Day-to-Day Evolution of Traffic Flow with Dynamic Rerouting in Degradable Transport Network

Random events like accidents and vehicle breakdown, degrade link capacities and lead to uncertain travel environment. And whether travelers adjust route or not depends on the utility dierence (dynamic rerouting behavior) rather than a constant. Considering travelers’ risk-taking behavior in uncertain environment and dynamic rerouting behavior, a new day-to-day trac assignment model is established. In the proposed model, an exponential-smoothing lter is adopted to describe travelers’ learning for uncertain travel time. ­e cumulative prospect theory is used to reect route utility and its reference point is adaptive and set to be the minimal travel time under a certain on-time arrival probability. Rerouting probability is determined by the dierence between expected utility and perceived utility of previously chosen route. Rerouting travelers choose new routes in a logit model while travelers who do not choose to reroute travel on their previous routes again. ­e proposed model’s several mathematical properties, including xed point existence, uniqueness, and stability condition, are investigated through theoretical analyses. Numerical experiments are also conducted to validate the proposed heuristic stability condition, show the eects of four main parameters on dynamic natures of the system, and investigate the dierences of the system based on expected utility theory and cumulative prospect theory and with static rerouting behavior and dynamic rerouting behavior.


Introduction
Transportation network modeling mainly examines the nal equilibrium state for the past few decades on an explicit or implicit assumption that if equilibrium exists, then it will also occur. e assumption is very ideal; in fact, it was quite contrary to happen [1]. To explain whether nal equilibrium state can be achieved and its attainment, a substantial stream of research has been developed to look into "day-to-day" ow dynamics. ey can be divided into two categories based on types of approaches, i.e., stochastic ones [2][3][4][5][6][7] or the deterministic [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. e everyday tra c ow is stochastic in stochastic one while it is deterministic in deterministic one. Under the deterministic framework, tra c ow adjustment has been formulated in either continuous time [8][9][10][11][12][13][14][15][16][17] or discrete time [18][19][20][21][22][23][24][25][26]. In reality, tra c ows are not continuously adjustable over days because of activity constraints [26]. Hence, it is more appropriate to model tra c ow dynamics as a discrete-time system with each discrete epoch representing a lag over which travels may be repeated, e.g., days or weekdays [26]. Horowitz [18] established a day-to-day (DTD) deterministic discrete time tra c assignment model in which perceived travel time is a weighted average of experienced travel times on previous days. Jha et al. [19] further considered the condence of the perception and established a perception update model based on Bayesian model. Huang et al. [20] modelled the adaptation of advanced traveler information system and its e ect on route choice. Under uncertain tra c network, Xu et al. [25] suggested adopting the route choice model based on cumulative prospect theory. Guo et al. [26] presented a link-based model to avoid the numeration of a large number of routes.
In these models, tra c ows are the result of travelers' perceived travel costs by learning experienced travel costs. e experienced travel costs depend on travelers' route choice and the tra c network. In reality, tra c networks frequently su er from some minor events including vehicle breakdown, accidents and so on. Such minor incidents occur randomly in our daily commutes and degrade the link capacity, leading to travel time uncertainty [27]. However, most of the DTD models previously introduced do not address travel time uncertainty led by randomly degradable link capacity.
Meantime, they commonly assume that travelers always adjust their routes to pursue the optimal utility or that a xed proportion of travelers adjust their route considering their inertia. However, both of them do not coincide with reality. For one thing, travelers do not always seek the optimal solution because they accept a satisfactory solution [28]. For another, travelers' inertia changes with the situation, which is validated through a set of laboratory experiments by Mahmassani and Liu [29]. For example, when the travel cost on a certain route signi cantly change, it is likely for travelers to give up their inertial behaviors and to choose another route.
is paper simultaneously considers the in uence of travel time uncertainty on route choice and travelers' dynamic rerouting behavior (DR) to construct a more realistic DTD tra c assignment model. In our model, travel time mean and variation of a route is updated by an exponential smooth lter. Perceived route utility is reference-dependent and follows cumulative prospect theory (CPT). Travelers' rerouting choice is dynamically determined by the di erence between expected travel utility and utility of previously chosen route. Based on the model, the natures of the dynamic system are further investigated, which provides the valuable information for network design and management. e remainder of this paper is organized as follows. e new day-to-day learning model will be elaborated in Section 2. In Section 3, we further discuss the existence, the uniqueness and the stability of the xed point of the evolutionary model. ese theoretical analyses are veri ed by numerical experiments in Section 4. Section 5 gives conclusions and some suggestions on the future direction. Figure 1 gives the framework of our day-to-day tra c assignment model. e model includes three main parts: perceived route travel time distribution, rerouting choice and route choice. A traveler will be able to obtain the information on travel time of his chosen route a er nishing a trip. Assuming that all travelers communicate the information with each other, each traveler will be able to know travel time distribution of all routes. Based on it, travelers will predict the route travel time distribution next day, which is described by the perceived route travel time distribution model. Travelers will further get know of attractiveness of all routes and decide whether to reroute and which route to choose if rerouting. e rerouting probability positively depends on the di erence between traveler's expected utility and the perceived utility of the previously chosen route. Based on it, the rerouting choice model is established. en, rerouting travelers will seek the maximal utility and choose the optimal route while travelers not to reroute will travel on their previous routes. e chosen probability of a route consists of them, which is exactly our route choice model. We speci cally illustrate these models in the next section.

Perceived Route Travel Time Distribution.
Assumed that all travelers communicate experienced travel time with each other a er nishing trips, travelers are able to get the route travel time distribution. Because route travel time distribution is normal regardless of link travel time distribution as long as Lindenberg's condition is satis ed [24,27], travelers t the route travel time with a normal distribution. Its mean and variance should be consistent with: where , is the experienced travel time distribution on route between origin and destination (O-D) pair on day . , and , are, respectively, in mum and supremum of route travel time. e former is the free-ow travel time of route while the latter is set as the 99.74 percentile route travel time on day . Λ , is the route-link index whose value is 1 if link belongs to route , zero otherwise. And is the travel time distribution of link on day .
On the condition that Bureau of Public Road (BPR) link performance function is used to describe link travel time, that link capacity distributions are independent of each other, that the link capacity distribution is independent of tra c ow volume on it and that the link capacity is uniformly distributed with the design capacity as its upper bound and the worst degraded capacity as its lower bound, the mean and variance of the link travel time can be derived as follows [27]: where subscript refers to a particular link; , and , respectively, are link ὔ free-ow travel time, design capacity and travel time with ow ; and are deterministic parameters; and is the worst capacity degradable rate of link .
Based on the newly obtained travel time distribution, travelers will update their perception. We assume that the update process is consistence with an exponential smoothing lter. Hence, the perceived travel time distribution can be expressed as: where +1 , is the perceived route travel time distribution on day + 1; (0 ≤ ≤ 1) is a constant that re ects travelers' learning rate.
According to the work of Lou and Cheng [24], the mean and variance of perceived travel time distribution can be calculated as follows:

Perceived Utility.
With di erent behavioral assumptions on travelers' risk-taking behaviors, various models have been proposed to describe the route utility in uncertain environment including the expected-utility-theory-based models [30,31], var + 1 , the travel time budget models [27], the late arrival penalty model [32], α-reliable mean-excess travel time model [33] and the cumulative-prospect-theory-based models [34,35]. Substantial evidence indicates that cumulative prospect theory (CPT) provides a well-supported descriptive paradigm for decision making under uncertainty because it captures riskattitude conversion and high estimation of low probability [36][37][38]. erefore, we use it to describe perceived utilities for alternative routes.
In the CPT, the reference point is a key parameter which directly determines the perceived gain and loss. Individual adjusts it according to the change [39]. Hence, we use an adaptive adjustment reference point, like Xu et al. [35]. e reference point is the time that travelers budget to ensure a desired on-time arrival probability. If the desired on-time arrival probability is set as , the reference point on the day + 1,

+1
, is the minimum of the th percentile travel times of all alternative routes. It adaptively changes with travelers' perception. Speci cally, it is expressed as: where +1 , is the cumulative distribution function of route travel time between O-D pair on day + 1.
e perceived prospect value for route between O-D pair on day + 1, denoted as +1 , ( ), can be formulated as: where represents the travel time of route between O-D pair on day . ℎ(⋅) and +1 , (⋅) in the Formula (12) are, respectively, the weighting function and the value function. In the context of route choice, they can be speci cally written as [24]: where is the probability; parameter refers to the adjustment level of the probability in the travelers' decision-making process; parameters and represent the degree of diminishing sensitivity of the value function. Typically, 0 < , < 1 and , ( ) , ∀ ∈ , ∈ , thus the value function exhibits risk-aversion over gains and risk-seeking over losses. And parameter ≥ 1 is loss-aversion coe cient that indicates that travelers are more sensitive to losses than gains.

Rerouting Choice.
A traveler has an expected utility for a trip and judges whether to reroute based on it. If the perceived utility of the route that he previously chose is higher than the expectation, he will not change route because satisfactory solution is good enough for travelers; otherwise, he will possibly choose to reroute. And the larger the di erence between his expected prospect value and the perceived prospect value of the route that he previously chose, it is more likely for the traveler to change its routes. Based on these, rerouting probability of travelers choosing route between O-D pair on on , is given by Equations (15) and (16).
where ∇ +1 , is the di erence between travelers' expected prospect value, denoted as +1 , and the perceived prospect value of route between O-D pair on day + 1; 0 is a constant that represents the maximal rerouting probability; And is a parameter.
If the expected prospect value for the routes between O-D pair is viewed as the expectation of maximal route prospect value through O-D pair [40], +1 is: where re ects the heterogeneity of travelers. e higher is, the greater is the consistency of perception characteristics among travelers.

Route Choice.
For travelers choosing to reroute, they will choose optimal route to travel, i.e., the route with the maximal prospect value. Because the parameters in CPT would vary across the population, such as risk aversion coe cient, travelers' perceived prospect values are di erent. e di erence can be re ected by adding a random error. If the random error complies with Gumbel distribution, the route choice probability of rerouting travelers, + 1 , , can be given in a logit-based formula: For travelers choosing not to reroute, they will continue to travel on the routes of day . Total chosen probability of route between O-D pair on day + 1, +1 , , is formulated as: e corresponding route ow pattern and link ow patterns are: where is the demand of O-D pair and +1 , is the ow of route between O-D pair .

Some Mathematical Properties of Proposed DTD Model
We can see that * , continuous with respect to the route prospect value, * . Further, we can infer that * , , ∇ * , and * , are continuous with * , . Based on them, we can obtain that * , is continuous with * , . erefore, the self-map of * are continuous and the xed point exists.

Fixed Point Uniqueness.
Fixed-point uniqueness can be analyzed by investigating the monotonicity of the self-map of * . However, the adopted route prospect value function, * , , usually cannot be ensured to be strictly monotone increasing due to endogenous reference point. erefore, the xed-point uniqueness cannot be guarantee.

Fixed Point Stability.
For any one xed point, its stability heavily depends on the parameters adopted in the DTD model. where Ψ covers all relations formulated in Section 2. Let denote the Jacobian matrix of the function Ψ. It is known that the evolutionary system de ned by our model will converge to a xed point if the 2R (R refers to the number of routes) eigenvalues of the matrix are within the unit circle [20,41]. However, it is very time-consuming, even impossible to derive the eigenvalues. Based on that the work of Cantarella and Cascetta [42], we give a heuristic condition for assuring the stability of the evolutionary system: e weight takes a small enough value, then the evolutionary system de ned by our model will likely converge to a xed point.
For the system with deterministic travel time, the condition is proved, see Appendix.

Numerical Experiments
e proposed DTD model is applied to the Nguyen-Dupuis network [43] with 13 nodes and 19 links, whose topology is illustrated in Figure 2 and the link characteristics are displayed in Table 1. ere are four O-D pairs in the network, 1-2, 1-3, 4-2 and 4-3. Table 2 gives their demands and all alternative routes. Basic numerical settings are given in Table 3. e tra c or 0.3, respectively, the system takes 69 days, 39 days or 32 days to evolve the equilibrium. Convergence duration decreases with the increase in their values because a larger represents that travelers are able to more quickly accept a new information and a larger 0 means that they more actively react to the information. ird, there are di erent equilibrium states when parameters and are set to be di erent values. However, parameter does in uence the equilibrium state, which is consistence with our mathematical analysis because it is not included in our xed point model. At the same time, it can also be seen that the equilibrium state has nothing to do with the parameter 0 . is is because 0 in * , can be eliminated.

Comparative Analysis of Dynamic Systems Based on EUT and CPT.
Under the framework of the proposed DTD model, we also conduct an experiment to investigate the di erence between the dynamic system based on CPT and the one based on EUT. It is easy to nd that the prospect value recovers to the expected utility when all parameters in the value function (⋅) and the weighting function ℎ(⋅) are set as 1. en we investigate the impacts of CPT and EUT on the dynamic natures of the proposed system by observing evolutionary processes under di erent values of parameters and . Figure 5(a) show the e ects of CPT and EUT on the equilibrium state of the dynamic system. Compared with the case of EUT, parameter in the case of CPT has more signi cant in uence on the equilibrium ow of route 13. e same phenomenon can also be seen on the other routes. is is explained in the following. Parameter in CPT not only in uences perceived gain and loss, but also further a ects risk-attitudes while the parameter in EUT just in uences the former.
We also check the sensitivity of two systems to parameter . As shown in Figure 5(b), when the value of parameter changes from 0.3 to 0.5, the system under EUT cannot converge to an equilibrium whereas the system under CPT still ow on day 0 over tra c network is consistent with the outcome of the all-or-nothing assignment.

Heuristic Stability Condition.
In the section, we validate our heuristic stability condition by comparing two cases. In Case I, the parameter is relatively small, i.e., = 0.1; in case II, it is ampli ed 6 times, i.e., = 0.6. Figure 3 shows the numerical results of these two cases with di erent maximal rerouting rate, 0 . For graph simplicity, only the tra c ow on route 13 (Link order: 2-17-7-10-16) is used to illustrate the evolution of the dynamic system, so are the below sections. It can be clearly observed from Figure 3(a) that the route ows gradually evolve to a stable value, 206.0 (Veh·h −1 ), regardless of the maximal rerouting rate in case I. In contrast with Case I, Figure 3(b) show with a larger learning rate , route ows intensively uctuate in the evolutionary process and the evolutionary system does not converge to a stable state. ese results demonstrate that the heuristic stability condition is applicable to our model.

Sensitivity Analysis.
We also conduct sensitivity analyses to investigate the in uence of four main parameters , 0 , , on dynamic natures of our evolutionary system.
In the analyses, we change the value of a parameter while keep the others unchanged each time. Figure 4 displays the route ow over time under di erent parameters.
We are able to clearly observe that parameters , 0 and a ect the stability of the evolutionary system because route ows cannot converge to an equilibrium when they are set to a certain value, which is evidenced by the simpli ed transformation formula. Although also is included in the transformation formula, it does not a ect the stability of the system. do. is demonstrates that the dynamic system under EUT is more sensitive to than the system under CPT.

Comparative Analysis of Dynamic Rerouting Behavior and
Static Rerouting Behavior. We also investigate the e ects of the di erent rerouting behaviors. In conducting static rerouting experiments, we set the rerouting rate as the maximal rerouting rate and keep it unchanged in the whole evolutionary process. Figure 6 shows the results with di erent rerouting behaviors. It can be observed from Figure 6(a) that the equilibrium states of the systems with static rerouting behavior (SR) are di erent from that of the systems with DR, which is consistent with our theoretical analysis. In the case with SR, the equilibrium route choice probability function is * while it is * , = ∑ ∈ * , * , * , + 1 − * , * , , ∀ ∈ , ∈ in the case with DR. As shown in Figure 6(b), the static rerouting behavior leads to the underestimation of the eciency of the tra c system due to the fact that it overlooks bounded rationality of travelers that they accept the suboptimal route and the bounded rationality makes the equilibrium state close to system equilibrium to some extent. Figure 6(c) show that the system with DR are more sensitive to the parameter than the system with SR. Intuitively, the reason might be that the dynamic rerouting probability depends on parameter and with larger value of , the in uence of new information on rerouting behavior becomes more signi cant and travelers' behavior more intensively varies, thus leading to a more unstable system.

Conclusion
In the framework of DTD deterministic discrete-time tra c assignment model, a new DTD model incorporating risktaking behavior and dynamic rerouting behavior is established. Based on it, we analytically study the existence and uniqueness of its xed point and give a heuristic stable condition of the xed point. en several numerical experiments are conducted on the Nguyen-Dupuis network to validate the heuristic stability condition, to show the e ects of the four main parameters, to investigate the di erence between two utility frameworks called EUT and CPT and nally to show the di erence between travelers' static rerouting behavior and dynamic rerouting behavior. e results show that: (1) Parameters , 0 and all a ect the stability of the dynamic system.  In the future, various research directions ensuing from this work can be explored. One of them would be to calibrate and validate the model with empirical data. Another line of research would be to analytically derive the su cient condition that assures the stability of the dynamic system. e third direction would be to incorporate the e ect of tra c information in the model because travelers are able to easily get various tra c information at current times.

Appendix
In the section, we assume that route travel time is deterministic. In the condition, our dynamic system's xed point is unique.
e system's Jacobian matrix is: where In order to simplify the expression of the eigenvalues of , let = α + − α Λ Λ , a × matrix independent of . For each of the eigenvalues ( ) of matrix , two eigenvalues of are de ned as a function of parameter : Based on the work of Cantarella and Cascetta, the system de nitely converges to the equilibrium if parameter is small enough, * * * α + * * − * * α is negative semide nite and * is positive de nite. By inspection, * * * α + * * − * * α of our model is negative semide nite and * is positive de nite.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that there are no con icts of interest regarding the publication of this paper.