2. Department of Computer Science, Tongji University, Shanghai 201804, China;
3. Department of Electrical and Computer Engineering, New Jersey Institute of Technology, Newark NJ 07102, USA;
4. Key Laboratory of Embedded System and Service Computing, Ministry of Education, Shanghai Electronic Transactions and Information Service Collaborative Innovation Center, Department of Computer Science, Tongji University, Shanghai 201804, China
Traffic incidents are any nonrecurring events including traffic crashes, disabled vehicles, roadway maintenance and reconstruction projects, and special nonemergency events, e.g., ball games, concerts, or any other events that significantly affect roadway operations ^{[1]}. They can cause a significant capacity reduction of roadways. Traffic incident management (TIM) makes a systematic effort to detect, respond to, and remove traffic incidents. It aims to offer the rapid recovery of traffic safety and capacity, and leads to many measurable benefits, such as decreases in fuel consumption, incident duration, secondary accidents, and traffic jams ^{[1, 2]}.
There are many traditional traffic control methods for TIM in highways, such as lane control ^{[3]}, variable speed limit control ^{[4]}, and ramp metering control ^{[5]}. So far, incidentbased urban traffic congestion is mostly controlled and prevented through traffic flow diversion with the help of the traffic police. Such a strategy is unfortunately laborintensive, inflexible, and costly. Intelligent transportation systems ^{[6]}^{[8]}, such as advanced traveler information systems (ATIS), can be employed to improve the network efficiency via direct or indirect recommendation of alternative routes ^{[9]}. Realtime traffic information can be sent to drivers through two main kinds of devices: incar ^{[10]} and roadside devices ^{[11]}. The type, such as radio GPSnavigators and Google Maps, helps drivers make sensible routing decisions at bifurcation nodes of the network. However, there are some disadvantages with these kinds of devices. On one hand, drivers who are familiar with the traffic conditions in a network may not use such agencies and thus optimize their individual routes based on past experiences. On the other hand, incident information is only useful to a finite number of selected drivers near the incident, and useless to others. The second kind of devices can be used to deliver information on major traffic events (e.g., incidents and congestion) and reduce incidentbased congestion or enhancing traffic safety. However, they are usually spatially and/or temporally limited and constrained in the amount of information delivered. Thus, to the best of our knowledge, we find no intelligent strategies that can decide which drivers should be informed of a particular traffic incident. Recently, the proliferation of mobile communication technologies and devices such as smartphones and onboard units of connected vehicles makes it possible to construct an accessible and costeffective platform for publicsector Traffic Operation Centers to deliver locationbased and personalized traveler information in a timely fashion ^{[12]}. In this work, we design a new strategy to deliver incident information to a finite number of selected drivers in urban areas. The Dijkstra's algorithm is used to generate a subnetwork where vehicles can receive the traffic incident information. This helps reduce incidentinduced congestion at a manageable communication cost. Simulations are conducted to give a quantitative result regarding traffic congestion reduction with the proposed strategy.
Many simulation models are proposed to model traffic jam formation due to incidents. Wright and Roberg propose an incidentbased jam growth model in ^{[13]} and discuss the impact of the length of the channelized part of roads and stopline width assignment on jam formation. Roberg et al. develop several alternative strategies in ^{[14]} to prevent gridlock of a network and dissipate traffic jams once they have been formed. Long et al. ^{[15]} extend a cell transmission model (CTM) and apply it to simulate incidentbased jam propagation in twoway rectangular grid networks. They also propose control strategies for dispersing incidentbased traffic jam and evaluate their efficiency ^{[16]}. CTMbased models can depict traffic flow at downstream road links well. However, the aforementioned simulation models do not contain any pathrelated information when studying travelers' detour behaviors and the incidentinduced congestion formulation. In this work we further extend the CTM, build a model to simulate incidentbased traffic jams in urban areas and illustrate the effectiveness of our proposed strategy.
The rest of this paper is organized as follows. Section Ⅱ reviews the related work. Section Ⅲ presents some definitions about the traffic network. Section Ⅳ gives traffic incident information delivery strategies. In this section, given some assumptions regarding traffic flow routing choices, a traffic flow subnetwork is generated based on Dijkstra's algorithm, and traffic flow in the network is modeled by an extended CTM. Section Ⅴ gives a case study and evaluates the effectiveness of the proposed detour strategies via simulation. Section Ⅵ concludes this paper.
Ⅱ. RELATED WORK A. Traffic Incident Management (TIM)There are many traditional traffic control methods for TIM. For example, lane control systems ^{[3]} deploy lane control signals in the context of lane closure. They are used to manage traffic at a lane level to facilitate a smooth lane change by informing drivers about an impending bottleneck. Variable speed limit control ^{[4]} in highways is verified to be able to increase workzone throughputs and decrease total vehicle delays. Traffic signals at onramps of freeway ^{[5]} can help manage the traffic inflow rate and reduce laneblocking incidentinduced traffic congestion. These methods are suitable for TIM in highways. So far incidentbased urban traffic congestion is mostly controlled and prevented through traffic flow diversion with the help of traffic police, which is unfortunately laborintensive, inflexible, and costly. Traffic light control ^{[6]} at road intersections is regarded as a major strategy to guarantee the safe crossing of conflicting streams of vehicles and pedestrians and lead to efficient network operations. They are suitable for nonsaturated and stable traffic conditions. However, changing conditions in a nonpredictable way such as an incident may lead to the invalidation of the aforementioned traffic light control strategies, and causes unexpected congestion. Some intelligent systems have recently been designed for preventing incidentinduced traffic congestion ^{[17, 18]}. In such systems, ban signals are used to notify road users of a ban situation that might not be readily apparent ^{[19]}^{[21]}.
B. Dynamic Traffic Assignment (DTA)DTA is used to assign timevarying traffic flow to different highways given the vehicular demand and certain behavioral rules ^{[22]}. It consists of two components: a travel choice principle and a trafficflow component. The former models how travelers decide whether to travel or not ^{[22]}, and if so, how they select their routes, departure time, modes, and destinations. The latter depicts how traffic propagates inside a transport network. DTA is an important research area because of its a wide range of applications in realtime traffic control and management ^{[23]}. In fact, DTA models are key components in developing sophisticated intelligent transportation systems (ITS) technologies such as advanced traveler information systems (ATIS) ^{[24]} and advanced traffic management systems (ATMS) ^{[25]}. In ATIS, DTA models can determine the best route and departure time and provide some anticipatory traffic information for travelers based on a forecasted traffic pattern.
DTA models can be developed by either an analytical ^{[26]} or simulationbased approach ^{[27]}. DTA problems can be formulated analytically in terms of mathematical problems ^{[23, 28]}, such as mathematical programming ^{[29]}, optimal control ^{[30]}, and variational inequality problems ^{[31]}. Most prior studies focus on determining in advance the solution properties of the models, such as solution existence and uniqueness. The simulation approach focuses on enabling practical deployment of the DTA models for realistic highway networks, their applicability to reallife highway networks, and their ability to adequately capture traffic dynamics and microscopic driver behavior such as lane changing. However, a simulationbased approach cannot guarantee the solution properties of the model in general ^{[23]}.
DTA problems are formulated as both pathbased ^{[31, 32]} and linkbased models ^{[28]}. An important feature of the former is that the path set has to be explicitly defined and can range from medium to largesized networks. Hence, for largescale network applications, path enumeration has not been used to obtain the path set. Linkbased models can avoid path enumeration in the solution procedure, and hence can be applied to largescale networks. However, they do not contain pathrelated information and cannot capture certain realistic traffic dynamics such as dynamic traffic intersections across multiple links.
The commonly adopted travel choice principle in DTA is the dynamic extension of Wardrop's Principle called the Dynamic User Optimal (DUO) principle ^{[23, 31]}. This principle assumes that travelers select their routes and/or departure times to minimize their travel costs such as travel time. Most existing planning and management procedures are developed with this notion. Virtually all network planning models adhering to Wardrop's principle require the following strong assumptions: 1) travelers know the travel time on all routes, and 2) travelers are able to select the paths costing the shortest travel time. While these assumptions may be reasonable in a static network, they are questionable for realworld networks because they are dynamic and can be highly stochastic. The current research is not presented as an operational model for actual applications. For example, we can deliver the incident information to the corresponding traveler if we know the path of each vehicle. However, we do not know such path information, and in this case we should design incident delivery strategy to travelers to deal with the cases in which an incident takes place.
C. Advanced Traveler Information Systems (ATIS)ATIS are any systems that acquire, analyze, and present information to assist surface transportation travelers in moving from their starting location (origin) to their desired destination. Relevant information may include locations of incidents, if any, weather and road conditions, optimal routes, recommended speeds, and lane restrictions ^{[24]}. ATIS are considered a powerful tool to enhance travelers' experience ^{[33]}. ATIS are also claimed to be useful under recurrent network congestion as they reduce the uncertainty of travelers with respect to travel time ^{[33]}. Moreover, they are useful for travelers who are unfamiliar with the network (e.g., tourists), as well as for all travelers when the network is temporarily affected by some significant disruptions and/or by unexpected or nonrecurrent traffic conditions ^{[34]}. Recently, the proliferation of mobile communication technologies and devices such as smartphones and onboard units of connected vehicles makes it possible to construct an accessible and costeffective platform for publicsector Traffic Operation Centers to deliver locationbased and personalized traveler information in a timely fashion ^{[12]}. Our work lies in the scope of the ATIS when it selectively delivers incident information to a finite number of selected drivers but not all vehicles. The effectiveness is verified through a simulation approach.
D. Cell Transmission Model (CTM)Daganzo ^{[35]} proposes a CTM to simplify the solution scheme of the LighthillWhithamRichards (LWR) model ^{[36, 37]} such that it can be used to depict the road link traffic which is consistent with the kinematic property of traffic flow. The CTM has been used to accurately describe realistic highway traffic. Daganzo's original development of the CTM is mainly intended to provide transportation planners with another way of predicting traffic behavior for a given roadway section. Researchers have employed the CTM in many realworld transportation applications such as dynamic traffic assignment ^{[38, 23]}, signal control ^{[39]}^{[41]}, ramp metering ^{[42, 43]}, and traffic prediction. The advantage of this approach is that traffic dynamics such as queue spillback and traffic interaction across links can be captured. The CTM has been used in the estimation of traffic flow density ^{[44]}^{[46]} and other traffic state variables such as flows and spacemean speeds ^{[47]}. Long et al. ^{[15]} extend the CTM and apply it to simulate incidentbased jam propagation in twoway rectangular grid networks. The interface of vehicles conducting different turns at urban road links can be well described ^{[15]}. They also propose control strategies for dispersing incidentbased traffic jams and evaluating their efficiency ^{[16]}. The above cellbased simulation models do not contain any pathrelated information when studying incidentinduced congestion formulation. In this work we further extend CTM ^{[15]} to simulate travelers' detour and incidentbased traffic jams in urban areas and illustrate the effectiveness of our proposed incident information delivery strategy.
Ⅲ. BASIC NOTATIONSFirst, some basic notations are presented:
Definition 1: A path
All concepts in Definition 1 can be found in the graph theory in ^{[48]}.
Definition 2:
Definition 3:
$ \begin{align} L(\sigma)=\sum\limits_{j=1}^{m1} {l_{j, j+1}} \end{align} $  (1) 
where
Definition 4:
Definition 5:
$ \begin{align} T(\sigma)=\sum\limits_{j=1}^{m1} {\tau (l_{j, j+1}, t_{j, j+1})} \end{align} $  (2) 
denotes the travel time that a vehicle needs to pass path
Definition 6:
Definition 7:
$ \begin{align} \forall \sigma \in \Gamma_{1, m}: T(\sigma_{1, m}^{\ast})\leq T(\sigma ) \text{ and }f(\sigma_{1, m}^{\ast})\leq f(\sigma) \end{align} $  (3) 
This section presents a dynamic strategy to deliver incident information to a finite number of selected drivers in urban areas in order to help them make detours. According to ^{[13, 14]}, and ^{[16]}, the boundary of incidentinduced traffic jams has an approximate diamond shape in grid networks with the first blocked junction as the center. We need to provide traffic incident information to drivers heading towards the blocked link. First, we make some assumptions regarding the traffic flow evolution. Based on them, we can simplify the network and focus on only those links where traffic flow is affected by the incident.
A. Drivers' Path Selection AssumptionsIn urban areas, for a traveler going from an origin to a destination, there are usually several paths. Regarding the path which is most selected by drivers, we make the following assumptions.
Assumption 1: Timedependent Shortest Path Selection: We assume that given two paths from an origin to destination, without any information regarding the traffic condition such as incident, drivers select the one requiring the least time. If two paths cost the same travel time, drivers will choose the one with the shortest path. Formally given two paths
Assumption 2: Fewestnode Path Selection: We assume that given two sametraveltime and samelength paths from the origin to destination, drivers usually select a path with fewer road intersections to prevent extra traffic delay induced by traffic signals. Formally given two paths
Assumption 3: Equal Probability Selection:) If there are
Given a network
Algorithm 1 Generate a Subnetwork 
Input: A network 
Output: A subnetwork 
Step 1. Initialization 
Set 
Set 
Set 
Set 
Step 2. Calculation and update 
Select a node 

such that 
For each path 

If 

Take node 
Put node 
End If 
End For 
If there exists no node 
Put all nodes in each path 

Put all links in each path 

End If 
Step 3. Iteration 
Repeat Step 2 until 
Return 
Assume that the number of nodes in
Daganzo ^{[35]} proposes a CTM to simplify the solution scheme of the LighthillWhithamRichards (LWR) model ^{[36, 37]} such that it can be used to depict the road link traffic which is consistent with the kinematic property of traffic flow. This work extends CTM to model urban network traffic flow. The pathrelated information will be contained in the model when studying the incidentinduced congestion formulation. We use a timestep method based on the extended CTM to simulate the formation and dissipation of incidentbased traffic jams and evaluate the efficiency of the proposed control strategies.
As shown in Fig. 1, each link
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Fig. 1 Link 
Note that traffic rules will not be changed and the vehicles in the downstream queue areas will not change lanes. Using the network model, we design a network traffic simulation model based on the timestep method. Traffic flow formulation can be classified into the following categories: inflow of upstream reservoir of the origin and the other nodes (
1) Inflow of Upstream Reservoir From an Origin: Let
$ \begin{align} y_{a}^{1} (t)=\min \left \{d_{a} (t), Q_{a}^{1} (t), \frac{w\left(N_{a}^{1} (t)n_{a}^{1} (t)\right)}{v}\right\}. \end{align} $  (4) 
The number of vehicles that enter the first cell of link
$ \begin{align} y_{ab}^{1} (t)=\phi _{ab} (t)y_{a}^{1} (t). \end{align} $  (5) 
The proportion of vehicles that enter the first cell of link
$ \begin{align} \phi _{a}^{1s} (t)={\frac {d_{a}^{s} (t)}{d_{a} (t)}}. \end{align} $  (6) 
The number of vehicles that enter the first cell of link
$ \begin{align} y_{a}^{1s} (t)=\phi _{a}^{1s} (t)y_{a}^{1} (t). \end{align} $  (7) 
The number of vehicles that enter the first cell of link
$ \begin{align} y_{ab}^{1s} (t)=\phi _{ab}^{1s} (t)y_{a}^{1} (t) \end{align} $  (8) 
where
2)Inflow of Upstream Cells:
$ \begin{align} &y_{a}^{i} (t)=\min \left\{n_{a}^{i1} (t), Q_{a}^{i} (t), \frac {w\left(N_{a}^{i} (t)n_{a}^{i}(t)\right)}{v}\right\}, \nonumber\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad 1 < i\leq \lambda 1 \end{align} $  (9) 
$ \begin{align} &\phi _{ab}^{i} (t)=\begin{cases} \dfrac {n_{ab}^{i1} (t)}{n_{a}^{i1} (t)},&\text{if }n_{a}^{i1} (t)\ne 0 \\ 0, &\text{else}\\ \end{cases} \end{align} $  (10) 
$ y_{ab}^{i} (t)=\phi _{ab}^{i} (t)y_{a}^{i} (t) $  (11) 
$ \begin{align} &\phi _{a}^{is} (t)=\begin{cases} \dfrac{n_{a}^{i1, s} (t)}{n_{a}^{i1} (t)},&\text{if }n_{a}^{i1} (t)\ne 0 \\ 0, &\text{else}\\ \end{cases} \end{align} $  (12) 
$ y_{a}^{is} (t)=\phi _{a}^{is} (t)y_{a}^{i} (t) $  (13) 
$ \begin{align} &\phi _{ab}^{is} (t)=\begin{cases} \dfrac{n_{ab}^{i1, s} (t)}{n_{a}^{i1} (t)},&\text{if }n_{a}^{i1} (t)\ne 0 \\ 0, &\text{else}\\ \end{cases} \end{align} $  (14) 
$ y_{ab}^{is} (t)=\phi _{ab}^{is} (t)y_{a}^{i} (t). $  (15) 
3) Inflow of Channelized Downstream Queue Area: The upper bound of inflow of the downstream queue area for vehicles travelling from link
$ \begin{align} y'_{ab} (t)=\min \left\{\alpha_{ab} Q_{a}^{\lambda} (t), \frac{w(\alpha_{ab} N_{a}^{\lambda} (t)n_{ab}^{\lambda} (t))}{v}\right\}. \end{align} $  (16) 
Because of interference between turning vehicles and straight vehicles ^{[30]}, the total inflow of the channelized queues area can be formulated as follows:
$ \begin{align} y_{a}^{\lambda} (t)=\min\limits_{b\in B(h_{a})} \left\{\frac{y'_{ab} (t)}{\alpha_{ab}}\right\}. \end{align} $  (17) 
The inflow of each direction can be calculated as follows:
$ \begin{align} &\phi _{ab}^{\lambda 1} (t)=\begin{cases} \dfrac{n_{ab}^{\lambda 1} (t)}{n_{a}^{\lambda 1} (t)}, &\text{if }n_{a}^{\lambda 1}(t)\ne 0 \\[2mm] 0, &\text{else}\\ \end{cases} \end{align} $  (18) 
$ y_{ab}^{\lambda} (t)=\min \left\{\phi _{ab}^{\lambda 1} (t)y_{a}^{\lambda} (t), \phi _{ab}^{\lambda 1} (t)n_{a}^{\lambda 1} (t)\right\}. $  (19) 
The proportion of vehicles that enter cell
$ \begin{align} \phi _{ab}^{\lambda s} (t)=\begin{cases} \dfrac{n_{a}^{\lambda 1, s} (t)}{kn_{a}^{\lambda 1} (t)}, &{\text{if }n_{a}^{\lambda 1} (t)\ne 0\text{ and } b\in A(h_{a})\cap {A}'} \\[2mm] 0, &\text{if }n_{a}^{\lambda1}(t)=0 \end{cases} \end{align} $  (20) 
where
The number of vehicles that enter cell
$ \begin{align} y_{ab}^{\lambda s} (t)=\begin{cases} {\phi _{ab}^{\lambda s} (t)y_{a}^{\lambda} (t)}, &\text{if }b\in A(h_{a})\cap {A}'\\[2mm] {0}, &\text{else}. \end{cases} \end{align} $  (21) 
4) Outflow of Channelized Downstream Queue Area
$ \begin{align} y_{ab}^{\lambda +1} (t)=\min \left\{n_{ab}^{\lambda} (t), \alpha_{ab} Q_{a}^{\lambda} (t), \frac{\gamma_{ab} w(N_{b}^{1} (t)n_{b}^{1} (t))}{v}\right\} \end{align} $  (22) 
where
$ \phi _{ab}^{\lambda +1, s} (t)=\frac{n_{ab}^{\lambda s} (t)}{n_{ab}^{\lambda} (t)} $  (23) 
$ y_{ab}^{\lambda +1, s} (t)=\phi _{ab}^{\lambda +1, s} (t)y_{ab}^{\lambda +1} (t) $  (24) 
$ y_{a}^{\lambda +1} (t)=\sum\limits_{b\in B(h_{a})} {y_{ab}^{\lambda +1} (t)}. $  (25) 
5) Inflow of Upstream Reservoir From a NoOrigin Node:
$ \begin{align} y_{a}^{1} (t)=\sum\limits_{b\in A(l_{a})\cap {A}'} {y_{ba}^{\lambda +1} (t)} +u_{a} (t) \end{align} $  (26) 
where
$ \begin{align} &y_{a}^{1, s} (t)=\sum\limits_{b\in A(l_{a})\cap {A}'} {y_{ba}^{\lambda +1, s} (t)} \\[2mm] &y_{ab}^{1} (t)\!=\!\begin{cases} {\max \left\{\phi _{ab} (t)y_{a}^{1} (t), \dfrac{y_{a}^{1, s} (t)}{k}\right\}}, \text{if }b\in A(h_{a})\cap{A}' \\[4mm] \displaystyle {k_{1} \bigg(y_{a}^{1} (t)\!\!\!\!\sum\limits_{c\in A(h_{a})\cap {A}'}\!\!\! {\max \left\{\phi _{ac} (t)y_{a}^{1} (t), \frac{y_{a}^{1, s} (t)}{k}\right\}}\bigg)}, \\\qquad\qquad\qquad\qquad\qquad\qquad\quad\ \text{if }b\in {A(h_{a})}/{{A}'}\\ \end{cases} \end{align} $  (27) 
where
As a result from the above formulae, the updated number of vehicles contained in each cell is formulated as follows. For
$ n_{a}^{i} (t+1)=n_{a}^{i} (t)+y_{a}^{i} (t)y_{a}^{i+1} (t) $  (29) 
$ n_{ab}^{i} (t+1)=n_{ab}^{i} (t)+y_{ab}^{i} (t)y_{ab}^{i+1} (t) $  (30) 
$ n_{a}^{is} (t+1)=n_{a}^{is} (t)+y_{a}^{is} (t)y_{a}^{i+1, s} (t) $  (31) 
$ n_{ab}^{is} (t+1)=n_{ab}^{is} (t)+y_{ab}^{is} (t)y_{ab}^{i+1, s} (t). $  (32) 
Traffic incidents are modeled by modifying the value of the corresponding flow capacity of the affected cells.
Traffic jam size is used to describe the effect of congestion. A cell is jammed if its density in the cells of upstream reservoir or in any direction of the downstream channelized areas is greater than 0.9
When drivers heading towards the incidentblocked link obtains information regarding the incident, they may detour at the following intersections along the path. In this section, we consider three detour strategies where the detour rates could be related to certain traffic conditions. Let
Assumption 4: Equal Detour Rate: If a vehicle in link
Assumption 5: Distancerelated Detour Rate: If a vehicle heading to the incidentblocked link
$ \begin{align} \beta_{a} =\frac{\displaystyle\frac{1}{L(\sigma_{h_{a}})}}{\displaystyle\sum\limits_{j\in \sigma_{h_{a} }} {\left(\frac{1}{L(\sigma_{j})}\right)}}. \end{align} $  (33) 
Assumption 6: 100% Detour: The proportion of vehicles that detour at
In the CTM model, given
$ \begin{align} n_{ab}^{i} (t)=\begin{cases} {n_{ab}^{i} (t)\dfrac{\alpha \beta_{a} n_{a}^{is} (t)}{k_{1}}}, &{b\in A(h_{a})\cap {A}'}\\[4mm] {n_{ab}^{i} (t)+\dfrac{\alpha \beta_{a} n_{a}^{is} (t)}{k_{2}}}, &{b\in A(h_{a})/{A}'} \end{cases} \end{align} $  (34) 
where
$ \begin{align} n_{a}^{is} (t)=(1\alpha \beta_{a})n_{a}^{is} (t), 1\leq i < \lambda. \end{align} $  (35) 
At downstream queue channel areas, the number of vehicles is changed as follows:
$ \begin{align} n_{ab}^{\lambda, s} (t)=(1\alpha \beta_{a})n_{ab}^{\lambda, s} (t). \end{align} $  (36) 
We also need to change inflow of the upstream reservoir from an origin for the next time interval. If
$ y_{a}^{1s} (t)=(1\alpha \beta_{a})y_{a}^{1s} (t) $  (37) 
$ \begin{align} y_{ab}^{i} (t)=\begin{cases} {y_{ab}^{i} (t)\dfrac{\alpha \beta_{a} y_{a}^{is} (t)}{k_{1}}}, &{b\in A(h_{a})\cap {A}'} \\[4mm] {y_{ab}^{i} (t)+\dfrac{\alpha \beta_{a} y_{a}^{is} (t)}{k_{2}}}, &{b\in A(h_{a})/{A}'} \end{cases} \end{align} $  (38) 
where
We also need to change inflow of the upstream reservoir of links that are not the origin for the next time interval. Suppose that
$ \begin{align} \bar{{y}}_{ba}^{\lambda +1, s} (t)=y_{ba}^{\lambda +1, s} (t){y}'\alpha \times (1\beta_{b})\beta_{a}. \end{align} $  (39) 
Given that
$ y_{ba}^{\lambda +1, s} (t)={y}'\times (1\alpha \beta_{b}) $ 
we have that
$ {y}'=\frac{y_{ba}^{\lambda +1, s} (t)}{1\alpha \beta_{b}}. $ 
Replacing
$ \begin{align} \bar{{y}}_{ba}^{\lambda +1, s} (t)=y_{ba}^{\lambda +1, s} (t)\times \frac{1\alpha (\beta_{b} +\beta_{a} \beta_{b} \beta_{a})}{1\alpha \beta_{b}}. \end{align} $  (40) 
As a result, we change the inflow of the upstream reservoir of link
$ \begin{align} y_{a}^{1, s} (t)=\sum\limits_{b\in A(l_{a})\cap {A}'} {y_{ba}^{\lambda +1, s} (t)\times \frac{1\alpha (\beta_{b} +\beta_{a} \beta_{b} \beta_{a})}{1\alpha \beta_{b}}}. \end{align} $  (41) 
Also, we change the following inflow value:
$ \begin{align} y_{ac}^{1} (t)=\begin{cases} \displaystyle {y_{ac}^{1} (t)\sum\limits_{b\in A(l_{a})\cap {A}'} {y_{ba}^{\lambda +1, s} (t)\times \frac{\alpha (1\beta_{b})\beta_{a}} {k_{1} (1\alpha \beta_{b})}}}, \\ \qquad\qquad\qquad\qquad\qquad\qquad\ \ \ {c\in A(h_{a})\cap {A}'}\\[3mm] \displaystyle {y_{ac}^{1} (t)+\sum\limits_{b\in A(l_{a})\cap {A}'} {y_{ba}^{\lambda +1, s} (t)\times \frac{\alpha (1\beta_{b})\beta_{a}} {k_{2} (1\alpha \beta_{b})}}}, \\ \qquad\qquad\qquad\qquad\qquad\qquad\ \ \ {c\in A(h_{a})/{A}'} \end{cases} \end{align} $  (42) 
where
We give a oneway grid network as an example as shown in Fig. 2 (a) where it is composed of oneway road links with adjacent rows or columns having opposite directions. Note that this kind of road networks is very common in major cities, for example, New York City. We install a single incident in the network: a single incident occurs on link (33, 34) in the network as shown in the figure. Note that in the grid network, the shortest paths are the timedependent shortest paths.
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Fig. 2 A traffic network with an incident in (a) and the generated subnetwork in (b) 
According to Algorithm 1, for each node in
Path 1: (3, 23, 28, 33, 34);
Path 2: (18, 31, 32, 33, 34);
Path 3: (7, 30, 29, 28, 33, 34);
Path 4: (14, 42, 37, 32, 33, 34);
Path 5: (1, 21, 22, 23, 28, 33, 34);
Path 6: (20, 21, 22, 23, 28, 33, 34);
Path 7: (5, 25, 30, 29, 28, 33, 34);
Path 8: (1, 21, 26, 31, 32, 33, 34);
Path 9: (20, 21, 26, 31, 32, 33, 34);
Path 10: (16, 41, 42, 37, 32, 33, 34).
Thus, we have
Now we evaluate the effectiveness of the proposed strategy by employing MATLAB software through simulation. We first set specific values for the parameters in our simulation. We set a single incident on the 5th cell of link (33, 34) in the network as shown in Fig. 2 (a). The corresponding values of the subnetwork are shown in Table Ⅰ. In the traffic network in Fig. 3, according to the special structure, the flow proportions for directions are set as:
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Fig. 3 Congestion formulation and dissipation under no traffic control strategy when traffic incident is from the 300 
With the simulation of our designed model, we study the traffic jam in the traffic network in Fig. 2 (b). After the incident messages are delivered, drivers will then detour. We identify the influences of some important parameters, i.e., the inflow rate, the detour rate of drivers, and the start time for providing traffic incident information. We provide a sensitivity test of the following parameters as shown in Table Ⅱ.
First, let
Then, we study the proportion of traffic demand at origins that head to the blocked link, with the results being shown in Fig. 4. Under the situation where
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Fig. 4 Congestion formulation and dissipation under no traffic control strategy and the proportion of traffic demand at origins that head to the blocked link is 
We further study the end time of the traffic incident and the results are shown in Fig. 5. Under the traffic conditions where
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Fig. 5 Congestion formulation and dissipation under no traffic control strategy and the incident is cleared at the 
We study when to provide incident information to drivers in the traffic network in Fig. 6. Under Assumption 4 and the traffic conditions
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Fig. 6 Congestion formulation and dissipation when traffic control strategy begins from the 
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Fig. 7 Congestion formulation and dissipation when traffic control strategy begins from the 
We study the detour proportion
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Fig. 8 Congestion formulation and dissipation when the strategy begins from the 
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Fig. 9 Congestion formulation and dissipation when the strategy begins from the 
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Fig. 10 Congestion formulation and dissipation under traffic conditions 
We study a link set
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Fig. 11 Congestion formulation and dissipation when 
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Fig. 12 Congestion formulation and dissipation when 
This paper presents a new strategy, which provides incident information to drivers and helps them make detours in urban areas. Traffic incident information is only transmitted to the affected vehicles that head towards the blocked link created by the incident. These vehicles are in a subnetwork that can be generated by the Dijkstra's algorithm. Simulations are done to test the effectiveness of the proposed strategy. The CTMbased model is used to estimate the congestion and promote the implementation of our strategy. Future work should consider real world traffic conditions when different links have different traffic density. We also need to design algorithms to accurately estimate the time for vehicles to pass road links, and thus, obtain the timedependent shortest path.
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