Mathematical Investigation of Traffic Dynamics on a Network

Abstract

Exploration of the diverse attributes of road traffic proves highly valuable in traffic management and in the strategic planning and design of roadway systems. Delving into the dynamics of traffic congestion and its progression is a complex endeavor that has been extensively researched in multidisciplinary fields, including mathematics, engineering, informatics, etc. The complexities of the traffic system due to its irregular and perplexing behavior have been extensively investigated from the perspective of statistical mechanics and nonlinear dynamics. Moreover, traffic is an intricate interplay between the large number of vehicles interacting on the highway, due to which traffic is congenitally a non-equilibrium process. To analyze these dynamic properties of traffic systems, they are modeled as the system of interacting vehicles derived at both micro- and macrolevels. Further, to understand the jamming phase transitions between uniformly moving traffic and congested traffic, the simplest hydrodynamic model has been introduced, which incorporates the properties of microscopic and macroscopic models, named the lattice hydrodynamic model. The model is investigated theoretically using the linear and nonlinear stability analysis, which is validated through the numerical simulations on a unidirectional road. In addition, the traffic networks are formed in a city. The urban road transportation system is investigated at the network level with the help of a macroscopic fundamental diagram. The concept of the macroscopic fundamental diagram (MFD) helped researchers explore and understand the dynamics of traffic flow in large-scale traffic networks. To analyze any changes to a network, one needs to analyze how the macroscopic fundamental diagram is affected by those changes. The simplest speed-matching model proposed generates the MFDs for all the density ranges. In transportation systems, various network structures are observed, including percolation-backbone fractal networks, star graphs, loop lines, etc. Therefore, in this thesis, traffic-related problems are discussed and broadly categorized using the lattice hydrodynamic and speed-matching approaches, with the investigation centered on the following key objectives: 1. Modeling, analysis, and simulation of traffic flow incorporating density-dependent passing. 2. Study the effect of heterogeneity in occupancy and transition rate on a closed traffic network. 3. Analyzing traffic density in a heterogeneous open percolation-backbone fractal network. Thus, The whole work is bifurcated into eight chapters: Chapter 1 presents the origins of traffic flow theory, followed by different modeling approaches to analyze and predict the traffic flow phenomena on road networks. ix In real traffic dynamics, passing significantly impacts the traffic flow. Passing/Overtaking is primarily influenced by the traffic density in the surroundings; therefore, considering passing as constant is impractical. In Chapter 2, we propose a lattice hydrodynamic model considering density-dependent passing for a unidirectional single-lane highway to examine the traffic system more realistically. Due to various experimental investigations, the passing behavior is considered similar to the flow-density curve. The passing increases with the density, and it decreases after achieving a maximum at a critical value. Thus, implementing the idea to model density-dependent passing is similar to the optimal velocity function. The impact of density-dependent passing on the lattice model is investigated through linear stability analysis, and it is shown that with an increase in passing, the stability region reduces significantly. Using nonlinear analysis, the kink-antikink soliton of the mKdV equation is obtained to describe the propagating behavior of the density wave near the critical point. For the small values of passing, there is a phase transition from the kink jam region to the free flow region, with decreasing sensitivity. On the other hand, for large values of passing, the phase transition occurs from the uniform to the kink jam region through the chaotic jam region, with increasing delay time. The influence of parameters involved in density-dependent passing is investigated theoretically and numerically. Recent developments in transportation systems have significantly accelerated the emergence of connected vehicles (CVs) within the V2V environment, coexisting with human-driven vehicles (HDVs). Understanding the traffic dynamics in the mixed environment of CVs and HDVs in disordered traffic where the vehicles do not follow lane discipline becomes excessively complex. Thus, a lattice hydrodynamic model is proposed in Chapter 3 that incorporates the area occupancy effect for the mixed traffic environment. Further, a multi-phase optimal velocity function is considered to portray the traffic flow characteristics more realistically as it considers the discontinuous accelerations occurring in real traffic. The traffic flow behavior is investigated through linear stability analysis, which depicts that the stability region narrows down as the fraction of CVs increases. Moreover, the mKdV equation is attained to study the slowly varying behavior of density waves near the critical point. It is observed that with an increase in the fraction of CVs, traffic flow stability increases significantly with increasing sensitivity. Notably, the theoretical results are validated through numerical simulation on unidirectional multi-phase traffic flow. Urban road networks play a prominent role in facilitating the movement of vehicles. The network is connected by different roads and has a heterogeneous nature. Road networks have different capacities and transition rates. In a real traffic scenario, the density of vehicles on the roads is primarily influenced by its capacity. When a road has less capacity which may occur X due to narrow lanes, bottlenecks, or restrictions, it accommodates less vehicles on the road, which in turn leads to the decrease in the vehicular density. Also, the transition rate (the number of vehicles entering the road) affects the traffic flow. Thus, Chapter 4 investigates the traffic flow in a percolation-backbone fractal with the consideration of occupancy of the road and transition rate. The traffic flow on the fractal network is presented with the help of a cell-transmission graph. The density equations are derived using the speed-matching model. The fundamental diagrams are obtained for urban-scale traffic networks by solving the density equations numerically. The steadystate vehicular densities and traffic currents are plotted for different occupancy and transition rates cases. It was found that the traffic flow increases significantly with the increase in road occupancy and transition rate. Chapter 4 discusses the concept of a closed fractal network. However, what happens if the network is open instead? How would this impact the traffic flow? Thus, in Chapter 5, an open fractal network is considered to study the heterogeneity of roads. The fractal network is described with the help of a cell-transmission graph. The density equations were presented using the speedmatching model with the consideration of capacity and transition rate. The fundamental diagrams were obtained, and it is observed how the macroscopic fundamental diagram (MFD) varies with different cases of capacity and transition rate. Traffic control serves as an indispensable component in optimizing the traffic flow, especially on networks. To analyze the varied complexity of traffic dynamics, the percolation backbone fractal network is characterized via a cell-transmission model. Taking into account a generalized flowdensity relation, Chapter 6 modifies the dynamic model to scrutinize the impact of transition rates on traffic flow in a conserved network. The macroscopic fundamental diagrams attained through numerical simulation are investigated for homogeneous as well as heterogeneous transition rates. For first-generation fractal networks, unimodal or bimodal traffic currents are observed with respect to mean density. Further, for a second-generation fractal network, two types of density waves are observed depending on the number of vehicles present: uniform equilibrium and oscillatory. It is reported that the transition rates corresponding to singly-connected nodes can control the traffic dynamics to ensure a uniform stationary flow, which cannot be achieved via the doubly-connected and quadruple-connected nodes. In transportation systems, various network structures are observed, one of which is loop lines. Therefore, in Chapter 7, a dynamic model for traffic flow is proposed to analyze the impact of occupancy in a multiple-loop network with a single intersection. The graph representation of multiple-loop lines is obtained utilizing the cell-transmission model, and consequently, the density xi equations are derived. The macroscopic fundamental diagrams are investigated for different cases of occupancy and the fraction of vehicles. Equilibrium densities, traffic currents, and travel time are obtained theoretically and validated via simulation for the double-loop lines. It is observed that the equilibrium densities increase linearly against the mean density for an equal fraction of vehicles while the traffic currents remain symmetric, unlike the case of an unequal fraction of vehicles. The travel time varies inversely with respect to the occupancy of the loop line. Additionally, when the fraction of vehicles and occupancy at both nodes are different, a critical value of mean density is obtained below (above), in which the travel time at one node is less (more) than the other. Stability analysis is conducted using the tools from cooperative theory and repelling boundaries to analyze the traffic flow on multiple-loop lines. As a result, the vehicular densities converge to a unique equilibrium point irrespective of the initial densities in the multiple-loop network. The theoretical results agree well with the simulation results. Chapter 8 provides a comprehensive summary of this study, highlighting its key findings and contributions. In addition, it outlines several significant directions for future work.