# Congestion-Free Ant Traffic: Jam Absorption Mechanism in Multiple Platoons
## 4 Model Validation  
For model validation, we compared data from the ATM simulations with the empirical data from [14], beginning with fundamental diagrams. The fundamental diagrams of the relationships between (a) the average velocity-density and (b) the flow-density, as shown in Figure 3, were obtained from ATM simulations for \(er=0.02\) (low \(er\)). Along with the \(er\) value, all parameter values used hereafter in the simulations are summarized in Table 2. As shown in Figure 3a, unlike a vehicular traffic system, for ATM, the \(v_{avg}\) remained constant for a wide range of \(d\) values (\(d=0\) to \(\approx 0.75\)). Consequently, the \(f\) on the trail increased linearly with \(d\). Thus, in the fundamental diagrams for the present simulation, the free-flow phase was observed up to \(d\approx 0.75\). These observations (i.e., a free-flow phase and constant \(v_{avg}\) up to \(d\approx 0.75\)) are in basic agreement with the empirical results from [14], which validates the fundamental diagrams of our model. On the other hand, the same fundamental diagrams indicated that after specific density (\(d\approx 0.75\)), due to the exclusion dynamics of the model, ants in the model experienced jamming, which was not observed in [14]. It is important to note that the data in [14] for \(d\) mentioned above (\(d>0.75\)) were limited and, therefore, cannot be used for validation (in the future, we should make an effort to validate the exclusion assumption, as well as the jamming phase observed in ATM by experimentation). Another interesting thing to note is that \(v_{avg}\) in Figure 3 is entirely independent of \(d\) in the free-flow phase, whereas in the case of empirical data from [14], the data showed a slight decrease in \(v_{avg}\) with an increasing \(d\). The models in [13; 15] replicated the above-mentioned decrease in \(v_{avg}\). In both models, agents' velocity decreased slightly at high densities (inside the platoon). In [13], the decrease was due to a progressive reduction of the headway, while in [15], it was due to a progressive increase of the pheromone concentration. In other words, both models suggested that there might be some weak force that slows down ants at high \(d\) values. Whether it is weak near-neighbor repulsion (as suggested in [13]) or a decrease in velocity due to the above saturation pheromone (as suggested in [15]) should be investigated in the future. However, as the \(v_{avg}\) decrease is minimal, for mathematical simplicity, the model in this paper did not consider the weak forces mentioned above. \begin{table}
\begin{tabular}{l l} \hline \hline
**Description** & **Simulation Values** \\ \hline Pheromone concentration saturation level & \(\sigma_{sat}=80\) \\ Resistance from the trail \(cell_{i}\) to the motion of an agent in the homogeneous & \(\Omega_{i}=0\) \\ trail scenario & Resistance from the trail \(cell_{i}\) in the high resistance section to the motion of an ant & \(\Omega_{i}=0.1\) \\ in the heterogeneous trail scenario & \(v_{min}=0.15\) \\ Minimum velocity of an ant towards the cell with no pheromone and no other ant & \(L=1000\) \\ Trail length & = 0.001 \\ Inflow rate & \(er=0.02\) \\ Evaporation rate & \(cell_{400}\)-\(cell_{800}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: An overview of the parameter values from the ATM simulations. 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 \(v_{min}=0.15\). Furthermore, as shown in Figure 4, we analyzed the distance headway distribution in the simulation, where at lower \(d\) values, large distance headways were predominantly found, which indicates a random-headway distribution. The distribution at lower \(d\) values was well described by a negative exponential distribution. On the other hand, at medium to higher \(d\) values [0.2, 0.7], the distribution became much sharper with an increasing \(d\) value, indicating platooning.