Nash flows over time with spillback

Abstract

Modeling traffic in road networks is a widely studied but challenging problem, especially under the assumption that drivers act selfishly. A common approach used in simulation software is the deterministic queuing model, for which the structure of dynamic equilibria has been studied extensively in the last couple of years. The basic idea is to model traffic by a continuous flow that travels over time from a source to a sink through a network, in which the arcs are endowed with transit times and capacities. Whenever the flow rate exceeds the capacity a queue builds up and the infinitesimally small flow particles wait in line in front of the bottleneck. Since the queues have no physical dimension, it was not possible, until now, to represent spillback in this model. This was a big drawback, since spillback can be regularly observed in real traffic situations and has a huge impact on travel times in highly congested regions. We extend the deterministic queuing model by introducing a storage capacity that bounds the total amount of flow on each arc. If an arc gets full, the inflow capacity is reduced to the current outflow rate, which can cause queues on previous arcs and blockages of intersections, i.e., spillback. We carry over the main results of the original model to our generalization and characterize dynamic equilibria, called Nash flows over time, by sequences of particular static flows, we call spillback thin flows. Furthermore, we give a constructive proof for the existence of dynamic equilibria, which suggests an algorithm for their computation. This solves an open problem stated by Koch and Skutella in 2010 [13].