Dynamics of a Transportation Network Model with Homogeneous and Heterogeneous Users

Abstract

This paper studies the dynamics of the traveler's day-to-day 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.

1. Introduction

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 [1]. At equilibrium, all travelers on the same origin-destination pair experience the same travel cost, while all unused routes have equal or higher travel costs. Wardrop equilibrium has been extended in different senses to address real world needs in transportation planning, analysis, and management, including stochastic user equilibrium (SUE) to allow imperfect information or user perception error in their route choice [2]; dynamic user equilibrium within-day to incorporate the departure time choice in route choice [3, 4].

To model the travelers' response to exogenous information, the growing interest in day-to-day dynamics has received much study after the original research of Horowitz [5], in which a two-link transportation network is adopted to explain the day-to-day dynamic link and path flow patterns and the stability issue. The evolution over days of travel choices or a learning rule based on the past experience associated with the traffic assignment model are the main methodologies to capture the day-to-day behavior. A comprehensive literature review on the day-to-day dynamics is made by Bie and Lo [6]. In the literature, travelers' day-to-day route choice adjustment behavior is modeled as either a deterministic dynamic process, such as the simplex gravity flow dynamics [7], the proportional-switch adjustment process [8, 9], the network tatonnement process [10], the projected dynamical system [11], and evolutionary traffic dynamics [12], or a stochastic dynamic process [13, 14]. Yang and Zhang [15] focused on the deterministic dynamic route adjustment process on modeling drivers dynamic route choice adjustment behavior and proved that many commonly used dynamic processes are so-called Rational Behavior Adjustment Process (RBAP) since the aggregated travel cost of the entire network decreases over days except that the path flow becomes stationary and is the user equilibrium path flow. He et al. [16] developed a link-based day-to-day dynamic traffic assignment model to copy with two essential pitfalls of the path-based model, that is, the initial path flow pattern is typically unidentifiable and different initial path flow patterns constituting the same link flow pattern generally give different day-to-day link flow evolutions; the path-based models ignore the interdependence among paths and thus can give very unreasonable results for networks with paths overlapping with each other. The day-to-day route adjustment process provides a useful tool for transportation management. Peeta and Yang [17] demonstrated that providing properly advanced travel information to the road users can result in the social optimal link flow pattern by assuming that the travelers' day-to-day route adjustment behavior follows the proportional-switch adjustment process. Using the potential games theory, Sandholm [12] proposed an evolutionary implementation mechanism for continuous dynamic route-based pricing policy to ensure the efficient use of the roadway system, while allowing for users to follow any reasonable myopic adjustment process. Apart from the unknown demand or demand function, he assumed that the individual traveler's day-to-day route choice adjustment process is not known to the planner either.

However, if the road pricing scheme is adopted in the transportation network as Sandholm [12], the day-to-day route adjustment process should incorporate the travelers' heterogeneous behavior since they value the monetary and time costs in different ways. With this consideration, we assume that all travelers in the transportation network can be classified into several discrete classes, the user in each class is captured by a parameter, value-of-time (VOT) which converts the travel time cost into the equivalent monetary cost [18]. It is easy to understand that the users in different user classes adjust their route choice in different ways according to their VOT values. In this paper, we develop a general model framework to capture the travelers' day-to-day route choice behavior, which considers the general transportation network with fixed or elastic demand, homogeneous or heterogeneous users. In this study, 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. The aggregate path flow dynamics induced by this model can be viewed as a natural extension of the so-called rational behavior adjustment process proposed by [15]. We call this path flow dynamics the excess travel cost dynamics, which corresponds to the excess payoff dynamics in evolutionary game [19]. The excess travel cost dynamics serves as a more general framework than the rational behavior adjustment process for modeling the travelers' dynamic route choice behavior in the transportation network with fixed or elastic demand, homogeneous or heterogeneous users. The Lyapunov function method is commonly used to study the stability of the equilibrium solutions [8, 10]. LaSalle's theorem addressing system stability over a set of equilibrium solutions rather than an isolated point [20] has been adopted by several researchers to study the stability of dynamic transportation network [6, 17, 21]. In this study, we use LaSalle's theorem to study the stability of the excess travel cost dynamics.

The paper is organized as follows. Section 2 introduces the definition of the excess travel cost dynamics. The stability issue for the proposed dynamic system is addressed in Section 3. Finally, conclusions are given in Section 4.

2. Day-to-Day Dynamical Route Adjustment Process

2.1. Excess Travel Cost Dynamics with Homogeneous Users

Consider a general transportation network, 𝐺(𝑁,𝐴), where 𝑁 is the set of nodes and 𝐴 is the set of links. Denote 𝑊 by the set of origin-destination (OD) pairs 𝑅_𝑤 by the set of routes between O-D pair 𝑤, and 𝑅=⋃_𝑤∈𝑊𝑅_𝑤. Let 𝑓^𝑤_𝑟, 𝑣_𝑎, and 𝜏_𝑎 be the path flow on route 𝑟, 𝑟∈𝑅_𝑤, the flow and toll of link 𝑎, 𝑎∈𝐴, |𝐴|=𝑚. The corresponding vectors of path flows, link flows and toll charges are denoted by 𝐟, 𝐯, and 𝜏. Each link 𝑎, 𝑎∈𝐴, is associated with a strictly increasing and convex link travel time cost function, 𝑡_𝑎(𝑣_𝑎) of the link flow, 𝑣_𝑎. The total aggregate link flow is 𝑣_𝑎=∑_𝑤∈𝑊∑_{𝑟∈𝑅𝑤}𝑓^𝑤_𝑟𝛿^𝑤_𝑎𝑟, where 𝛿^𝑤_𝑎𝑟 equals 1 if route 𝑟 between OD pair 𝑤 uses link 𝑎; 0 otherwise. Without confusion, let 𝑇^𝑤_𝑟 and 𝐶^𝑤_𝑟 be the total travel time and total cost (travel time plus the monetary cost, if any) of path 𝑟∈𝑅_𝑤 between OD pair. Each OD pair 𝑤∈𝑊 is associated with a demand, which is fixed or is elastic. In the latter case the demand is given by an invertible strictly decreasing function of the generalized travel cost 𝑐_𝑤 alone, 𝑞_𝑤=𝐷_𝑤(𝑐_𝑤). The set of all feasible path flow patterns is denoted by Ω=𝐟≥0∶𝑞_𝑤=_{𝑟∈𝑅𝑤}𝑓^𝑤_𝑟,𝑤∈𝑊. (2.1) Denote the day-to-day path flow dynamics by 𝐟(̇𝑠)=(𝑓^𝑤_𝑟̇(𝑠),𝑟∈𝑅_𝑤,𝑤∈𝑊), say, the derivative of path flow vector 𝐟(𝑠) with respect to day to day time 𝑠.

Three assumptions on travelers' route adjustment process are proposed and compared by Nagurney and Zhang [11]. The third one among those assumptions is suggested to be the most reasonable and defined by Yang and Zhang [15] as the rational behavior adjustment process (RBAP) with fixed demand. The day-to-day route choice adjustment process is the RBAP with fixed travel demand if the aggregated travel cost of the entire network decreases based on the previous day's path travel costs when path flows change from day to day; otherwise, if path flow becomes stationary over day, then it is equivalent to the user equilibrium path flow. That is to say, if the current path flow pattern is not UE path flow, then ∑_𝑤∈𝑊∑_{𝑟∈𝑅𝑤}𝐶^𝑤_𝑟(𝐟)𝑓^𝑤_𝑟̇<0. The RBAP is mild and includes various commonly used route choice adjustment process as its special cases, such as the proportional-switch adjustment process [17], the projected dynamical system [11], and the Brown-von Neumann-Nash dynamics [22].

Suppose that a well-defined 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 OD-specific travel cost, namely, _{𝑤∈𝑊𝑟∈𝑅𝑤}𝐶^𝑤_𝑟(𝐟)−𝐶^𝑤(𝐟)𝑓^𝑤_𝑟̇=_{𝑤∈𝑊𝑟∈𝑅𝑤}𝐶^𝑤_𝑟(𝐟)𝑓^𝑤_𝑟̇<0. (2.2) since ∑_{𝑟∈𝑅𝑤}𝑓^𝑤_𝑟̇=0 and 𝐶^𝑤(𝐟)=(1/𝑞_𝑤)∑_{𝑟∈𝑅𝑤}𝐶^𝑤_𝑟(𝐟)𝑓^𝑤_𝑟. In fact, the reference travel cost level, 𝐶^𝑤(𝐟), may be altered as the current minimal travel cost. A more general definition of the excess travel cost is given in [19], in which, at least one traveler travels with a positive excess travel cost whenever the stationary equilibrium is not achieved. For simplification of our representation, we only refer the excess travel cost to the difference between the travel cost and the average travel cost associated a specific OD pair for the case with fixed demand, namely, ETC^𝑤_𝑟(𝐟)=𝐶^𝑤_𝑟(𝐟)−𝐶^𝑤(𝐟).

For the case with elastic demand, notably, the inverse demand function, 𝐷_𝑤⁻¹(𝑞_𝑤), is the current travel benefit between OD pair 𝑤 if the demand is 𝑞_𝑤. We thus define the excess travel cost of path 𝑟∈𝑅_𝑤 as 𝐶^𝑤_𝑟(𝐟)−𝐷_𝑤⁻¹(𝑞_𝑤),𝑤∈𝑊 or ETC^𝑤_𝑟(𝐟)=𝐶^𝑤_𝑟(𝐟)−𝐷_𝑤⁻¹(𝑞_𝑤). To incorporate the cases with both fixed and elastic demands, we now extend the definition of rational behavior adjustment process proposed by Yang and Zhang [15]. Denote the path flow dynamics as the following equation: 𝐟̇=𝐹(𝐟). (2.3) The day-to-day route adjustment process (2.3) is the excess travel cost dynamics if the following three conditions are satisfied:(A1) dynamic system (2.3) admits a unique solution trajectory for any given initial condition, and the trajectory is Lipschitz continuous;(A2)∑_𝑤∈𝑊∑_{𝑟∈𝑅𝑤}ETC^𝑤_𝑟𝐹^𝑤_𝑟<0 whenever 𝐹(𝐟)≠0 where the excess travel cost, ETC^𝑤_𝑟, of path 𝑟∈𝑅_𝑤 between 𝑤∈𝑊 is a well-defined function of link flow vector 𝐟;(A3)𝐟^∗ is an equilibrium point of system (2.3) if 𝐹(𝐟^∗)=0.

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 −ETC, then the excess travel cost dynamics is the excess payoff dynamics, which is a special case of the well-behaved dynamics proposed by Sandholm [19].

2.2. Excess Travel Cost Dynamics with Heterogeneous Users

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 𝑀 denote such user classes, and 𝛽_𝑚, 𝛽_𝑚>0, be the average VOT for users of class 𝑚∈𝑀. The travelers of each class adjust their routes according to their owner perception for any given and fixed link toll pattern, 𝜏={𝜏_𝑎,𝑎∈𝐴}. Let 𝑇^𝑤_𝑟(𝐟) and 𝜏^𝑤_𝑟 denote the travel time and toll of path 𝑟∈𝑅_𝑤 between OD pair 𝑤∈𝑊, where we further suppose that each OD pair and user class is associated with a fixed or elastic demand. In the latter case, the classic-specific demand is given by a strictly decreasing function 𝑞^𝑚_𝑤=𝐷^𝑚_𝑤(𝜇^𝑚_𝑤), where 𝜇^𝑚_𝑤 is the generalized travel cost in time unit for user class 𝑚∈𝑀 between OD pair 𝑤∈𝑊. The set of the feasible path flow pattern is now expressed as Ω=𝐟≥0∶𝑞^𝑚_𝑤=_{𝑟∈𝑅𝑤}𝑓_𝑟^𝑤,𝑚,𝑤∈𝑊,𝑚∈𝑀. (2.4) Notably, the travelers with different VOT may estimate their excess travel cost differently. The average travel cost of OD pair 𝑤∈𝑊 estimated by the users of class 𝑚 is 𝐶^𝑚_𝑤(𝐟)=(1/𝑞^𝑚_𝑤)∑_{𝑟∈𝑅𝑤}(𝑇^𝑤_𝑟(𝐟)+(𝜏^𝑤_𝑟/𝛽_𝑚))𝑓^𝑤_𝑟. The excess travel cost for user class 𝑚 to use path 𝑟∈𝑅_𝑤 between OD pair 𝑤∈𝑊 with fixed demand can be expressed as ETC_𝑟^𝑤,𝑚(𝐟)=𝐶_𝑟^𝑤,𝑚(𝐟)−𝐶^𝑚_𝑤. As before for the case with elastic demand, the excess travel cost of path 𝑟∈𝑅_𝑤 between OD pair 𝑤∈𝑊 for user class 𝑚∈𝑀 is ETC_𝑟^𝑤,𝑚(𝐟)=𝐶_𝑟^𝑤,𝑚(𝐟)−𝐷⁻¹_𝑤,𝑚(𝑞^𝑚_𝑤). Note that, for the multiclass day-to-day dynamic route choice adjustment process (2.3), the term, 𝐟̇=(𝑓̇_𝑟^𝑤,𝑚,𝑟∈𝑅_𝑤,𝑤∈𝑊,𝑚∈𝑀), is class-specific path flow derivative. Similarly, the aggregate class-specific link flow 𝐯^𝑀, 𝐯^𝑀=(𝑣^𝑚_𝑎,𝑎∈𝐴,𝑚∈𝑀), where the aggregate link flow of user class 𝑚 is 𝑣^𝑚_𝑎=∑_𝑤∈𝑊∑_{𝑟∈𝑅𝑤}𝑓_𝑟^𝑤,𝑚𝛿^𝑤_𝑎𝑟. The multiclass day-to-day route adjustment process (2.3) is rational if assumptions (A1), (A3), and the following (A2) are satisfied:(A2′)∑_𝑤∈𝑊∑_{𝑟∈𝑅𝑤}∑_𝑚∈𝑀ETC_𝑟^𝑤,𝑚𝐹_𝑟^𝑤,𝑚<0 whenever 𝐹(𝐟)≠0.

Assumption (A2′) says that, for any given and fixed link toll pattern, 𝜏={𝜏_𝑎,𝑎∈𝐴}, each user adjusts his/her route choice according to his/her own preference, which results in the decrease in aggregated excess travel cost of the entire network in time unit in comparison with that on previous day when path flows change from day to day. A stronger behavior assumption is that the flow on those paths on which travelers experienced positive excess travel cost strictly decreases, and vice versa. Assumption (A2′) is a direct extension of the homogeneous case, in which all travelers have an identical VOT.

Example 2.1. The proportional-switch adjustment process (PSAP) was first introduced by Smith [8] to suppose that the drivers on a higher cost path switch to other lower cost paths in a rate that is proportional to the cost difference between this path and the other lower cost paths. Peeta and Yang [17] modeled the dynamical traffic flow switch rate of the PSAP with fixed demand as: 𝑓^𝑤_𝑟̇=_{𝑟′∈𝑅𝑤}𝑓^𝑤_𝑟′𝐶^𝑤_𝑟′(𝐟)−𝐶^𝑤_𝑟(𝐟)₊−𝑓^𝑤_𝑟𝐶^𝑤_𝑟(𝐟)−𝐶^𝑤_𝑟′(𝐟)₊, (2.5) where 𝐶^𝑤_𝑟 is the travel cost of path 𝑟∈𝑅_𝑤 between OD pair 𝑤, and [𝑦]₊=max{0,𝑦}. By introducing the excess travel cost, ETC^𝑤_𝑟, for each path, dynamic system (2.5) can be rewritten as: 𝑓^𝑤_𝑟̇=_{𝑟′∈𝑅𝑤}𝑓^𝑤_𝑟′ETC^𝑤_𝑟′(𝐟)−ETC^𝑤_𝑟(𝐟)₊−𝑓^𝑤_𝑟ETC^𝑤_𝑟(𝐟)−ETC^𝑤_𝑟′(𝐟)₊. (2.6) Furthermore, the PSAP with fixed demand, given by (2.6), can be extended to the multiclass user case. Given a toll pattern 𝜏=(𝜏_𝑎,𝑎∈𝐴), the travel cost and excess travel cost in time unit for the travellers with VOT 𝛽_𝑚 are 𝐶_𝑟^𝑤,𝑚=𝑇^𝑤_𝑟+𝜏^𝑤_𝑟/𝛽_𝑚 and 𝐶^𝑤,𝑚(𝐟)=(1/𝑞_𝑤)∑_{𝑟∈𝑅𝑤}(𝑇^𝑤_𝑟+𝜏^𝑤_𝑟/𝛽_𝑚)𝑓^𝑤_𝑟. The multiclass PSAP with fixed demand, for all 𝑚∈𝑀, 𝑟∈𝑅_𝑤 and 𝑤∈𝑊, is 𝑓̇_𝑟^𝑤,𝑚=_{𝑟′∈𝑅𝑤}𝑓_𝑟^𝑤,𝑚_′ETC_𝑟^𝑤,𝑚_′(𝐟)−ETC_𝑟^𝑤,𝑚(𝐟)₊−𝑓_𝑟^𝑤,𝑚_{𝑟′∈𝑅𝑤}ETC_𝑟^𝑤,𝑚(𝐟)−ETC_𝑟^𝑤,𝑚_′(𝐟)₊. (2.7)
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, 𝑄_𝑤. The dummy link, 𝑟⁻_𝑤, is associated with the travel cost of 𝐷_𝑤⁻¹(𝑞_𝑤) and flow 𝑞_𝑤, where 𝑞_𝑤 and 𝑞_𝑤 denote the revealed and latent demand between OD pair, 𝑤∈𝑊, respectively. Namely, 𝐶^𝑤_𝑟−𝑤=𝐷_𝑤⁻¹(𝑞_𝑤) and 𝑓^𝑤_𝑟−𝑤=𝑞_𝑤=𝑄_𝑤−𝑞_𝑤. Let the average travel cost be 𝐶^𝑤(𝐟)=(1/𝑄_𝑤)(∑_{𝑟∈𝑅𝑤}𝐶^𝑤_𝑟𝑓^𝑤_𝑟+𝐷_𝑤⁻¹(𝑞_𝑤)𝑞_𝑤). Like the case with fixed demand, the excess travel cost is defined as the difference between the travel cost and the average travel cost associated with the specific path. The PSAP with elastic demand can be revised, for 𝑟∈𝑅_𝑤∪{𝑟⁻_𝑤} and 𝑤∈𝑊, as 𝑓^𝑤_𝑟̇=_{𝑟′∈𝑅𝑤∪𝑟−𝑤}𝑓^𝑤_𝑟′ETC^𝑤_𝑟′(𝐟)−ETC^𝑤_𝑟(𝐟)₊−𝑓^𝑤_𝑟ETC^𝑤_𝑟(𝐟)−ETC^𝑤_𝑟′(𝐟)₊. (2.8) Like the case with fixed demand again, the PASP with elastic demand, given by (2.8), can be extended to the multiclass PSAP indexed by user class 𝑚∈𝑀. Following the method described above and the definition of the excess travel cost with multiclass users, the Brown and von Neumann process can be extended to the cases with elastic demand and user heterogeneity.

3. Stability of the Excess Travel Cost Dynamics

The stability is a central issue for the dynamic system analysis. Peeta and Yang [17] investigated the system stability over a set of desirable states rather than an isolated point, applying LaSalle's invariant set theorem for the dynamic traffic assignment problem with fixed demand by assuming the proportional-switch adjustment process, given by (2.5). The objective functions of the system optimum and user equilibrium problem are adopted as the Lyapunov functions to conduct the stability analysis, which is physically meaningful. Similar method of constructing Lyapunov function is also used by Jin [21] to examine the system stability of the dynamic traffic assignment problem under OD first-in-first-out violation situation. In this section, in a similar spirit of [17], the system stability of the excess travel cost dynamics proposed in Section 2 is addressed.

A set Π is an invariant set of dynamic system (2.3) if every system trajectory which starts from a point in Π remains in Π for all future time [20]. For instant, any equilibrium point of system (2.3) or, 𝐹(𝐟)=𝟎, as a set is an invariant set. LaSalle's invariant set theorem is stated as the follows.

Lemma 3.1 (LaSalle’s invariant set theorem [20]). Consider the autonomous system 𝐱̇=𝐹(𝐱). Let 𝑉(𝐱) be a scalar continuously differentiable function such that(1)lim_|𝐱|→∞𝑉(𝐱)=∞, or, function 𝑉(𝐱) is radically unbounded; (2)𝑉̇(𝐱)≤0 for all 𝐱∈Ω.Let 𝑆={𝐱∈𝑅^𝑛|𝑉̇(𝐱)=0} and Π is the largest invariant set contained in 𝑆, namely, Π is the union of all invariant sets within 𝑆. Then every solution 𝐱(𝑠) is bounded and converges to Π.

The standard LaSalle's invariant set theorem described above is in general not applicable in the case of discontinuous right-hand side, such as, projected dynamic system [11] and the network tatonnement process [10]. In this case, the generalized invariant set theorem [23] is applicable, only requiring the existence, uniqueness, and continuity of the solution of dynamic system that are guaranteed by assumption (A1). In this study, the standard LaSalle's invariant set theorem is adopted to investigate the stability issues of dynamic system (2.3).

3.1. Stability Property of the Dynamic System with Homogeneous Users

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 min_𝐟∈Ω𝐿=_𝑎∈𝐴^𝑣𝑎₀𝑡_𝑎(𝜔)𝑑𝜔−_𝑤∈𝑊^𝑞𝑤₀𝐷_𝑤⁻¹(𝜔)𝑑𝜔 (3.1) or min_𝐟∈Ω𝐿=_𝑎∈𝐴^𝑣𝑎₀𝑡_𝑎(𝜔)𝑑𝜔. (3.2) Under the assumptions that 𝐷_𝑤⁻¹(⋅) is strictly decreasing and 𝑡_𝑎(⋅) is strictly increasing, we know that the Wardrop user equilibrium link flow pattern is unique for the case with fixed demand, and the UE demand level is also unique for the case with elastic demand. Denote the set of user equilibrium path flow patterns as Π=𝐟≥Ω∶_{𝑤∈𝑊𝑟∈𝑅𝑤}𝑓^𝑤_𝑟𝛿^𝑤_𝑎𝑟=𝑣_𝑎,𝑎∈𝐴;_𝑤∈𝑊𝑓^𝑤_𝑟=𝑞_𝑤,𝑤∈𝑊, (3.3) where 𝐯={𝑣_𝑎,𝑎∈𝐴} and 𝐪={𝑞_𝑤,𝑤∈𝑊} are unique UE link flow pattern and corresponding OD demand vector, respectively. It is must be pointed out that, for the case with fixed demand, 𝐪 is the OD demand matrix, and thus, definition (3.3) is suit for both cases with fixed and elastic demand. We first prove that the set (3.3) is an invariant set of system (2.3), that is to say, if any solution with an initial state in set Π will reside in Π for ever, which is depicted in the following lemma.

Lemma 3.2. The set Π, defined by (3.3), is an invariant set of system (2.3), namely, 𝐟(0)∈Π implies that 𝐟(𝑠)∈Π, for all 𝑠≥0.

Proof. From assumptions (A2)-(A3), we know that 𝐹(𝐟)≠0⇒ETC(𝐟)∘𝐹(𝐟)<0 and, conversely, ETC(𝐟)∘𝐹(𝐟)=0⇒𝐹(𝐟)=0, which implies that 𝐟 is an equilibrium point. It is clear that Π={𝐟∈Ω∶ETC(𝐟)=0}={𝐟∈Ω∶𝐹(𝐟)=0} is an invariant set of system (2.3). The proof is completed.

From Lemma 3.2, we now investigate the stability of system (2.3).

Theorem 3.3. Under assumptions (A1)–(A3), any solution of system (2.3) converges to the invariant set, Π, defined by (3.3), and, thus, the dynamic path flow pattern with any initial state induces the user equilibrium link flow pattern of static traffic assignment problem.

Proof. Consider the following continuously differentiable function (we only consider the case with elastic demand): 𝑉(𝐟)=_𝑎∈𝐴^𝑣𝑎₀^(𝐟)𝑡_𝑎(𝜔)𝑑𝜔−_𝑤∈𝑊^𝑞𝑤₀^(𝐟)𝐷_𝑤⁻¹(𝜔)𝑑𝜔−min{𝐿}, (3.4) where 𝐿 is the objective function of mathematical programming problem (3.1); 𝑞_𝑤, 𝑤∈𝑊, and 𝑣_𝑎, 𝑎∈𝐴, are functions of path flow pattern 𝐟. Let 𝑆={𝐟∈𝑅^𝑛∣𝑉̇(𝐟)=0}, and from Lemma 3.2, the largest invariant set contained in 𝑆 is Π. In fact, Π=𝑆 since 𝑆 is the solution set of the minimization problem (3.1). Furthermore, note that 𝑉(𝐟)>0 for all 𝐟∈Ω⧵𝑆, and 𝑉(𝐟)=0 for all 𝐟∈𝑆. Therefore 𝑉(𝐟) is a Lyapunov function candidate of system (2.3). Taking derivative of 𝑉(𝐟) in day-to-day time, and from assumption (A2), we get, for all 𝐟∈Ω, 𝑉̇(𝐟)=𝜕𝑉𝜕𝑓^𝑤_𝑟,𝑟∈𝑅_𝑤,𝑤∈𝑊𝐟̇=_{𝑤∈𝑊𝑟∈𝑅𝑤}ETC^𝑤_𝑟𝐹^𝑤_𝑟(𝐟)≤0, (3.5) which implies that 𝑉(𝐟) is a Lyapunov function of system (2.3). According to the LaSalle's invariant set theorem, we know that all solutions of system (2.3) converge to Π, which induces the UE link flow pattern. For the case with fixed demand, we can also show that the derivative of 𝑉(𝐟) is nonpositive since ∑_{𝑟∈𝑅𝑤}𝑓^𝑤_𝑟̇=0 according to the flow conservation condition. The proof is completed.

The rational behaviour adjustment process proposed by Yang and Zhang [15] is a useful concept frame to model the rational behaviour of the travellers under which the route adjustment process must reduce the aggregate total travel cost of the whole network day by day. However, the definition is suitable only for the case with fixed demand. More generally, the excess travel cost dynamics, or, excess payoff dynamics introduced by Sandholm [19] can incorporate the cases with fixed and elastic demand. Theorem 3.3 depicts that for the excess travel cost dynamics satisfying assumptions (A1)–(A3), any dynamic path flow pattern must move to a certain UE path flow pattern, which induces the UE link flow pattern. Therefore, the set of all UE path flow patterns is a stable invariant set.

Furthermore, for any given link toll pattern, 𝜏={𝜏_𝑎,𝑎∈𝐴}, the travel cost of path 𝑟∈𝑅_𝑤 between OD pair 𝑤∈𝑊 is the full travel cost (the time cost plus the monetary cost), and the excess travel cost can be defined similarly as the case 𝜏=0, namely, 𝐶^𝑤_𝑟(𝐟)=∑_𝑤∈𝑊∑_{𝑟∈𝑅𝑤}(𝑡_𝑎(𝑣_𝑎(𝐟))+𝜏_𝑎/𝛽)𝛿^𝑤_𝑎𝑟. Then the set of the UE path flow patterns corresponding to the toll vector 𝜏={𝜏_𝑎,𝑎∈𝐴} is a stable invariant set of system (2.3) or any dynamic path flow pattern must converge to the set. Specially, if the system optimal and fixed link toll pattern, 𝜏={𝜏_𝑎=MCP(𝑣_𝑎(𝐟)̃),𝑎∈𝐴} is imposed, where 𝐟̃ is the SO path flow pattern for fixed- or elastic-demand case, then any dynamic path flow pattern converges to a certain SO path flow pattern, in this case, the invariant set is Π=𝐟≥Ω∶_{𝑤∈𝑊𝑟∈𝑅𝑤}𝑓^𝑤_𝑟𝛿^𝑤_𝑎𝑟=𝑣_𝑎,𝑎∈𝐴;_𝑤∈𝑊𝑓^𝑤_𝑟=𝑞_𝑤,𝑤∈𝑊, (3.6) where 𝐯={𝑣_𝑎,𝑎∈𝐴} and 𝐪={𝑞_𝑤,𝑤∈𝑊} are unique SO link flow pattern and corresponding OD demand vector, respectively.

Example 3.4. Consider that the following simple network example, shown in Figure 1, consists of 4 links, 4 nodes, and 2 OD pairs. The OD pairs and the corresponding path are listed in Table 1. Each OD pair has one unit fixed demand. Since all links have the same link travel time function, 𝑡_𝑖(𝑣_𝑖)=𝑣_𝑖, it is clear that the user equilibrium path flow pattern is not unique although there is a unique link flow pattern.

Table 1 

OD pairs and path sets.

Figure 1 

The network for Example 3.4.

By simple calibration, we know that all nonnegative path flow pattern satisfying condition 𝑓₁+𝑓₂=1, 𝑓₁=𝑓₄ and 𝑓₂=𝑓₃ are user equilibrium path flow pattern, which induces the unique link flow pattern, 𝑣_𝑖=1 for all links, 𝑖∈{1,2,3,4}. The proportional-switch adjustment process (2.5) is expressed as 𝑓₁̇𝑓₂̇=[𝑌]₊−[𝑌]₊+[−𝑌]₊𝑓₁[𝑌]₊−[𝑌]₊+[−𝑌]₊𝑓₂, (3.7) where 𝑌=2−2𝑓₁+𝑓₂. (3.8)

Using MATLAB Simulink, the behavior of the dynamic system (3.7) starting from 500 randomly selected initial states in the all feasible domain is shown in Figure 2. It is clear that, for the path flow dynamic system (3.7), the equilibrium states consist of a set rather than an isolated equilibrium point is an invariant set, and no equilibrium state is stable for path flow pattern in the sense of Lyapunov stability although the link flow is stable.

Figure 2 

State trajectories demonstrating stability of equilibria.

3.2. Stability Property of the Dynamic System with Heterogeneous Users

We now move to examine the system stability when the travelers are heterogeneous in valuation of travel time. It is well known [24] that the static multiclass traffic assignment problem with fixed demand can be formulated as the following equivalent minimization problem min_𝐟∈Ω𝐿=_𝑎∈𝐴^𝑣𝑎₀^(𝐟)𝑡_𝑎(𝜔)𝑑𝜔+_{𝑎∈𝐴𝑚∈𝑀}𝜏_𝑎𝛽_𝑚𝑣^𝑚_𝑎(𝐟), (3.9) where the feasible path flow set Ω is given by (2.4). The objective function (3.9) is strictly convex in aggregate link flow 𝐯 for monotone link travel time function 𝑡_𝑎(𝑣_𝑎). Thus, under a given link toll pattern 𝜏, the equilibrium aggregate link flow is unique. However, the class-specific link flow 𝐯^𝑀 is generally not unique (neither is the UE path flow).

Consider the following continuously differentiable function: 𝑉(𝐟)=_𝑎∈𝐴^𝑣𝑎₀^(𝐟)𝑡_𝑎(𝜔)𝑑𝜔+_{𝑎∈𝐴𝑚∈𝑀}𝜏_𝑎𝛽_𝑚𝑣^𝑚_𝑎(𝐟)−min{𝐿}. (3.10) From the proof of Lemma 3.2, we know that 𝑆={𝐟∈𝑅^𝑛∣𝑉̇(𝐟)=0} is the largest invariant set of multiclass dynamic system (2.3), which is the solution of minimization problem (3.9). Note that 𝑉(𝐟)>0 for all 𝐟∈Ω, and 𝑉(𝐟)=0 for all 𝐟∈𝑆. Therefore 𝑉(𝐟) is a Lyapunov function candidate of system (2.3). Furthermore, from the assumption (A2′) and the flow conservation ∑_{𝑟∈𝑅𝑤}𝑓̇_𝑟^𝑤,𝑚=0, we know, for all 𝐟∈Ω, that 𝑉̇(𝐟)=𝜕𝑉𝜕𝑓_𝑟^𝑤,𝑚,𝑚∈𝑀,𝑟∈𝑅_𝑤,𝑤∈𝑊𝐟̇=_{𝑤∈𝑊𝑟∈𝑅𝑤𝑚∈𝑀}ETC_𝑟^𝑤,𝑚𝐹_𝑟^𝑤,𝑚(𝐟)≤0, (3.11) which implies that 𝑉(𝐟) is a Lyapunov function of system (2.3). In (3.11), ETC_𝑟^𝑤,𝑚 is the excess travel cost of path 𝑟∈𝑅_𝑤 between OD pair 𝑤∈𝑊 for user class 𝑚∈𝑀 defined in Section 2.2. Similar to (3.3), we still denote Π as the set of multiclass user equilibrium path flow patterns as Π=𝐟≥Ω∶_{𝑤∈𝑊𝑟∈𝑅𝑤𝑚∈𝑀}𝑓_𝑟^𝑤,𝑚𝛿^𝑤_𝑎𝑟=𝑣_𝑎,𝑎∈𝐴;_𝑤∈𝑊𝑓_𝑟^𝑤,𝑚=𝑞^𝑚_𝑤,𝑤∈𝑊, (3.12) where 𝐯={𝑣_𝑎,𝑎∈𝐴} and 𝐪^𝑀={𝑞^𝑚_𝑤,𝑤∈𝑊,𝑚∈𝑀} are unique aggregate UE link flow pattern and class-specific OD demand vector, respectively. According to the LaSalle's invariant set theorem, we know that all solutions of system (2.3) converge to Π, which induces the UE link flow pattern. Therefore, dynamic class-specific path flow pattern with any initial state induces the multiclass user equilibrium link flow pattern by the day-to-day route adjustment process.

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 [24]: min_𝐟∈Ω𝐿=_𝑎∈𝐴^𝑣𝑎₀^(𝐟)𝑡_𝑎(𝜔)𝑑𝜔+_{𝑎∈𝐴𝑚∈𝑀}𝜏_𝑎𝛽_𝑚𝑣^𝑚_𝑎(𝐟)−_{𝑤∈𝑊𝑚∈𝑀}^𝑞𝑚𝑤₀𝐷⁻¹_𝑤,𝑚(𝜔)𝑑𝜔, (3.13) where 𝐷⁻¹_𝑤,𝑚(⋅) is the inverse demand function for user class 𝑚∈𝑀 between OD pair 𝑤∈𝑊. Define the candidate Lyapunov function as 𝑉(𝐟)=_𝑎∈𝐴^𝑣𝑎₀^(𝐟)𝑡_𝑎(𝜔)𝑑𝜔+_{𝑎∈𝐴𝑚∈𝑀}𝜏_𝑎𝛽_𝑚𝑣^𝑚_𝑎(𝐟)−_{𝑤∈𝑊𝑚∈𝑀}^𝑞𝑚𝑤₀𝐷⁻¹_𝑤,𝑚(𝜔)d𝜔−min{𝐿}. (3.14) The multiclass UE path flow 𝑆={𝐟∈𝑅^𝑛∣𝑉̇(𝐟)=0} is the largest invariant set. It is clear that, 𝑉(𝐟)>0 for all 𝐟∈Ω⧵𝑆, and 𝑉(𝐟)=0 for all 𝐟∈𝑆. Therefore 𝑉(𝐟) is a Lyapunov function candidate of system (2.3). By direct calibration, we know, for all 𝐟∈Ω, 𝑉̇(𝐟)=_{𝑤∈𝑊𝑟∈𝑅𝑤𝑚∈𝑀𝑎}𝑡_𝑎𝑣_𝑎+𝜏_𝑎𝛽_𝑚𝛿^𝑟_𝑎−𝐷⁻¹_𝑤,𝑚𝑞^𝑚_𝑤𝑓̇_𝑟^𝑤,𝑚=_{𝑤∈𝑊𝑟∈𝑅𝑤𝑚∈𝑀}ETC_𝑟^𝑤,𝑚𝐹_𝑟^𝑤,𝑚(𝐟)≤0, (3.15) which implies that 𝑉(𝐟) is a Lyapunov function of system (2.3). From LaSalle's invariant set theorem, we have the following stability property for the dynamic system with heterogeneous users.

Theorem 3.5. Under assumptions (A1), (A2′), and (A3), for any given and fixed link toll pattern, any solution of system (2.3) with heterogeneous users converges to a certain user equilibrium path flow pattern, which results in the multiclass user equilibrium link flow pattern.

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 3.5 also provides a method to calculate the multiclass UE link flow pattern.

Example 3.6. Consider the network example [18], shown in Figure 3, consisting of 5 links, 4 nodes, and 2 OD pairs. The link cost functions are given by 𝑡₁=20+2𝑣₁,𝑡₂=𝑣₂,𝑡₃=𝑣₃,𝑡₄=20+𝑣₄,𝑡₅=2𝑣₅. (3.16) There are two classes of travelers with different values of time, class 1 consists of travelers with a lower value of time of 𝛽₁=1.0 (Yuan/min) and class 2 consists of travelers with a higher value of time of 𝛽₂=2.0 (Yuan/min). The OD pairs and the corresponding path are listed in Table 2. Let 𝑓_𝑟^𝑤,𝑚 denote the flow of path 𝑟∈{1,2,3,4} between OD pair 𝑤∈{1,2}, with user class 𝑚∈{1,2}. The demand for each user class between each OD pair is 𝑞¹₁=10, 𝑞²₁=10, 𝑞¹₂=20, and 𝑞²₂=10. Given the link toll pattern of 𝜏₁=𝜏₃=𝜏₄=0, and 𝜏₂=𝜏₅=10 Yuan, which results in the static multiclass user equilibrium link flow pattern, 𝑣₁=10, 𝑣₂=10, 𝑣₃=20, 𝑣₄=10 and 𝑣₅=20. Here we still adopt the proportional-switch adjustment process (PSAP) to model the multiclass path flow dynamics. From the flow conservation, we only consider four free variables, 𝑓₁^1,1, 𝑓₁^1,2, 𝑓₂^2,1, and 𝑓₂^2,2. The PSAP (2.5) is expressed as: 𝑓̇₁^1,1𝑓̇₁^1,2𝑓̇₂^2,1𝑓̇₂^2,2=10−𝑓₁^1,1𝑌₁₊−𝑓₁^1,1−𝑌₁₊10−𝑓₁^1,2𝑌₂₊−𝑓₁^1,2−𝑌₂₊20−𝑓₂^2,1𝑌₃₊−𝑓₂^2,1−𝑌₃₊10−𝑓₂^2,2𝑌₄₊−𝑓₂^2,2−𝑌₄₊, (3.17) where 𝑌₁=𝑇¹₃(𝐟)−𝑇¹₁(𝐟)+𝜏¹₃−𝜏¹₁𝛽₁𝑌₂=𝑇¹₃(𝐟)−𝑇¹₁(𝐟)+𝜏¹₃−𝜏¹₁𝛽₂𝑌₃=𝑇²₄(𝐟)−𝑇²₂(𝐟)+𝜏²₄−𝜏²₂𝛽₁𝑌₄=𝑇²₄(𝐟)−𝑇²₂(𝐟)+𝜏²₄−𝜏²₂𝛽₂, (3.18) and 𝑇^𝑤_𝑟(𝐟) and 𝜏^𝑤_𝑟 are the total travel time and toll of path 𝑟∈𝑅_𝑤 between OD pair 𝑤∈{1,2}, respectively. Using MATLAB Simulink, the path flow pattern of the dynamic system (2.5) starting randomly from selected initial state converges to the unique solution, in which, 𝑓₁^1,1=0, 𝑓₁^1,2=10, 𝑓₂^2,1=10 and 𝑓₂^2,2=0. Denote 𝐶^𝑚_𝑤 as the class-specific average travel cost in time unit between OD pair 𝑤∈{1,2}, such as, for 𝑚∈{1,2}, 𝐶^𝑚₁=𝑇¹₁(𝐟)+𝜏¹₁/𝛽_𝑚𝑓¹₁+𝑇¹₃(𝐟)+𝜏¹₃/𝛽_𝑚𝑓¹₃𝑞₁, (3.19) The day-to-day aggregate excess travel cost in time unit can be calculated as ΔETC=_{𝑤=1,2𝑟∈𝑅𝑤𝑚=1,2}𝑇^𝑤_𝑟(𝐟)+𝜏^𝑤_𝑟𝛽_𝑚−𝐶^𝑚_𝑤𝑓̇_𝑟^𝑤,𝑚 (3.20) which approaches zero. The behavior of the dynamic system (3.17), starting from 200 randomly selected initial states in all the feasible domain, is shown in Figures 4 and 5. Observed from the two figures for the multiclass case, the state trajectories become chaotic since the individual traveler adjusts his/her route choice based on his/her own value of time. However, the trajectories with any initial states solution for the path flow dynamic system (3.17) converge to the UE path flow pattern.

Table 2 

OD pairs and path sets.

Figure 3 

The network for Example 3.6.

Figure 4 

State trajectories demonstrating stability of UE flow on Path 1.

Figure 5 

State trajectories demonstrating stability of UE flow on Path 4.

4. Conclusion

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 [15], which corresponds to the excess payoff dynamics in evolutionary game [19]. The excess travel cost dynamics serves as a more general framework than the rational behavior adjustment process for modeling the travelers' dynamic route choice behavior in the transportation network with fixed or elastic demand, homogeneous or heterogeneous users. LaSalle's theorem is adopted to study the dynamic system stability. We proved that the trajectories of the excess travel cost dynamics with any initial state converge to the user equilibrium link flow pattern in various scenarios (fixed or elastic demand, homogeneous or heterogeneous users).

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' day-to-day 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.

Acknowledgments

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 self-determined and innovative research funds of HUST (2011QN103).