IEEE/CAA Journal of Automatica Sinica  2018, Vol. 5 Issue(5): 934-945   PDF    
A Dynamic Road Incident Information Delivery Strategy to Reduce Urban Traffic Congestion
Liang Qi1,2,3, Mengchu Zhou3, Wenjing Luan4     
1. Department of Computer Science and Technology, Shandong University of Science and Technology, Qingdao 266590, China;
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
Abstract: Advanced information and communication technologies can be used to facilitate traffic incident management. If an incident is detected and blocks a road link, in order to reduce the incident-induced traffic congestion, a dynamic strategy to deliver incident information to selected drivers and help them make detours in urban areas is proposed by this work. Time-dependent shortest path algorithms are used to generate a subnetwork where vehicles should receive such information. A simulation approach based on an extended cell transmission model is used to describe traffic flow in urban networks where path information and traffic flow at downstream road links are well modeled. Simulation results reveal the influences of some major parameters of an incident-induced congestion dissipation process such as the ratio of route-changing vehicles to the total vehicles, operation time interval of the proposed strategy, traffic density in the traffic network, and the scope of the area where traffic incident information is delivered. The results can be used to improve the state of the art in preventing urban road traffic congestion caused by incidents.
Key words: Cell transmission model (CTM)     intelligent transportation systems (ITS)     traffic incident management (TIM)     urban traffic congestion    
Ⅰ. INTRODUCTION

Traffic incidents are any non-recurring events including traffic crashes, disabled vehicles, roadway maintenance and reconstruction projects, and special non-emergency 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, incident-based urban traffic congestion is mostly controlled and prevented through traffic flow diversion with the help of the traffic police. Such a strategy is unfortunately labor-intensive, 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]. Real-time traffic information can be sent to drivers through two main kinds of devices: in-car [10] and road-side devices [11]. The type, such as radio GPS-navigators 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 incident-based 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 on-board units of connected vehicles makes it possible to construct an accessible and cost-effective platform for public-sector Traffic Operation Centers to deliver location-based 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 incident-induced 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 incident-based 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 incident-based jam propagation in two-way rectangular grid networks. They also propose control strategies for dispersing incident-based traffic jam and evaluate their efficiency [16]. CTM-based models can depict traffic flow at downstream road links well. However, the aforementioned simulation models do not contain any path-related information when studying travelers' detour behaviors and the incident-induced congestion formulation. In this work we further extend the CTM, build a model to simulate incident-based 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 sub-network 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 work-zone throughputs and decrease total vehicle delays. Traffic signals at on-ramps of freeway [5] can help manage the traffic inflow rate and reduce lane-blocking incident-induced traffic congestion. These methods are suitable for TIM in highways. So far incident-based urban traffic congestion is mostly controlled and prevented through traffic flow diversion with the help of traffic police, which is unfortunately labor-intensive, 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 non-saturated and stable traffic conditions. However, changing conditions in a non-predictable 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 incident-induced 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 time-varying 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 traffic-flow 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 real-time 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 simulation-based 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 real-life highway networks, and their ability to adequately capture traffic dynamics and microscopic driver behavior such as lane changing. However, a simulation-based approach cannot guarantee the solution properties of the model in general [23].

DTA problems are formulated as both path-based [31, 32] and link-based models [28]. An important feature of the former is that the path set has to be explicitly defined and can range from medium to large-sized networks. Hence, for large-scale network applications, path enumeration has not been used to obtain the path set. Link-based models can avoid path enumeration in the solution procedure, and hence can be applied to large-scale networks. However, they do not contain path-related 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 real-world 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 non-recurrent traffic conditions [34]. Recently, the proliferation of mobile communication technologies and devices such as smartphones and on-board units of connected vehicles makes it possible to construct an accessible and cost-effective platform for public-sector Traffic Operation Centers to deliver location-based 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 Lighthill-Whitham-Richards (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 real-world 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 space-mean speeds [47]. Long et al. [15] extend the CTM and apply it to simulate incident-based jam propagation in two-way 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 incident-based traffic jams and evaluating their efficiency [16]. The above cell-based simulation models do not contain any path-related information when studying incident-induced congestion formulation. In this work we further extend CTM [15] to simulate travelers' detour and incident-based traffic jams in urban areas and illustrate the effectiveness of our proposed incident information delivery strategy.

Ⅲ. BASIC NOTATIONS

First, some basic notations are presented: $\mathbb R$ is a real number set, $\mathbb R^{+}$ is the set of positive real numbers, $\mathbb{N}= \{0, 1, 2, \ldots \}$ is a natural number set, $\mathbb{N}^{+}= \mathbb{N}/\{0\}$ is the positive integer set, $\mathbb{N}_{m}= \{0, 1, 2, \ldots, m\}$ and $\mathbb{N}_{m}^{+}= \{1, 2, \ldots, m\}$ where $m$ $\in$ $\mathbb{N}^{+}$ . We consider a multi-destination network as a directed and connected graph $ G(N, A)$ , where $N= \{1, 2, $ $\ldots, $ $ m\}$ denotes the set of nodes and $A$ denotes the set of links. Link $a$ $=(l_{a}$ , $h_{a})\in A$ is a link with a tail node $l_{a}$ and a head node $h_{a}$ . $A(j)$ and $B(j)$ are the set of links leaving node $j$ and heading to node $j$ , respectively. We give some definitions regarding a multi-destination traffic network $G(N, A)$ . Usually a path is composed of a sequence of links. We assume that there is at most one link from a node to another. Thus, a path can be expressed by a sequence of nodes. We give the formal definition of a path as follows.

Definition 1: A path $\sigma_{1, m}= (1, 2, \ldots, m)$ is defined as a sequence of nodes labeled from 1 to $m$ , where $(j, j+ 1) \in A$ , $\forall j$ $\in$ $\mathbb{N}_{m-1}^{+}$ , 1 is its origin and $m$ is its destination. The set of all paths in $G$ is denoted as $\Gamma$ . The set of all paths from node 1 to node $m$ is denoted as $\Gamma_{1, m}$ . If $(j, j+ 1)$ is a link in path $\sigma_{1, m}$ , we denote that $(j, j+ 1) \in \sigma_{1, m}$ , else $(j, j+ 1) \notin \sigma_{1, m}$ ; if $j$ is a node in path $\sigma_{1, m}$ , we denote that $j\in \sigma _{1, m}$ , else $j$ $\notin$ $\sigma_{1, m}$ ; and $\sigma_{j}$ denotes a path with node $j$ as its origin. We define a connection between two paths as $\sigma_{1, m}=\sigma_{1, j}+\sigma _{j, m}$ , where $j\in \sigma_{1, m}$ .

All concepts in Definition 1 can be found in the graph theory in [48].

Definition 2: $l$ : $A\to \mathbb{R}^{+}$ is a link length function, where $l(j, $ $j+1)$ is the length of the link $(j, j+ 1)$ ; and $l_{j, j+1}$ denotes the length between two nodes $j$ and $j+ 1$ where $l_{j, j+1}=l(j, j+ 1)$ if $(j, j+ 1) \in A$ , and $ l_{j, j+1}=+\infty $ if $(j, j + 1) \notin A$ .

Definition 3: $L$ : $\Gamma \to \mathbb{R}^{+}$ is a path length function, where

$ \begin{align} L(\sigma)=\sum\limits_{j=1}^{m-1} {l_{j, j+1}} \end{align} $ (1)

where $\sigma = (1, 2, \ldots, m)\in \Gamma$ .

Definition 4: $\tau $ : $(A, t)\to \mathbb{R}^{+}$ is a link travel-time function, where $\tau (a, t)$ denotes the travel time for a vehicle to pass link $a$ when it stays at the tail of $a$ at time $t$ . $\tau _{i, j}$ denotes the time needed from nodes $i$ to $j$ .

Definition 5: $T$ : $\Gamma \to \mathbb{R}^{+}$ is a path travel-time function, where

$ \begin{align} T(\sigma)=\sum\limits_{j=1}^{m-1} {\tau (l_{j, j+1}, t_{j, j+1})} \end{align} $ (2)

denotes the travel time that a vehicle needs to pass path $\sigma \in \Gamma$ .

Definition 6: $f$ : $\Gamma \to \mathbb{N}^{+}$ is a node-count function, where $f(\sigma)$ $=|\sigma |-1$ denotes node count in path $\sigma \in \Gamma$ .

Definition 7: $\sigma_{1, m}^{\ast} $ denotes a time-dependent shortest path with the fewest nodes from node 1 to node $m$ , if

$ \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)

$T_{1, m} =T(\sigma_{1, m}^{\ast})$ denotes the travel time to pass the time-dependent shortest path with the fewest nodes from node 1 to node $m$ ; and $\Gamma_{1, m}^{\ast} $ is the set of all time-dependent shortest path with the fewest nodes from 1 to $m$ . $\Gamma ^{*}$ denotes the set of all time-dependent shortest path with the fewest nodes.

Ⅳ. INCIDENT INFORMATION DELIVERY STRATEGY

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 incident-induced 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 Assumptions

In 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: Time-dependent 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 $\sigma_{1}$ , $\sigma_{2}$ $\in$ $\Gamma_{1, m}$ with $T(\sigma_{1}) < T(\sigma_{2})$ , or $T(\sigma_{1})=T(\sigma _{2})$ and $L(\sigma_{1})$ $ < $ $L(\sigma_{2})$ , then drivers select path $\sigma_{1}$ from origin 1 to destination $m$ .

Assumption 2: Fewest-node Path Selection: We assume that given two same-travel-time and same-length 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 $\sigma_{1}$ , $\sigma_{2}\in \Gamma _{1, m}$ with $T(\sigma_{1})=T(\sigma_{2})$ , $L(\sigma _{1})=L(\sigma_{2})$ and $f(\sigma_{1})$ $ < $ $f(\sigma_{2})$ , drivers select path $\sigma_{1}$ from origin 1 to destination $m$ .

Assumption 3: Equal Probability Selection:) If there are $N$ same-travel-time, same-length and same-number-of-node paths from the origin to destination, the probability of each path to be selected is equal, i.e., a driver has the probability of $1/N$ to select each path.

B. Sub-network Construction

Given a network $G(N, A)$ and a link $s=(l_{s}, h_{s})$ where an incident happens and blocks the link, we adopt the Dijkstra's algorithm to generate a sub-network $G(N', A')$ which contains all paths with each satisfying the following conditions: it is a time-dependent shortest path from the origin to its destination $h_{s}$ ; it has fewer nodes than other paths from the origin to $h_{s}$ ; and it contains link $s$ . The traffic flow in the sub-network could be affected by the incident on $s$ under Assumptions 1-3, because some vehicles are being driven to the blocked link. The detailed steps to obtain $G(N', A')$ are shown as follows.

Algorithm 1  Generate a Sub-network $G(N', A')$
Input: A network $G(N, A$ ) and a link $s=(l_{s}, h_{s}$ ) where an incident occurs.
Output: A sub-network $G(N', A'$ ).
Step 1. Initialization
                Set $S: = \{h_{s}, l_{s}\}$ ; //$ S$ contains the nodes whose timedependent shortest paths to $ h_{s}$ have been obtained
                Set $R: = N-S$ , $N' : = \{h_{s}, l_{s}\}$ , $A' : = \{(l_{s}, h_{s})\}$ and $\sigma_{l_{s}}: =s$ ;
                Set $T_{j, h_{s}} : =+\propto $ and $ f_{l_{s}, h_{s}} : =0$ ;
                Set $\Gamma_{1, m}^{\ast}:=\Gamma^{*}:= \{s\}$ and $\Gamma_{j, h_{s}} ^{\ast}:= {\emptyset}$ ;
Step 2. Calculation and update
                Select a node $j\in {R}$ such that the paths from $j$ to $l_{s}$ and then $ h_{s}$ are the time-dependent shortest paths of those from a node in ${R}$ to $l_{s}$ and then $h_{s}$ , i.e.,
                $\exists \sigma_{k}= (k, \ldots, l_{s}, h_{s})\in \Gamma^{*} {\rm and} (j, k)\in A$
                such that $\forall \sigma_{k'} =(k', \ldots, l_{s}, h_{s})\in \Gamma^{*}$ , $j'\in T$ and ($j', k')\in A: T(\sigma_{k})+\tau_{j, k} \leq T(\sigma _{k'})+\tau_{j', k'} $ , $L(\sigma_{k})$ $+$ $l_{j, k} \leq L(\sigma_{k' })+l_{j', k'} $ ;
                For each path $\sigma_{j}$
                     $f(\sigma_{j}):=f(\sigma_{k})+ 1$ if $\sigma_{j}=(j, k)+\sigma_{k}$ .
                     If $\sigma_{j}$ is a time-dependent shortest path with the fewest nodes from $j$ to $h_{s}$ , then
                          $\Gamma_{j, h_{s}} ^{\ast}:= \Gamma_{j, h_{s}}^{\ast} \cup \{\sigma_{j}\}$ ;
                          Take node $j$ out of $R$ , i.e., $R:=R-\{j\}$ ;
                          Put node $j$ into $S$ , i.e., $S:=S +\{j\}$ ;
                     End If
                End For
                If there exists no node $j' $ such that $(j', j)\in A$ , then
                     Put all nodes in each path $\sigma_{j} \in \Gamma_{j, h_{s}} ^{\ast} $ into $N'$ , i.e.,
                     $N': =N' + \{r| r\in \sigma_{j}\}$ ;
                     Put all links in each path $\sigma_{j} \in \Gamma_{j, h_{s}} ^{\ast}$ into $A'$ , i.e.,
                     $A' :=A' +$ {($r$ , $k) |$ ($r$ , $k)\in \sigma_{j}$ };
                End If
Step 3. Iteration
                Repeat Step 2 until $S=N$ ;
Return

Assume that the number of nodes in $N$ is $n$ . For each node in $N$ , all paths with the shortest distance from it to $h_{s} $ should be computed. Therefore, the complexity of the algorithm of generating such a sub-network is $O(n^{2})$ .

C. Traffic Network Model Based on CTM

Daganzo [35] proposes a CTM to simplify the solution scheme of the Lighthill-Whitham-Richards (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 path-related information will be contained in the model when studying the incident-induced congestion formulation. We use a time-step method based on the extended CTM to simulate the formation and dissipation of incident-based traffic jams and evaluate the efficiency of the proposed control strategies.

As shown in Fig. 1, each link $a$ in the networks is divided into two distinct zones: downstream queue storage areas $L$ , $S$ and $R$ where vehicles are organized into separate turning movements (left in $ L$ , straight in $S$ , and right in $R$ ), and an upstream reservoir where the turning movements are mixed. Notice that there may be less than three storage areas, e.g., the queue storage areas are composed of $S$ and $R$ where there is no left turn for the traffic outflow of link $a$ . We assume that the upstream reservoir is composed of $\lambda $ cells indexed from 1 to $\lambda$ , and the channelized queue area occupies a cell indexed $\lambda+1$ . We adopt $\phi _{L}$ , $\phi _{S}$ , and $\phi _{R}$ to denote the proportions of vehicles travelling in the left turning, straight, and right turning directions, respectively. Stopline width assignment variables $\alpha_{L}$ , $\alpha_{S}$ , $\alpha_{R}$ , respectively, denote the proportions of the segregated queue areas devoted to the left turning queue storage area, to the straight queue storage area and to the right turning queue storage area. According to the definition, we have $\phi _{L}+\phi _{S}+\phi _{R}= 1$ and $\alpha_{L}+\alpha _{S}+\alpha_{R}= 1$ . We adopt a ``balanced'' layout of stop line assignment [13] such that the stop line widths devoted to the straight and turning directions are in exactly the same ratio as the demands, i.e., $\phi _{L}=\alpha_{L}$ , $\phi _{S}$ $=\alpha_{S}$ , and $\phi _{R}=\alpha_{R}$ .

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Fig. 1 Link $a$ in networks

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 time-step method. Traffic flow formulation can be classified into the following categories: inflow of upstream reservoir of the origin and the other nodes ($i= 1$ ), inflow of upstream cells $(1 < i\leq \lambda -1)$ , and inflow and outflow of channelized downstream queue area ($i=\lambda$ ). In the following equations, $v$ (miles per hour) is the free flow speed and $w$ (miles per hour) is the speed of all backward moving waves. $y^{i}(t)$ is the number of vehicles that enter cell $i$ during time interval $t$ , $n^{i}(t)$ is the number of vehicles in cell $i$ before $t$ , $N^{i}(t)$ denotes the maximum number of vehicles that can be contained in cell $i$ during $t$ , and $ Q^{i}(t)$ denotes the inflow capacity in cell $i$ during $t$ . More details regarding CTM can be referred to [35]. The inflow formulation under normal traffic conditions, i.e., without incidents, is presented as follows:

1) Inflow of Upstream Reservoir From an Origin: Let $d_{a}(t)$ denote traffic demand rate from an origin node $l_{a}$ to link $a$ in time interval $t$ , $d_{a}^{s} (t)$ traffic demand rate from node $l_{a}$ through link $a $ to link $s$ at time $t$ , $\phi _{a}^{s} (t)$ and the proportion of vehicles entering link $a$ at time $t$ with the destination to $s$ . The inflow of upstream reservoir of the origin can be calculated as

$ \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 $a$ in time interval $t$ and choose $b$ as the next link is calculated as

$ \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 $a$ and head to link $s$ in time interval $t$ is computed as

$ \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 $a$ in time interval $t$ with the destination to $s$ is calculated as

$ \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 $a$ in time interval $t$ , choose $b$ as the next link and head to $s$ is calculated as

$ \begin{align} y_{ab}^{1s} (t)=\phi _{ab}^{1s} (t)y_{a}^{1} (t) \end{align} $ (8)

where $\phi _{ab}^{1s} (t)$ is determined according Assumptions 1-3.

2)Inflow of Upstream Cells: $n_{a}^{is} (t)$ denotes the number of vehicles contained in cell $i$ of $a$ and head to $s$ at the start of time interval $t$ ; $\phi _{ab}^{i} (t)$ denotes the proportion of vehicles contained in cell $i$ of link $a$ and choose $b$ as the next link; $\phi _{a}^{is} (t)$ denotes the proportion of vehicles that enter cell $i$ of link $a$ and head to link $s$ in time interval $t$ ; $y_{a}^{is} (t)$ denotes the number of vehicles that enter cell $i$ of link $a$ , choose $b$ as the next link and head to $s$ in time interval $t$ ; $\phi _{ab}^{is} (t)$ and $y_{ab}^{is} (t)$ respectively denote the proportion and number of vehicles that enter cell $i$ of link $a$ , where link $b$ is the next link, and the time to link $s$ is time interval $t$ . The inflow of upstream cells can be calculated as

$ \begin{align} &y_{a}^{i} (t)=\min \left\{n_{a}^{i-1} (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}^{i-1} (t)}{n_{a}^{i-1} (t)},&\text{if }n_{a}^{i-1} (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}^{i-1, s} (t)}{n_{a}^{i-1} (t)},&\text{if }n_{a}^{i-1} (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}^{i-1, s} (t)}{n_{a}^{i-1} (t)},&\text{if }n_{a}^{i-1} (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 $a$ to link $b$ is computed as follows:

$ \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 $\lambda $ of link $a$ , choose link $b$ as the next link, and head to link $s$ in time interval $t$ is computed as

$ \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}^{\lambda-1}(t)=0 \end{cases} \end{align} $ (20)

where $k=|A(h_{a})\cap {A}'|$ .

The number of vehicles that enter cell $\lambda $ of link $a$ , choose link $b$ as the next link, and head to link $s$ in time interval $t$ is computed as

$ \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 $y^{\lambda+1}$ is defined as the outflow of the terminal cell $\lambda$ . The outflow of the channelized downstream queue area can be calculated as follows:

$ \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 $\gamma_{ab} ={\alpha_{ab}^{\lambda} (t)}/{\sum_{a\in B(l_{b})\cap A} {\alpha_{ab}^{\lambda} (t)}}$ .

$ \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 No-Origin 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 $u_{a}(t)$ denotes the inflow rate from node $l_{a}$ through link $a $ at time $t$ .

$ \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 $k=|A(h_{a})\cap {A}'|$ and $k_{1} ={\phi _{ab} (t)}/{\sum_{b\in A(h_{a})/{A}'} {\phi _{ab} (t)}}$ .

As a result from the above formulae, the updated number of vehicles contained in each cell is formulated as follows. For $1$ $\leq$ $i \leq \lambda$ , we have

$ 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. $Q_{a}^{i} (t)=0$ if $t$ belongs to the period with an obstruction on cell $i$ . We assume that $Q_{a}^{i} (t)$ and $N_{a}^{i} (t)$ are independent of time and cells' indices. Hence, they are constants denoted as $Q_{a}^{i} (t)=Q$ and $N_{a}^{i} (t)=N$ , where $N$ , $Q\in \mathbb{R}^{+}$ .

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$N$ [16], where $N$ is the maximal number of vehicles that a cell in the upstream reservoir or an area in the downstream channelized areas can contain. The size of the traffic jam is described in terms of the total number of jammed cells.

D. Drivers' Detour Model

When drivers heading towards the incident-blocked 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 $s$ be a road link blocked by an incident. In the following discussion, $\alpha $ denotes the proportion of vehicles heading to link $s$ that will change their direction and release from a node along its path; $\beta_{a}$ represents the proportion of flows that change their direction to the incident-blocked link and release link $a$ 's head node. There are assumptions that we will adopt to define the detour rate:

Assumption 4: Equal Detour Rate: If a vehicle in link $a$ heading to the incident-blocked link $s$ receives the incident information and decides to detour, it has equal probability to detour at each of the following nodes on its path to link $s$ . Formally, let $\sigma_{h_{a}} $ be the path of vehicles from the head node of link $a$ to $s$ , and let the proportion of vehicles that detour at $a$ 's head node be $\beta_{a} =1/f(\sigma_{h_{a}})$ .

Assumption 5: Distance-related Detour Rate: If a vehicle heading to the incident-blocked link $s$ receives the incident information and decides to detour, its probability to detour is related with the distance to $s$ , i.e., the detour probability is bigger when the distance is shorter. Formally, let $\sigma_{h_{a}} $ be the path of vehicles from the head node of link $a$ to $s$ ; the proportion of vehicles detour at node $a$ then equals

$ \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 $a$ 's head node is equal to 100% when there is another equal-length time-dependent path containing no incidental link from the $a$ 's head node to $s$ 's head node.

In the CTM model, given $\alpha $ and $\beta_{a}$ , we first modify the number of vehicles in each cell as follows:

$ \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 $k_{1} =|A(h_{a})\cap {A}'|\ne 0, k_{2} =|A(h_{a})/{A}'|\ne 0$ , and $1\leq$ $i$ $ < $ $\lambda$ ,

$ \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 $y_{a}^{1s} (t)$ enters the 1st cell, then $\alpha $ proportion of flows leading to link $s$ will change their direction immediately. Thus, we have

$ 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 $k_{1} =|A(h_{a})\cap {A}'|\ne 0, k_{2} =|A(h_{a})/{A}'|\ne 0$ , and $1\leq$ $i$ $< $ $\lambda$ .

We also need to change inflow of the upstream reservoir of links that are not the origin for the next time interval. Suppose that $y_{ba}^{\lambda +1, s} (t)$ vehicles enter the 1 st cell from $b$ . Then $\alpha $ portion of flows change their direction immediately. Suppose the outflow from link $b\in A(l_{a})\cap A' $ to $s$ without any traffic incident information from equals $y'$ . Then, after receiving the incident information, the amount of $y' \alpha $ vehicles will change their directions while $y' \alpha \beta_{b}$ vehicles change their directions and leave from link $b$ , and $y' \alpha (1-\beta_{b})$ vehicles change directions at upcoming nodes in the path. In the next link $a$ , the number of vehicles that change their directions is $y' \alpha (1-\beta_{b})\beta_{a}$ . We use $\bar{{y}}_{ba}^{\lambda +1, s} (t)$ to denote the outflows of the terminal cell $\lambda $ of link $a$ that choose link $b$ as the next link and head to link $s$ in time interval $t$ . Thus, we have the following equations:

$ \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 $y' $ in (39), we have

$ \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 $a$ during each next time interval as follows:

$ \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 $k_{1} =|A(h_{a})\cap {A}'|$ , $k_{2} =|A(h_{a})/{A}'|$ , $k_{2} \times k_{2} \ne 0$ , and $1$ $\leq$ $i < \lambda$ .

Ⅴ. CASE STUDY A. An Example to Construct a Sub-Network

We give a one-way grid network as an example as shown in Fig. 2 (a) where it is composed of one-way 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 time-dependent shortest paths.

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Fig. 2 A traffic network with an incident in (a) and the generated sub-network in (b)

According to Algorithm 1, for each node in $G$ , we compute all the shortest paths leading to node 34 with (33, 34) as the last link. After that, we obtain the following paths to generate the sub-network.

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 $N' =$ {34, 33, 28, 23, 3, 32, 31, 18, 29, 30, 7, 37, 42, 14, 22, 21, 1, 20, 25, 5, 26, 41, 16}, $A' =$ {(33, 34), (28, 33), (23, 28), (3, 23), (32, 33), (31, 32), (18, 31), (29, 28), (30, 29), (7, 30), (37, 32), (42, 37), (14, 42), (22, 23), (21, 22), (1, 21), (20, 21), (25, 30), (5, 25), (26, 31), (21, 26), (41, 42), (16, 41)}. After the above steps, the generated sub-network is shown in Fig. 2 (b). According to Assumptions 1-3, when a traffic incident happens, we only need to provide the incident information to drivers in the subnetwork. Drivers from other links of the network in Fig. 2 (a) will not head to the blocked link, and thus the incident information is useless for them.

B. Simulation and Results

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 sub-network 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: $\phi _{T}=\alpha_{S}= 0.5$ where $\phi _{T}\in \{\phi _{L}, \phi _{R}\}$ . The analysis period of interest is divided into 2000 intervals (i.e., 2.78 h). The network is assumed to be empty initially. Traffic starts to enter the network from origins 1-20 at the first time interval. Several time intervals are required to allow the system to stabilize. The incident occurs at the $t_{1}= 301{\rm st}$ interval. Note that here we do not consider the effect of traffic light signal strategies regulating the traffic flow. This can be easily realized by modifying the value of the corresponding flow capacity of the affected cells. In the simulation, in order to keep the balance of the traffic flow, we set all inflows as well as all outflows of links to be equal. We have the value of inflow and outflow of the nodes.

Table Ⅰ
PARAMETERS FOR THE CTM [16]
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Fig. 3 Congestion formulation and dissipation under no traffic control strategy when traffic incident is from the 300${\rm th}$ to 1000${\rm th}$ interval, and the proportion of traffic demand heading to the blocked link is $p= 0.1$

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.

Table Ⅱ
PARAMETERS TO TEST

First, let $z$ denote a constant traffic demand at origins, i.e., $\forall t$ , $a$ , $z=d_{a}(t)$ , and $p$ denote the proportion of traffic flow heading to the blocked link s at origins. We study the effect of $z$ on jam formation and dissipation if there is no traffic control strategy. The incident occurs at the $301{\rm st}$ interval and is cleared at the $1000{\rm th}$ interval. We set $p= 0.1$ . Some simulation results of congestion formation and dissipation are shown in Fig. 3. We learn that when the traffic demand increases, the traffic jam forms more quickly. We have a result that if $z\leq 1.3$ , the traffic jam can dissipate in 100 intervals (8.33 minutes). In the following simulation, we set the traffic demand $z= 0.8$ .

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 $z= 0.8$ , with a increase of $p$ , the number of jammed cells grows sharply. There is little congestion when $p$ is below 0.01.

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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 $p=$ 0.01, 0.05, 0.1, 0.2, and 0.3, respectively, and $z= 0.8$

We further study the end time of the traffic incident and the results are shown in Fig. 5. Under the traffic conditions where $z= 0.8$ and $p= 0.1$ , if the incident is cleared and traffic capacity of the blocked link is recovered in 200 intervals (16.67 minutes), the number of jammed cells can be kept below 10.

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Fig. 5 Congestion formulation and dissipation under no traffic control strategy and the incident is cleared at the $400{\rm th}$ , $500{\rm th}$ , $600{\rm th}$ , $700{\rm th}$ , and $1000{\rm th}$ intervals, respectively, where $z= 0.8$ and $p= 0.1$

We study when to provide incident information to drivers in the traffic network in Fig. 6. Under Assumption 4 and the traffic conditions $z= 0.8$ and $p= 0.1$ , if the proportion of flows that change their direction to link $s$ is 0.9, there is little congestion when we start to deliver the incident information in less than 100 intervals (8.33 minutes). Under Assumption 5, we have similar results as shown in Fig. 7.

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Fig. 6 Congestion formulation and dissipation when traffic control strategy begins from the $300{\rm th}$ , $400{\rm th}$ , $410{\rm th}$ , $420{\rm th}$ , $430{\rm th}$ , $440{\rm th}$ , and $450{\rm th}$ intervals, respectively, under Assumption 4 and where $z=0.8$ and $p= 0.1$
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Fig. 7 Congestion formulation and dissipation when traffic control strategy begins from the $300{\rm th}$ , $400{\rm th}$ , $410{\rm th}$ , $420{\rm th}$ , $430{\rm th}$ , $440{\rm th}$ , and $450{\rm th}$ intervals, respectively, under Assumption 5 and where $z= 0.8$ and $p= 0.1$

We study the detour proportion $\alpha $ and have the results as shown in Fig. 8. We can find there is little congestion formed when $\alpha > 0.8$ , which means that if more than 80 % vehicles can obtain the incident information and then detour when $z$ $\leq$ $0.8$ and $p\leq 0.1$ , no traffic jam occurs. Under Assumption 5, we have similar results as shown in Fig. 9. Lastly we study the detour models given by Assumptions 4 and 5, respectively. Under the traffic conditions that $z= 0.8$ and $p= 0.1$ , we set $z$ $=2.5$ , $p= 0.4$ , $ t_{3}= 450$ . The results in Fig. 10 show that there is more congestion under the traffic detour strategy of Assumption 5 compared to that of Assumption 4. It means that drivers approaching the blocked link need to leave the path earlier to help reduce the traffic congestion induced by the incidents.

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Fig. 8 Congestion formulation and dissipation when the strategy begins from the $300{\rm th}$ to $1000{\rm th}$ interval, and the detour proportion $\alpha =$ 0.5, 0.6, 0.7, 0.8, and 0.9, respectively, under Assumption 4 and where $z= 0.8$ and $p= 0.1$
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Fig. 9 Congestion formulation and dissipation when the strategy begins from the $300{\rm th}$ to $1000{\rm th}$ interval, and the detour proportion $\alpha =$ 0.5, 0.6, 0.7, 0.8, and 0.9, respectively, under Assumption 5 and where $z= 0.8$ and $p= 0.1$
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Fig. 10 Congestion formulation and dissipation under traffic conditions $z$ $=$ $2.5$ , $p= 0.4$ , and when the traffic control strategy begins at $ t_{3}= 450$ , and under the two detour strategies of Assumptions 4 and 5, respectively

We study a link set $A ''$ and a constant $\eta $ where the number of nodes from a link in $A ''\subseteq A' $ to link $s$ is no more than $\eta $ while the number of nodes from a link in $A' -A ''$ to $s$ is bigger than $\eta$ . We only deliver the incident information to drivers at road links in $A ''$ instead of $A'$ . Obviously, $\eta $ represents the scope of area where incident information is delivered. Some simulation results are shown in Figs. 11 and 12: when $\alpha =$ 0.95, we only deliver information to drivers in link (28, 33) and (32, 33) such that at most one cell is congested as shown in Fig. 11. With the decrease of $\alpha$ , when delivering the incident information to a larger scope of road links, we can have a better result with fewer jammed cells.

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Fig. 11 Congestion formulation and dissipation when $f=$ 1-5 under Assumption 5, and where $\alpha = 0.95$ , $z= 0.8$ and $p= 0.1$
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Fig. 12 Congestion formulation and dissipation when $f=$ 1-5 under Assumption 5, and where $\alpha = 0.60$ , $z= 0.8$ and $p=0.1$
Ⅵ. CONCLUSIONS

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 sub-network that can be generated by the Dijkstra's algorithm. Simulations are done to test the effectiveness of the proposed strategy. The CTM-based 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 time-dependent shortest path.

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