User-Equilibrium Properties of Fixed Points in Dynamic Traffic Assignment∗

This paper considers the problem of dynamic traffic assignment under the principle that individual drivers will choose fastest paths, in the dynamic situation where path durations consist of time-dependent link travel times. Rather than constructing a unified model encompassing traffic dynamics and route choice, we decompose the model into an assignment mapping, which identifies the link travel times resulting from an input routing policy, and a routing mapping, which yields fastest-path routings associated with input link travel times. Since time-dynamic link travel times are influenced by route choice, this dynamic situation therefore encompasses predictive routing strategies. We establish that user-equilibrium routing policies are fixed points of the composition of the routing and assignment functions. After discussing difficulties associated with establishing existence of fixed points under discrete-time modeling and all-ornothing routing, we present instead new iterative routing mappings for continuous-time multipath routing (the splitting of a single-class flow onto multiple paths), which adjust routing policies more incrementally. We provide sufficient conditions for existence of fixed points in various routing policy domains and offer some suggestions on the computation of these fixed-point policies. ∗This research was supported in part by the University of Michigan ITS Center. †Business Operations Analysis, AT&T, Room 2B19, 379 Campus Drive, Somerset, NJ 08873 ‡Department of Industrial and Operations Engineering, University of Michigan, Ann Arbor, MI 481092117 §Mitretek Systems, Inc., 600 Maryland Avenue, SW, Suite 755, Washington, DC 20024