right)open="\{"\; close>\begin{array}{cc}({\sigma}_{i}^{\prime}(t+1)+\tau ),\hfill & \phantom{\rule{1.em}{0ex}}if\phantom{\rule{1.em}{0ex}}{s}_{i}\left(t\right)=1\phantom{\rule{1.em}{0ex}}and\phantom{\rule{1.em}{0ex}}{\sigma}_{i}^{\prime}(t+1){\sigma}_{sat}\phantom{\rule{1.em}{0ex}}and\phantom{\rule{1.em}{0ex}}{v}_{j}\left(t\right)0\hfill \\ {\sigma}_{sat},\hfill & \phantom{\rule{1.em}{0ex}}if\phantom{\rule{1.em}{0ex}}{s}_{i}\left(t\right)=1\phantom{\rule{1.em}{0ex}}and\phantom{\rule{1.em}{0ex}}{\sigma}_{i}^{\prime}(t+1){\sigma}_{sat}.\hfill \end{array}$$

#### 2.2. Simulation Scenarios

#### 2.2.1. Periodic Boundary Conditions and Introduction of New Ants

#### 2.2.2. Trail Scenarios

#### Homogeneous Trail

#### Heterogeneous Trail

## 3. Fundamental Diagrams and Evaporation Rate

#### 3.1. High to Medium Evaporation Rate ($0.1<er\le 1$)

#### 3.2. Meager Evaporation Rate ($0\le er<0.001$)

#### 3.3. Low Evaporation Rate ($0.01<er<0.1$)

## 4. Model Validation

## 5. Analysis of Jam-Free Ant Traffic

#### 5.1. Intra-Platoon Analysis

#### 5.2. Inter-Platoon Analysis

#### 5.3. Analysis of Platoon Headway and Density

## 6. Concluding Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Variables in the Ant Trail Model

#### Appendix A.1. Minimum Velocity in ATM (v_{min})

**Figure A1.**(

**a**) Average velocity and (

**b**) flow of agents plotted against their densities for different ${v}_{min}$ values (indicated in the legend). Parameters other than ${v}_{min}$ were kept constant: L = 1000 cells, ${\sigma}_{sat}$ = 80, $er$ = 0.02.

#### Appendix A.2. Pheromone Saturation Level (σ_{sat})

**Figure A2.**The (

**a**) capacity flow and (

**b**) capacity flow density in the simulation are plotted against ${\sigma}_{sat}$. Parameters other than ${\sigma}_{sat}$ were kept constant: L = 1000 cells, ${v}_{min}$ = 0.15, $er$ = 0.02.

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**Figure 1.**Schematic representation of the ant-trail model (ATM), showing a single-lane unidirectional ant trail from left to right. Each cell is indexed by i, and each ant is indexed by j. At any given time, each cell can contain only one ant.

**Figure 2.**A fundamental diagram of the ATM simulation for different $er$ values is plotted: (

**a**) average velocity–density relationship (

**b**) flow–density relationship. Parameters other than $er$ were kept constant: $L=1000$ cells, ${\sigma}_{sat}$ = 80, ${v}_{min}$ = 0.15.

**Figure 3.**(

**a**) Average velocity and (

**b**) flow of agents plotted against their density. Simulation scenario: $er=0.02$, L = 1000 cells, ${\sigma}_{sat}$ = 80, and ${v}_{min}$ = 0.15.

**Figure 4.**The distance headway distribution in different density ranges is plotted: $\left(\mathbf{a}\right)\phantom{\rule{1.em}{0ex}}d\phantom{\rule{1.em}{0ex}}\in \phantom{\rule{1.em}{0ex}}[0,0.2]$, $\phantom{\rule{1.em}{0ex}}\left(\mathbf{b}\right)\phantom{\rule{1.em}{0ex}}d\phantom{\rule{1.em}{0ex}}\in \phantom{\rule{1.em}{0ex}}[0.2,0.4],\phantom{\rule{1.em}{0ex}}\left(\mathbf{c}\right)\phantom{\rule{1.em}{0ex}}d\phantom{\rule{1.em}{0ex}}\in \phantom{\rule{1.em}{0ex}}[0.4,0.7]$. For all density ranges, the corresponding log normal (red) ($(Frequency\left(HW\right))=\frac{1}{HW{\sigma}_{log}\sqrt{2\pi}}{e}^{\frac{{(D-ln(HW))}^{2}}{2{\sigma}_{log}^{2}}})$) and negative-exponential (green) $(Frequency(HW)={e}^{(}\frac{-HW}{\lambda})$ distributions are also shown. Simulation scenarios: $er=0.02$, L = 1000 cells, ${\sigma}_{sat}$ = 80, ${v}_{min}$ = 0.15.

**Figure 5.**The velocity distribution of agents in different densities is plotted: $(\mathbf{a})\phantom{\rule{1.em}{0ex}}d\phantom{\rule{1.em}{0ex}}\in \phantom{\rule{1.em}{0ex}}[0,0.2]$, $(\mathbf{b})\phantom{\rule{1.em}{0ex}}d\phantom{\rule{1.em}{0ex}}\in \phantom{\rule{1.em}{0ex}}[0.2,0.4],(\mathbf{c})\phantom{\rule{1.em}{0ex}}d\phantom{\rule{1.em}{0ex}}\in \phantom{\rule{1.em}{0ex}}[0.4,0.7]$. For all density ranges, the corresponding Gaussian distribution $(Frequency(v))=\frac{1}{\sigma \sqrt{2\pi}}{e}^{\frac{-{(V-v)}^{2}}{2{\sigma}^{2}}})$ is also shown. Simulation scenarios: $er=0.02$, L = 1000 cells, ${\sigma}_{sat}$ = 80, ${v}_{min}$ = 0.15.

**Figure 6.**The instantaneous velocities of (

**a**) $an{t}_{0}$, (

**b**) $an{t}_{50}$, and (

**c**) $an{t}_{100}$ in the intra-platoon analysis plotted against the positions of the same ants. The presented data were extracted from an ATM computer simulation of a heterogeneous trail. Simulation scenarios: $er=0.02$; high-resistance section $(\mathsf{\Omega}=0.1)$ of trail = $cel{l}_{400}$–$cel{l}_{800}$, L = 1000 cells, ${\sigma}_{sat}$ = 80, ${v}_{min}$ = 0.15.

**Figure 7.**(

**a**) Instantaneous velocities of ants plotted against the positions of the same ants for prototypical inter-platoon analysis. (

**b**) The positions of the ants plotted against time for prototypical inter-platoon analysis. Data were obtained from an ATM computer simulation of a heterogeneous trail. Ants observed: last ant in leading platoon (red); leader of the following platoon (black). Simulation scenarios: $er=0.02$, high-resistance section $(\mathsf{\Omega}=0.1)$ of trail = $cel{l}_{400}$–$cel{l}_{800}$, L = 1000 cells, ${\sigma}_{sat}$ = 80, ${v}_{min}$ = 0.15.

**Figure 8.**Headway (jam absorption buffer) of the leader of the following platoon for different densities $\left(d\right)$ plotted against the position of the same ant for inter-platoon analysis. Simulation scenarios: $er=0.02$, high-resistance section $(\mathsf{\Omega}=0.1)$ of trail = $cel{l}_{400}$–$cel{l}_{800}$, L = 1000 cells, ${\sigma}_{sat}$ = 80, ${v}_{min}$ = 0.15.

Description | Symbol |
---|---|

Unique identity of a cell in the trail | i |

Presence or absence of an ant in the trail $cel{l}_{i}$ at time t | ${s}_{i}\left(t\right)$ |

Pheromone concentration in the trail $cel{l}_{i}$ at time t | ${\sigma}_{i}\left(t\right)$ |

Resistance by the trail $cel{l}_{i}$ to the motion of an ant | ${\mathsf{\Omega}}_{i}$ |

Pheromone concentration saturation level | ${\sigma}_{sat}$ |

Unique identity of an ant in the simulation | j |

Velocity of the $an{t}_{j}$ at time t | ${v}_{j}\left(t\right)$ |

Position of the $an{t}_{j}$ at time t | ${p}_{j}\left(t\right)$ |

Minimum velocity of an ant towards the cell with no pheromone and no other ant | ${v}_{min}$ |

Trail length | L |

Evaporation rate | $er$ |

Description | Simulation Values |
---|---|

Pheromone concentration saturation level | ${\sigma}_{sat}=80$ |

Resistance from the trail $cel{l}_{i}$ to the motion of an agent in the homogeneous trail scenario | ${\mathsf{\Omega}}_{i}=0$ |

Resistance from the trail $cel{l}_{i}$ in the high resistance section to the motion of an ant in the heterogeneous trail scenario | ${\mathsf{\Omega}}_{i}=0.1$ |

Minimum velocity of an ant towards the cell with no pheromone and no other ant | ${v}_{min}=0.15$ |

Trail length | $L=1000$ |

Inflow rate | = 0.001 |

Evaporation rate | $er=0.02$ |

High resistance trail section in the heterogeneous trail scenario | $cel{l}_{400}$–$cel{l}_{800}$ |

Density | Number of Platoons |
---|---|

0.1 | 28 |

0.2 | 21 |

0.3 | 14 |

0.4 | 8 |

0.5 | 3 |

0.6 | 3 |

0.7 | 2 |

0.8 | 1 |

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**MDPI and ACS Style**

Kasture, P.; Nishimura, H.
Congestion-Free Ant Traffic: Jam Absorption Mechanism in Multiple Platoons. *Appl. Sci.* **2019**, *9*, 2918.
https://doi.org/10.3390/app9142918

**AMA Style**

Kasture P, Nishimura H.
Congestion-Free Ant Traffic: Jam Absorption Mechanism in Multiple Platoons. *Applied Sciences*. 2019; 9(14):2918.
https://doi.org/10.3390/app9142918

**Chicago/Turabian Style**

Kasture, Prafull, and Hidekazu Nishimura.
2019. "Congestion-Free Ant Traffic: Jam Absorption Mechanism in Multiple Platoons" *Applied Sciences* 9, no. 14: 2918.
https://doi.org/10.3390/app9142918