Mathematical Problems in Engineering

Volume 2018, Article ID 8478910, 10 pages

https://doi.org/10.1155/2018/8478910

## Network Traffic Flow Evolution Model Based on Disequilibrium Theory

^{1}School of Traffic and Transportation Engineering, Changsha University of Science and Technology, Changsha 410114, China^{2}School of Information Science and Technology, Hunan Institute of Science and Technology, Yueyang 414006, China^{3}School of Traffic and Transportation Engineering, Central South University, Changsha 410075, China

Correspondence should be addressed to Jianhui Wu; moc.361@tsinh_hjw

Received 25 March 2018; Accepted 19 July 2018; Published 9 August 2018

Academic Editor: Dong Ngoduy

Copyright © 2018 Zhongxiang Huang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The disequilibrium theory in economics is used to depict the network traffic flow evolution process from disequilibrium to equilibrium. Three path choice behavior criteria are proposed, and the equilibrium traffic flow patterns formed by these three criteria are defined as price regulation user equilibrium, quantity regulation user equilibrium, and price-quantity regulation user equilibrium, respectively. Based on the principle of price-quantity regulation user equilibrium, the method of network tatonnement process is used to establish a network traffic flow evolution model. The unique solution of the evolution model is proved by using Picard’s existence and uniqueness theorem, and the stability condition of the unique solution is derived based on stability theorem of nonlinear system. Through numerical experiments, the evolution processes of network traffic flow under different regulation modes are analyzed. The results show that all the single price regulation, single quantity regulation, and price-quantity regulation can simulate the evolution process of network traffic flow. Price-quantity regulation is the combination of price regulation user equilibrium and quantity regulation user equilibrium, which thus can simulate the evolution process of network traffic flow with multiple user class.

#### 1. Introduction

Network traffic flow evolution model is mainly used to study the network traffic flow evolution process and traffic fluctuations, which is an important part of traffic network design and traffic management strategy optimization. In order to give a more accurate and meticulous depiction of the formation and evolution of network traffic flow, it is necessary to start with the traveler’s travel choice behavior. There are many factors affecting a traveler’s travel choice behavior and the decision-making process is complex. Since the establishment of two important principles of user equilibrium and system optimization put forward by Wardrop [1], scholars have done a lot of research on this basis. Smith [2] supposed that travelers would consider changing their routes in order to obtain lower travel time and proposed a continuous day-to-day evolution model. Friesz et al. [3] studied day-to-day adjustment process of network traffic flow and analyzed the impact of information integrity for network traffic flow. Zhang and Nagurney [4] proposed a deterministic day-to-day evolution model by using the method of projected dynamical system. Wei et al. [5] presented a day-to-day route choice model by using reinforcement learning and multiagent simulation.

At present, the theories used to study the evolution model of network traffic flow mainly include random utility theory (e.g., Wen et al. [6] and Chen and Pan [7]), expected utility theory (e.g., Savage [8] and Blavatskyy [9]), prospect theory (e.g., Li et al. [10], Liu [11], and Wang et al. [12]), regret theory (e.g., Chorus [13] and Ramos et al. [14]), and disequilibrium theory (e.g., Zhang and Monden [15] and Huang et al. [16]). Random utility theory and expected utility theory are based on the assumption of traveler’s complete rationality, and they are widely applied in the analysis of travel choice behavior and the description of traffic flow evolution law. In the face of many uncertain factors, people’s judgment and decision-making behavior are not completely rational, but greatly influenced by personal habit preference, risk attitude, and so on, so it is bounded rationality. Both prospect theory and regret theory are based on the hypothesis of bounded rationality and can better describe the evolution process of network traffic flow in uncertain scenarios. Avineri [17] has applied the prospect theory to stochastic user equilibrium and studied the influence of reference point on stochastic user equilibrium. Based on regret theory, scholars like Chorus [18] proposed a stochastic regret minimum model based on the assumption that the traveler’s choice of one path depends only on the regret value obtained by comparing the path with the best path in other alternative paths. As a matter of fact, travel choice is usually repeated daily decision, while prospect theory and regret theory mainly involve one decision and do not take the travelers’ learning and behavior adjustment of decision results into consideration.

The disequilibrium theory regards road network system as an economic market, the potential traveler of road network as the travel demand-side, and the road network itself as the travel supply-side. The decision behaviors of travelers are explained from two factors, price, and quantity, so as to describe the dynamic evolution of traffic flow. Huang et al. [19] proposed an idea of disequilibrium transportation planning based on disequilibrium theory and believed that travelers will not only consider the travel time (price) of the current route, but also the traffic flow (quantity) obtained from traffic information and historical experience. In the actual traffic management, congestion charges (e.g., Wu et al. [20], Chiou and Fu [21], and Jia et al. [22]) and parking charges (e.g., Washbrook et al. [23] and Simićević et al. [24]) are to change the travel decisions of some travelers through typically price regulation. Ramp control (e.g., Hasan et al. [25] and Frejo and Camacho [26]) and public transport departure frequency optimization (e.g., Berrebi et al. [27] and Wang et al. [28]) are to change the travel decision of some travelers through typically quantity regulation, so travelers are subject to the double constraints of price and quantity. Based on this, this paper assumes that some travelers choose the lowest time path, and some travelers choose the most comfortable path. A concept of path-residual capacity is proposed to describe the driving comfort. Based on disequilibrium theory, path-travel time is regarded as price, path-residual capacity is regarded as quantity, and the influence of price-quantity on the distribution of traffic flow on the road network is also analyzed.

The remainder of this paper is organized as follows. In the next section, we propose three path choice behavior criteria based on disequilibrium theory. Section 3 establishes a network traffic flow evolution model. Section 4 proves the existence, uniqueness, and stability of solution to the evolution model. Section 5 verifies the evolution process and stable state of network traffic flow under different regulation modes. Section 6 is the conclusions of this research.

#### 2. Path Choice Behaviors

The different criterion of path choice naturally leads to the different distribution of network traffic flow. This paper proposes three types of path choice behavior criterion based on disequilibrium theory and defines the stable state of network traffic flow formed by these three criteria as price regulation user equilibrium, quantity regulation user equilibrium, and price-quantity regulation user equilibrium, respectively.

##### 2.1. Price Regulation User Equilibrium

Suppose that travelers have complete travel time (price) information and choose travel path according to the minimum travel time. We call this kind of travelers’ path choice behavior to follow price regulation completely.

The relationship between travel time on path and travel time on link can be written aswhere represents the set of links, is the traffic flow on link at time , is the traffic flow on path at time , and is a link-path incidence relationship, specifically if and otherwise.

We define price regulation user equilibrium as the travel time on all used paths on OD pair is equal and is less than or equal to the travel time on any unused paths. It can be expressed asSubject towhere represents the set of nodes which are trip origins, is the set of nodes which are trip destinations, is the set of paths on OD pair , , is the minimum travel time on OD pair at time , and travel demand is a function of .

Evidently, the price regulation user equilibrium is Wardrop’s user equilibrium.

##### 2.2. Quantity Regulation User Equilibrium

In the process of travel decision-making, travelers may not always choose travel path according to the minimum travel time even they have got complete traffic information. For example, from the northeast corner of North 3rd Ring Road in Beijing to the Beijing Capital International Airport, there are two alternative routes: Airport Expressway and Jingcheng Expressway. Because Jingcheng Expressway has smaller vehicle flow, better driving comfort, and higher travel time reliability, some travelers would like to choose Jingcheng Expressway rather than Airport Expressway with high vehicle flow and short travel time.

Suppose that travelers cannot get travel time information or do not choose travel path according to the travel time, but choose travel path according to the vehicle flow (quantity) on the optional paths. We call this kind of travelers’ path choice behavior to follow quantity regulation completely.

This paper uses the method of residual capacity to define the quantity regulation variable of vehicle flow on the link. The more the residual capacity, the better the drive comfort, and the higher the travel time reliability. We express as the capacity on link , as the residual capacity on link , ; hence the residual capacity on path can be written as

We define quantity regulation user equilibrium as the residual capacity on all used paths on OD pair is equal and is greater than or equal to the residual capacity on any unused paths. It can be expressed assubject towhere is the maximum residual capacity on OD pair at time and travel demand is a function of .

##### 2.3. Price-Quantity Regulation User Equilibrium

Based on the price-quantity regulation principle of disequilibrium theory in economics, it is assumed that travelers choose travel path according to the comprehensive travel cost on the optional paths. We define the comprehensive travel cost on path aswhere is weight factor, reflecting the sensitivity of travelers to travel time and residual capacity .

We define the minimum comprehensive travel cost on OD pair as

We define price-quantity regulation user equilibrium as the comprehensive travel cost on all used paths on OD pair is equal and is less than or equal to the comprehensive travel cost on any unused paths. It can be expressed assubject towhere travel demand is a function of and .

For the above user equilibrium, if , travelers choose travel path only according to travel time, and price-quantity regulation user equilibrium becomes single price regulation user equilibrium; if , travelers choose travel path only according to residual capacity, and price-quantity regulation user equilibrium becomes single quantity regulation user equilibrium.

#### 3. Network Traffic Flow Evolution Model

According to price-quantity regulation user equilibrium, this paper uses the method of network tatonnement process to simulate travelers’ path choice behavior and constructs a network traffic flow evolution model.

To give a mathematical state of this model we express the excess travel time on path asThe excess residual capacity on path is expressed asWe define excess comprehensive travel cost on path as comprehensive travel cost minus minimum comprehensive travel cost , that is, formula (7) minus (8). It can be expressed asEvidently

When excess comprehensive travel cost at time is positive (comprehensive travel cost more than minimum comprehensive travel cost), path flow will decrease because some travelers will automatically move to the lower comprehensive travel cost path; otherwise path flow will increase. To this end, the adjustment principle of and can be expressed asConsidering , formula (15) can be rewritten aswhere . Supposing that is a continuous differentiability function of , we haveFormula (16) into (17), we also have

The excess travel demand on OD pair is expressed asWhen excess travel demand at time is positive, the minimum travel time will increase and the maximum residual capacity will decrease to reduce the potential travel demand; otherwise the minimum travel time will decrease and the maximum residual capacity will increase. To this end, the adjustment principle of , , and can be expressed asSupposing that is a continuous differentiability function of , we knowFormula (20) into (21), we haveSupposing that is a continuous differentiability function of , we also have

Furthermore, we impose the initial conditions , and . Using formulas (18) (23) (24), network traffic flow evolution model can be expressed aswhere , , , , , , , , , .

#### 4. Existence, Uniqueness, and Stability

##### 4.1. Existence and Uniqueness

For the convenience of analysis, we make the following definitions:Network traffic flow evolution model (25) can be reexpressed aswhere , , , is the path-OD pair incidence matrix, specifically if path connects OD pair and zero otherwise.

*Hypothesis 1. * is a positive, continuous, and strictly monotonically increasing function of , is a positive, continuous, and strictly monotonically decreasing function of , and is a continuous, bounded, and strictly monotonic function of and .

Theorem 2. *For any , there exists a unique solution to network traffic flow evolution model (25).*

*Proof. *Let , then we introduce a Picard’s existence and uniqueness theorem (Dupuis and Nagurney [29]).*Picard’s Existence and Uniqueness*. If is continuous in the feasible domain ( is constant), and there exists constant with values to positive ensure that for any .(1), that is, local Lipschitz continuous.(2), that is, linearly bounded. Then there exists a unique solution to formula (27).

We first prove that is continuous in the feasible domain . According to the incidence relationship between path and link, is a continuous function of . Since is continuous on and , and is continuous on , then is a continuous function of . Evidently, is a continuous function of ; thus is continuous in the feasible domain .

Second, we prove that satisfies the condition of local Lipschitz continuous. For any one hasLet , , , , , , ; thenSince is bounded, then is a nonempty, bounded, and closed convex set. Moreover, according to the definition of travel time and residual capacity on the path, and are also a nonempty, bounded, and closed convex set; thus is bounded. Let , formula (29) can be expressed asFormula (30) into (28), we havewhere . Hence is a local Lipschitz continuous function.

Finally, we prove that satisfies the condition of linearly bounded. Since , then . Let , , ; thus . Moreover for any where . Hence satisfies the condition of linearly bounded.

In conclusion, there exists a unique solution to formula (27); that is, is a unique solution to network traffic flow evolution model (25).

##### 4.2. Analysis of Steady State

Theorem 3. *If there exists a unique solution to network traffic flow evolution model (25) and , then the unique solution is asymptotically stable.*

*Proof. *We introduce a stability theorem of nonlinear system (Cho and Hwang [30]).*Stability Theorem of Nonlinear System.* If there exists a continuous differentiable function for formula (27) and satisfies the following conditions,(1), and ,(2), then the unique solution is asymptotically stable.

Constructing a continuous differentiable function , it can be written asAccording to formula (33), we know that satisfies condition (1) in the stability theorem of nonlinear system, and then we only need to prove . We knowSince , thenFormula (30) into (35), we haveFormula (36) into (34), we knowLet ; formula (37) can be reexpressed asSince , then . In addition, since for , then ; thus . Hence, the unique solution of network traffic flow evolution model (25) is asymptotically stable.

#### 5. Calculations and Analysis of Numerical Example

##### 5.1. Test Road Network

A road network with nine nodes and twelve links illustrated as Figure 1 is used to show the numerical results of this established model. There is only one OD pair (1, 2) which is connected by six paths. The incidence relationship between path and link is shown in Table 1.