A Variational Inequality Formulation of the Dynamic Network User Equilibrium Problem

Abstract

In the present paper we are concerned with developing more realistic dynamic models of route choice and departure time decisions of transportation network users than have been proposed in the literature heretofore. We briefly review one class of models that is a dynamic generalization of the static Wardropian user equilibrium, the so-called Boston traffic equilibrium. In contrast, we then propose a new class of models that is also a dynamic generalization of the static Wardropian user equilibrium. In particular, we show for the first time that there is a variational inequality formulation of dynamic user equilibrium with simultaneous route choice and departure time decisions which, when appropriate regularity conditions hold, preserves the first in, first out queue discipline.

Probabilistic travel time progression and its application to automatic vehicle identification data

“…AVI) is concerned, the road travel time is best estimated from the sum of link travel times. Recovering the road travel time from the individual links is typically carried out through a method known as travel time progression (Friesz et al, 1993, Nanthawichit et al, 2013, Shim et al, 2011. Using the formalisation of Friesz et al (1993), given a road defined as a sequence of N connected nodes, we will let τ n,n−1 be the travel time for link (n − 1, n); and ε n (t) the departure time from node n ∈ {1, .…”

The existence, uniqueness and computation of an arc-based dynamic network user equilibrium formulation

“…This approach is originally proposed by Friesz et al (1993) and extended by Astarita (1996), Wu et al (1998) and Xu et al (1999). The advantage of this approach is to ensure internal consistencies among arc traffic dynamics, flow propagation constraints and arc delay functions under plausible regularity conditions.…”

“…In the continuous-time models of Friesz et al (1993), Astarita (1996), Wu et al (1998), andXu et al (1999), the arc travel time for vehicles entering the arc at time t is treated as a function of the number of vehicles present on the arc at time t. However, because we are working in discrete time, we must include the arc inflow in the travel time function to capture the average number of vehicles on the arc during each time period. Otherwise, it would be possible in a particular time period to load the network heavily without the delay that reflects the traffic volume accumulated during that time period.…”

“…This paper is mainly concerned with a particular class of dynamic traffic assignment problem, that is, a user equilibrium simultaneous departure time and route choice problem with elastic demands. A similar problem with fixed demands has been studied in the works of Friesz et al (1993) and Wie et al (1995a,b) that present the route-based variational inequality models and algorithms and in the work of that presents an arc-based variational inequality model.…”

“…It should be noted that our arc-based approach to formulate a dynamic user equilibrium model is a significant departure from the path-based formulation used in Friesz et al (1993). It is not trivial how to handle the arc dynamics in discrete time in a manner in which the outflows and the arc travel times are consistent.…”