Anticipating Tipping Points for Disordered Traffic: Critical Slowing Down on the Onset of Congestion

TL;DR

This work extends the theory of early warning signals by applying a lattice hydrodynamic area-occupancy model to heterogeneous disordered traffic and demonstrating that critical slowing down manifests as rising variance and autocorrelation before congestion. By combining linear and nonlinear stability analyses with stochastic simulations, the authors identify two distinct tipping-point regimes and show that generic EWSs can anticipate both kink and chaotic jams in a non-lane-based setting. The study provides a quantitative framework to forecast traffic regime shifts in complex, real-world traffic with overtaking and heterogeneity, offering potential guidance for proactive traffic management. The results underscore the robustness of EWSs across jam types and lay groundwork for extending analysis to networked traffic and resilience of predictive indicators.

Abstract

Regime shifts are quite common in complex systems like cell regulations, disease transmissions, ecosystems, marine ice instability, etc. Several statistical indicators known as early warning signals (EWS) have been theorized to anticipate these abrupt transitions in advance. These regime shifts happen because they cross some critical value of the parameter that influences the overall dynamics. This critical threshold is known as tipping point. In the vicinity of a tipping point, perturbations gradually increases, and as a consequence, system-state extensively swing around the quasi-static attractor, and the local dynamics become progressively slow, which is known as critical slowing down (CSD). Because of this CSD, statistical measures known as early warning signals (EWS) such as variance and lag-1 autocorrelation increase. From the point of view of physics, a free flow can become congested when the mean car density crosses its tipping point. Recently, for lane-based traffic system using continuum model, study reveals that analysis of the generic EWSs serve as a good measure to predict upstream stop-and-go traffic jams. Now, we introduce EWSs to anticipate traffic jam for heterogeneous disordered traffic relevant for non-lane-based systems. We have analyzed a lattice hydrodynamic area occupancy model with passing and through numerical simulations, we have shown emergence of kink or chaotic jam. Also, we provided proper framework for prediction of traffic jams via different EWSs. From simulated data, we demonstrated that EWSs are sensitive as tipping is approached.

Figures (4)

• Figure 1: (a)There are two common types of heterogeneous traffic systems found on roads, which involve several vehicles with distinct physical properties and behavioral characteristics. (i) The concept of an ideal lane-based traffic system involves the implementation of strict lane discipline, where vehicles of various types adhere to designated lanes. (ii) On the other hand, a realistic traffic system may not always maintain perfect lane order, allowing vehicles of different sizes to travel alongside each other and enabling overtaking maneuvers by utilizing the lateral width of the road. In the context of a disordered traffic system, a surveillance device is employed to capture and quantify the movement of traffic, afterward converting it into a signal that can be subjected to analysis. (b) This graphic presents a systematic representation of the critical slowing down phenomenon observed in a traffic flow system when it approaches a tipping point. As the tipping point is near, the possible landscape becomes increasingly level. The stable attractor is represented by the local minima of the potential well, while the state of the system is shown by the presence of the red ball. As the tipping point approaches, the ball becomes increasingly susceptible to external disturbances. The acquisition of moving window statistical indicators is derived from the analysis of observed traffic patterns. An abrupt increase in the indications before the occurrence of the regime transition serves as an early indication of the imminent regime shift.
• Figure 2: Phase diagrams of density $\rho^*_j(t)$$\bigl(=B\rho_j(t)\bigr)$ versus sensitivity $a$ for (a) different values of area occupancy factor $B$ when $C=0.7$ and $\gamma=0.05$ and (b) for different values of passing rate $\gamma$ when $C=0.7$ and $B=1.6$. (c) Sensitivity diagram in parameter space $(\rho^*,B,a)$ space when $C=0.7.$ (d) Phase diagrams of passing rate $\gamma$ versus sensitivity $a$ for different values of area occupancy factor $B$ when $C=0.7$ and $\rho_0=0.2$.
• Figure 3: Spatiotemporal evolutions of density waves $\rho_j^*(t)$ ($=B\rho_j(t)$) when (a) $a=3.5$, (d) $a=5$, typical density distribution over the $100$ at $N=25200$ simulation time when (b) $a=3.5$, (e) $a=5$, density vs density difference plot between $16000-25200$ simulation time for (c) $a=3.5$, (f) $B=5$. Plots (a), (b), and (c) are associated with kink jam, and (d), (e), and (f) are associated with chaotic jam instability. Other parameters are $B=1.6,$$C=0.7,$$\gamma=0.4$ and $\rho_0=\rho_c=0.2.$
• Figure 4: (a) The spatiotemporal pattern of traffic flow density. The mean vehicular density on the road remains constant during the initial two-hour simulation period and afterward exhibits a linear increase until it reaches its critical threshold. Patterns of chaotic jam and kink jam are detected when the system's dynamics surpasses its stability threshold $\rho_{c_1}^*=0.1573$ for two different values of $a$, namely $a=5$ and $a=3.5$. Detrended density was retrieved from five specific lattice sites located in the center of the system to determine EWSs. The early warning signals were derived from stochastic fluctuations observed in the detrended mean density of vehicles at certain lattice sites, for both the (b) chaotic scenario (c) and the kink scenario.