Dynamical properties of a compartmental model for traffic networks

Abstract

In this paper, we propose a traffic network flow model particularly suitable for qualitative analysis as a dynamical system. Flows at a junction are determined by downstream supply of capacity (lack of congestion) as well as upstream demand of traffic wishing to flow through the junction. This approach is rooted in the celebrated Cell Transmission Model for freeway traffic flow, and we analyze resulting equilibrium flows and convergence properties. I. INTRODUCTIONCompartmental systems are a broad modeling paradigm to study fluid-like flow of a single substance among interconnected "compartments" [1], [2]. The main contribution of this paper is to propose and analyze a compartmental model of freeway traffic networks that is amenable to analysis as a dynamical system. Existing approaches are often well-suited for simulations or for validation/fitting with empirical data, but the available literature often gives little insight into the network-level, qualitative properties of the dynamics. For example, models such as [3], [4] and the celebrated Cell Transmission Model (CTM) [5], [6] were primarily developed for simulation with few analytical results available. The primary exception is [7] which provides a thorough investigation of the CTM when modeling a stretch of highway with onramp queues but does not consider more general networks.We propose a model that encompasses the CTM and extends the model to general nonlinear supply and demand functions and to more general network topologies. In our proposed model, we consider a traffic network composed of road links interconnected at junctions. In keeping with the philosophy of the CTM, the flow of traffic through a junction is determined by the available supply of downstream road space into which vehicles can flow and upstream demand of vehicles wishing to flow into a given link.Our work is related to the dynamical flow networks recently proposed in [8], [9] and further studied in [10]. In [8], [9], downstream supply is not considered and therefore the flow exiting a link is equal to the link's demand. Thus downstream congestion does not affect upstream flow, an unrealistic assumption for traffic modeling. In [10], the authors allow flow to depend on the density of downstream links, but the paper focuses on throughput optimality of a particular class of routing policies that does not accomodate most models of traffic flow, including the proportional-priority, first-in-first-out rule considered in this paper, and limited theoretical results are given for general routing policies.

Separability of Lyapunov functions for contractive monotone systems

“…The matrix measures given in (19) and (20) provide the connection to sum-separable and max-separable Lyapunov functions which are the focus of this paper. We are particularly interested in vector norms and matrix measures that arise from a scaling of another norm.…”

Formal Synthesis of Control Strategies for Positive Monotone Systems

“…The primary objective in our traffic management approach is finding control policies such that the state is restricted to Π, which not only eliminates congestion, but also ensures that the system is monotone hence the methods of this paper become applicable. It is worth to note that the traffic system becomes non-monotone when flow is congested in diverging junctions, as shown in [49]. This phenomena is attributed to the first-infirst-out (FIFO) nature of the model.…”

“…Specification: As mentioned earlier, the primary objective is keeping the state in the congestion-free set. In addition, since the demand for the north-south side roads (links [49][50][51][52][53] is smaller than the traffic in the east-west roads, we add a timed liveness requirement for the traffic flow on links 49-53: ψ = l=49,50,··· ,53…”

Global exponential stabilization of freeway models

“…(and by allowing them to be time-varying), we have taken into account all possible cases for the relative priorities of the inflows (and we also allow the priority rules to be time-varying); see [6,14,16,19,20] for freeway models with specific priority rules, which are special cases of our general approach.…”