# Congestion-Free Ant Traffic: Jam Absorption Mechanism in Multiple Platoons
## 2 Model and Simulation Scenario
### Model
#### 2.1.1 Stage I: Ant Motion  
The first stage of the update represents the behavior of ants, which depends on the interaction of a given ant with its surroundings (surroundings of an ant include trail and other ants in the simulation). In this stage, based on the information about pheromones and the presence of ants in the cell ahead, the value of the instantaneous ant velocity \(v_{j}(t)\) is generated for a given time step \(t\). At the end of Stage I, we obtain the value of \(p_{j}(t+1)\) for each ant and the scan value of \(s_{i}(t+1)\) for each cell at time \(t+1\). The ant positions and parameters are updated according to the following rules. If ant \(j\) is in cell \(i\), the instantaneous velocity of that ant from cell \(i\) towards cell \(i+1\) depends on \(s_{i+1}(t)\), \(\sigma_{i+1}(t)\), and \(\Omega_{i+1}\), as follows:  
\[v_{j}(t)= \begin{cases}0,&\text{if }s_{i+1}(t)=1\\ \begin{cases}\max(v_{j}(t-1)-0.1,v_{\min})-\Omega_{i+1}&\text{with probability P}\\ v_{j}(t-1)-\Omega_{i+1}&\text{with probability (P-1)}\end{cases}&\text{if }s_{i+1}(t)=0\text{ and }\sigma_{i+1}(t)<1\\ (v_{\min}+\mathbf{a}\times\sigma_{i+1}(t)-\Omega_{i+1}),&\text{if }s_{i+1}(t)=0\text{ and }1\leq \sigma_{i+1}(t)<\sigma_{\text{sat}}\\ (v_{\min}+\mathbf{a}\times\sigma_{\text{sat}}-\Omega_{i+1}),&\text{if }s_{i+1}(t)=0\text{ and } \sigma_{i+1}(t)\geq\sigma_{\text{sat}}.\end{cases} \end{cases} \tag{1}\]  
\[p_{j}(t+1)=p_{j}(t)+v_{j}(t). \tag{2}\]  
For the time step given in Equation (1), the following four cases are possible. The first case represents the action of an ant in cell \(i\) if the next cell is occupied by another ant. In that case, the former ant cannot move forward, which is represented by \(v_{j}(t)=0\). In the second case, the next cell contains no ant \((s_{i+1}(t)=0)\), but also no pheromone. Ants are sensitive to the pheromone concentration; they can detect a pheromone even at an extremely low concentration. For the simulations in this paper, we assumed that ants cannot detect pheromones below \(\sigma_{i+1}(t)<1(<<\sigma sat)\). When no pheromones are present, the ant will not have the trail; thus, to avoid wastage of energy due to high velocity, we assumed that the ants reduce their velocity or move with \(v_{min}\), whichever is higher, and the probability \(P\) gives the probability of this event (change in velocity). Conversely, the agents maintain the same velocity as \(t\) with the probability \((1-P)\). In the second case, we assumed that \(P=0.7(>0.5)\), which represents high sensitivity of the ant to absence of the pheromone. After a velocity reduction, the final velocity is calculated with consideration of \(\Omega_{i+1}\). For the simulations in this paper, \(v_{min}\) was chosen in such a way that it would avoid backwards motion due to \(\Omega_{i+1}\). Therefore, we needed to choose an \(v_{min}\) value that was always greater than \(\Omega\). On the other hand, a larger value of \(v_{min}\) led to a smaller velocity range. Therefore, to maximize the velocity range, we chose \(v_{min}=0.15\) (for further explanation on \(v_{min}\), check Appendix A.1). \begin{table}
\begin{tabular}{l l} \hline \hline
**Description** & **Symbol** \\ \hline Unique identity of a cell in the trail & \(i\) \\ Presence or absence of an ant in the trail \(cell_{i}\) at time \(t\) & \(s_{i}(t)\) \\ Pheromone concentration in the trail \(cell_{i}\) at time \(t\) & \(\sigma_{i}(t)\) \\ Resistance by the trail \(cell_{i}\) to the motion of an ant & \(\Omega_{i}\) \\ Pheromone concentration saturation level & \(\sigma_{sat}\) \\ Unique identity of an ant in the simulation & \(j\) \\ Velocity of the \(ant_{j}\) at time \(t\) & \(v_{j}(t)\) \\ Position of the \(ant_{j}\) at time \(t\) & \(p_{j}(t)\) \\ Minimum velocity of an ant towards the cell with no pheromone and no other ant & \(v_{min}\) \\ Trail length & \(L\) \\ Evaporation rate & _er_ \\ \hline \hline \end{tabular}
\end{table} 
Table 1: An overview of the parameters in the ant-trail model (ATM).