This paper studies the dynamics of the traveler's daytoday route adjustment process in the general transportation network with fixed or elastic demand and homogeneous or heterogeneous users. Each traveler is assumed to adjust his/her route choice according to the excess travel cost between the instantaneous experienced travel cost and a certain referred level, which induces an aggregate path flow dynamics. We call the path flow dynamics the excess travel cost dynamics, which corresponds to the excess payoff dynamics in evolutionary games and serves a general framework of modeling the homogeneous or heterogeneous route choice behavior of travelers.
Network equilibrium models have a long history, both in the transport research literature and in practical studies of scheme assessment, since the notion of user equilibrium (UE) was proposed [
To model the travelers' response to exogenous information, the growing interest in daytoday dynamics has received much study after the original research of Horowitz [
However, if the road pricing scheme is adopted in the transportation network as Sandholm [
The paper is organized as follows. Section
Consider a general transportation network,
Three assumptions on travelers' route adjustment process are proposed and compared by Nagurney and Zhang [
Suppose that a welldefined excess travel cost is associated with each path, such as the difference between the travel cost and the average travel cost corresponding to a specific OD pair. It is easy to check that, both the aggregated travel cost and excess travel cost of the entire network decreases based on the previous day's path travel costs when path flow pattern changes from day to day, whenever the process is RBAP. The logic is intuitive since the flow process follows the conservation constraint if the excess travel cost is defined as the difference between the path cost and average ODspecific travel cost, namely,
For the case with elastic demand, notably, the inverse demand function,
dynamic system (
With the excess travel cost dynamics, the aggregated excess travel cost of the entire network must decrease from the previous day's path travel costs when path flows change from day to day; otherwise, if path flow becomes stationary over days, then it is equivalent to the user equilibrium path flow. Define the excess payoff for using each path as
The excess travel cost dynamics, described above, assumes that all travelers perceive the path travel cost identically (travel time plus a discount monetary cost). However, it is well known that travelers may value travel time differently, depending on their income levels or travel purposes. Thus, when a toll policy is introduced into the network, heterogeneous users with different value of time (VOT) have to be considered in converting travel time into equivalent monetary cost. In this subsection, we focus our discussion on the behavior assumption of the route choice adjustment process in presence of the user heterogeneity. Furthermore, we consider a discrete set of user classes corresponding to the groups of users with different socioeconomic characteristics, such as income level.
Let
Assumption (A2′) says that, for any given and fixed link toll pattern,
The proportionalswitch adjustment process (PSAP) was first introduced by Smith [
We now consider the elastic demand based on the simple model of PSAP. Suppose that there is a dummy link associated with each OD pair and the latent noncar travelers traverse from the origin to the destination by the dummy link. Suppose there is a large potential demand for each OD pair,
The stability is a central issue for the dynamic system analysis. Peeta and Yang [
A set
Consider the autonomous system
The standard LaSalle's invariant set theorem described above is in general not applicable in the case of discontinuous righthand side, such as, projected dynamic system [
It is well known that the user equilibrium path flow pattern with elastic (fixed) demand can be identified by solving the following typical elastic (fixed) demand traffic assignment problem
The set
From assumptions (A2)(A3), we know that
From Lemma
Under assumptions (A1)–(A3), any solution of system (
Consider the following continuously differentiable function (we only consider the case with elastic demand):
The rational behaviour adjustment process proposed by Yang and Zhang [
Furthermore, for any given link toll pattern,
Consider that the following simple network example, shown in Figure
OD pairs and path sets.
OD pair  Path 


Path 1: 

Path 3: 
 
2 
Path 2: 

Path 4: 
The network for Example
By simple calibration, we know that all nonnegative path flow pattern satisfying condition
Using MATLAB Simulink, the behavior of the dynamic system (
State trajectories demonstrating stability of equilibria.
We now move to examine the system stability when the travelers are heterogeneous in valuation of travel time. It is well known [
Consider the following continuously differentiable function:
Similar analysis can be conducted for the case with elastic demand. In this case, the static multiclass traffic assignment problem with elastic demand can be formulated as the following equivalent minimization problem [
Under assumptions (A1), (A2′), and (A3), for any given and fixed link toll pattern, any solution of system (
Although the route adjustment behavior assumption is relaxed to consider the heterogeneity of travelers, that is to say the individual traveler adjusts his/her route choice based on his/her own perception, the multiclass UE link flow pattern is obtained. Theorem
Consider the network example [
OD pairs and path sets.
OD pair  Path 

1 
Path 1: 

Path 3: 
 
2 
Path 2: 

Path 4: 
The network for Example
State trajectories demonstrating stability of UE flow on Path 1.
State trajectories demonstrating stability of UE flow on Path 4.
We presented an excess travel cost dynamics to model travelers' route choice behavior in a transportation network with fixed or elastic demand and homogeneous or heterogeneous users. The excess travel cost dynamics is a natural extension of the rational behavior adjustment process proposed by Yang and Zhang [
It is an interesting challenge to design the system optimum pricing scheme in a transportation network by taking into account both the hidden actions of travelers' daytoday route choice adjustment and the true but unknown demands, specially, with heterogeneous users. Another research direction is to model the heterogeneous behavior of the travelers on their perceived travel cost except the way valuing the monetary and time costs, which requires a stochastic dynamic route choice process incorporating the user heterogeneity.
The authors are very grateful to the referees for their valuable suggestions, which helped to improve the paper significantly. The work described in this paper was supported by the Research Fund for the Doctoral Program of Higher Education of China (20110142120086), the selfdetermined and innovative research funds of HUST (2011QN103).