On the Existence of Solutions to the Dynamic User Equilibrium Problem

This paper is concerned with the existence of solutions to a dynamic network equilibrium problem modelled as an innnite dimensional variational inequality. Our results are based on properties of operators that map path ow departure rates to consistent time-dependent path ows and other link performance functions. The existence result requires the introduction of a novel concept that strengthens the familiar concept of FIFO (`First-In-First-Out'). Following Ben-Akiva 3], one can identify four main elements in any dynamic network equilibrium model (DNEP): arc performance functions, path choice criterion, a demand model, ow conservation relationships at the nodes of the network. According to this author, the extension of a static to a dynamic setting requires the determination of time-dependent arc performance functions, temporal origin-destination demand, and the assessment of queueing eeects on time-varying arrival rates. In general, these relationships are not available in closed form, and consequently analytical properties of the model are diicult, if not impossible, to derive. Several models of dynamic equilibrium have been proposed, that involve only route choice ((6] 10] 12] 16] 17]) or route choice and departure time choice ((2] 5] 10] 14] 18]). In all the abovementioned models, the issue of existence of a dynamic equilibrium has not been fully addressed. In this paper we give an existence result for the dynamic User Equilibrium { Route Choice model introduced by Friesz et al 5], thus giving a theoretical foundation for the numerical algorithms that have been proposed for this model. The analysis is performed under the assumption that ow throughout the network obeys the FIFO (`First-In-First-Out') rule. In the absence of such a queue discipline, ow propagation can be irrealistic (see Astarita 1]). We believe that analytic models should ensure that FIFO is satissed. A byproduct of our analysis will be a theoretical analysis of the loading procedure (see also Xhu et al. 19]) that relates the link performance functions ((ow volumes, entry rates,