a novel vehicle sequencing algorithm with ... - Semantic Scholar
8th International Conference of Modeling and Simulation - MOSIM’10 - May 10-12, 2010 - Hammamet - Tunisia ”Evaluation and optimization of innovative production systems of goods and services”
A NOVEL VEHICLE SEQUENCING ALGORITHM WITH VEHICULAR INFRASTRUCTURE INTEGRATION FOR AN ISOLATED INTERSECTION F. YAN, M. DRIDI and A. EL MOUDNI Laboratoire Syst`emes et Transports Universit´e de Technologie de Belfort-Montb´eliard 90010 Belfort cedex, France {fei.yan, mahjoub.dridi, abdellah.el-moudni}@utbm.fr
ABSTRACT: In this paper, we study a novel vehicle sequencing algorithm for an isolated intersection with a new control method. The control method is based on the utilization of Vehicular Infrastructure Integration and the consideration that each vehicle arrives individually. We consider knowledge of each vehicle’s arrival time and the time each vehicle needs to traverse the intersection. Our objective is to schedule all approaching vehicles to pass this intersection in the shortest duration, the throughput of the intersection will then be increased. A Branch and Bound algorithm and a heuristic are proposed to evacuate the approaching vehicles as soon as possible. Structural properties of the problem are carefully investigated to simplify the search procedure of an optimal passing sequence. Simulation results in evacuation time, average waiting time and average queue size demonstrate the performance gain obtained when using the proposed schemes. KEYWORDS: Isolated intersection, Vehicular Infrastructure Integration, Vehicle arrival time, Branch and bound.
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INTRODUCTION
Traffic congestion at intersections is one of the main issues to be addressed by today’s traffic management schemes. Countless efforts have been made directed toward efficiently improving the traffic situation at intersections. However, although there are well-planned road management schemes, sufficient infrastructure for transportation, we still face the congestion of traffic, especially at the intersections. The construction of additional infrastructure may be considered as one of the solution for the problem, but it is less feasible owing to political and environmental concerns. This urges researchers to find other methods to improve the efficiency of traffic control at intersections. The enhancement of high-tech information and communication technology such as the low cost wireless connectivity, miniaturization of computing devices, and availability of Global Positioning System (GPS) provoked the emergence of intelligent vehicles equipped with In-vehicle Information System (IVIS). The Vehicular Infrastructure Integration (VII) technology came forth under this background. Many realtime message dissemination systems and reliable information exchange protocols were proposed to cater to the development of new traffic control systems. For example, a wireless traffic light system is pro-
posed, where information about current lights status, location of intersection, and a reference point are broadcasted periodically (Huang Q.-F. and R. Miller, 2003). Intelligent devices that can be embedded in next generation vehicles were invented to provide the drivers with a real-time view of the road traffic far beyond what they can physically see (Nadeem T. et al., 2004). Vehicles equipped with this device can disseminate traffic information using short-range wireless communication. Besides, approaches and algorithms were proposed focusing on scheduling the traffic signal to maximize the traffic throughput while minimizing the average latency with various theories. For instance, based on new embedded devices and communication technologies, car platooning investigations aiming at increasing the capacity of roads by reducing temporal distances between vehicles were extensively discussed (Pan Y.-J., 2008), (Contet J.-M. et al., 2007). A reservation based system for alleviating traffic congestion at intersections is proposed in (Dresner K. and P. Stone, 2004) and (Dresner K. and P. Stone, 2006). Vehicles are controlled by agents under the assumption. Queuing theory is implemented in traffic control, and the problem of scheduling traffic at an intersection was addressed by structuring the problem as a Markov decision process (Yu X.-H. and A. R. Stub-
MOSIM’10 - May 10-12, 2010 - Hammamet - Tunisia
berud, 1997). A new approach of traffic control at intersection is proposed that requires communication with a center controller at intersection (Wunderlich R. et al., 2008). This approach needs the information about the intended route of each vehicle that reaches the intersection. There are also researches to predict vehicle’s accurate arrival time at the intersection with both the historical and real-time GPS vehicle location data (Tan C.-W. et al., 2008). Based on the knowledge of each vehicle’s arrival time at intersection, algorithms in (Wu J. et al., 2009) aim at scheduling approaching vehicles to pass the intersection taking into account each vehicle arrives individually, but authors only considered the control strategy for a 2-way intersection. All these efforts in wireless communication and schedule algorithms share a common goal: to improve the traffic situation at controlled intersection while guaranteeing the driver safety. We study in this paper a more general problem to guide vehicles through the intersection with the consideration of Vehicular-to-Infrastructure communications. We assume that there is a controller in the center of the intersection and all vehicles equipped with IVIS are capable of communicating with the center controller in some fashion. The vital telemetry data of each vehicle can be obtained by the controller once the vehicle enters the controller’s control range. Basically, we assume knowledge of the vehicle’s accurate arrival time at the intersection, which is possible in very near future with development of Intelligent Transportation System. Decision of vehicle passing sequence will then be made and broadcasted to all vehicles. A Branch and Bound algorithm and a heuristic are proposed for finding an optimal vehicle passing sequence. Structural properties of the problem are carefully studied to simplify the search procedure of the Branch and Bound algorithm. The rest of this paper is structured as follows: In Section 2, we specify the studied problem, along with some useful notations that will be used in the sequel. In Section 3, the structural properties of the problem are carefully investigated. Algorithms are presented in the section follows. In Section 5, various traffic scenarios are simulated to compare the proposed control strategies with a traditional fix-cycle time traffic lights technique. Conclusions are drawn in the last section. 2 2.1
PROBLEM DEFINITION System model
To specify the problem we study, an example of the intersection under consideration is presented in figure. 1. This intersection appears frequently in realword traffic networks. It is a four-approach intersection with through lanes (some also serve as right-turn
lanes) and exclusive left-turn lanes. The controller is located in the center of intersection with same control range for the four directions. Vehicles approaching the intersection should inform the center controller their data after enter the control range, and vehicle overtaking on same lane is not allowed. According to frequently used traffic convention, lanes on which vehicles can pass through intersection without conflicts are compatible. For example, lane 1 and lane 2 in figure. 1 are compatible lanes. We define all the vehicles on compatible lanes as one Vehicle Class (VC). Thus, all the vehicles in the control range can be divided into several vehicle classes according to the lanes they are running on. Vehicles on incompatible lanes (or vehicles in different vehicle classes) can not traverse the intersection at same time. For the reason of safety, there is usually a time duration between two vehicles from different vehicle classes during which no vehicles behind the stop lines can pass intersection. We define this time as the integral stop time and we assume that this time duration is a constant only decided by the vehicle class that will get the authorization of using the intersection. For instance, the integral stop time when changing authorization from vehicles on lane 1 to vehicles to lane 3 is only decided by the vehicles class that contains lane 3. Suppose at time t0 = 0, there are n vehicles in the control range approaching the intersection from different directions, and the data of all vehicles have been received by the controller in a very short time. The information of each vehicle contains four parts: • Vehicle identification (ID): used to identify the records belonging to different vehicles. • Lane number: which lane it is moving on. • Vehicle arrival time: the precise time it arrives at the stop line from time t0 . • Vehicle passing time: the time it needs to accelerate from stop line until it reaches a safe distance to its follower on same lane (only related to the type of vehicles). The vehicle passing time depends on the type of vehicles, for example, cars and trucks. Normally, trucks are slower to change speed and, therefore, use more time to accelerate until it can reach a safe distance for the vehicle following it on same lane to start. Our objective is to decide an vehicle passing sequence to minimize the evacuation time (the time all vehicles passing the intersection) of all vehicles. The minimization of the evacuation time will naturally increase the throughput of the intersection in consequence. It should be noted that the control technique may be applied to any intersection layout.
MOSIM’10 - May 10-12, 2010 - Hammamet - Tunisia • li , the number of lanes in VC i . • l(i,l) , the lth lane in VC i , where 1 6 l 6 li . • n(i,l) , the number of vehicles on lane l(i,l) . • v(i,l,j) , the j th vehicle on lane l(i,l) , the j is indexed according the arrival precedence of vehicles on lane l(i,l) . • a(i,l,j) , the arrival time of v(i,l,j) , i.e., the time v(i,l,j) needs to arrive at the stop lane (or the waiting queue) from time t0 . • s(i,l,j) , the time vehicle v(i,l,j) starts to pass the intersection. Figure 1: Schematic of intersection control with Control Device
• p(i,l,j) , the passing time of v(i,l,j) , i.e., the time v(i,l,j) needs to accelerate from stop line until it reaches a safe space to the vehicle following it on same lane.
As a conclusion, the constraints of the problem can be described as:
• C(i,l,j) , the time that vehicle v(i,l,j) have a safe space to the vehicle following it on the same lane.
1. All vehicles approaching the intersection are partitioned into several vehicle classes.
Thus, for n vehicles detected at time t0 , the objective function to minimize can be described as
4. Before a vehicle pass the intersection, the vehicle after it on same lane cannot start to pass even it has arrived.
max{C(i,l,j) } 1 6 i 6 m, 1 6 l 6 li , 1 6 j 6 n(i,l) s > a(i,l,j) (i,l,j) C(i,l,j) = s(i,l,j) + p(i,l,j) s.t. s > C(i,l,j) (i,l,j+1) C − C(i0 ,l0 ,j 0 ) > si0 , if i 6= i0 and (i,l,j) C(i,l,j) > C(i0 ,l0 ,j 0 )
5. Vehicles in same class but not the same lane may pass the intersection simultaneously.
In order to design an efficient algorithm, we need to study the structural properties of the problem first.
2. There are several lanes in each vehicle class. 3. Vehicles on each lane need to pass the intersection in First In First Out (FIFO)-way.
6. There is a time duration (integral stop time) between two vehicles from different vehicle classes during which no vehicles behind the stop line can pass intersection. 2.2
Formulation
In the following, the formal description of the problem is given. At first, some notations that will be used are defined. Suppose there are n vehicles approaching the intersection and all vehicles are partitioned into m vehicle classes according to the lanes where they are running on. In each class, vehicles are separated on different lanes.
3
PROPERTIES OF THE PROBLEM
In order to derive an algorithm that can find an optimal vehicle passing order from time t0 = 0, we need study the structural properties of vehicle data. These properties will be used to simplify the search procedure of the algorithms we are going to present. First, we need some definition about the vehicle group. A vehicle group G is defined as a set of vehicles from same VC that pass intersection without the interruption of vehicles in other VC. Specifically, a vehicle group will have the following properties: 1. A vehicle class has at least one group of vehicles. 2. Each vehicle should be in one and only one group.
• VC i , the i
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vehicle class, where 1 6 i 6 m.
• ni , the number of vehicles in VC i . • si , the integral stop time of VC i .
3. There may be idle time waiting for some vehicles to arrive in a group, but there is no integral stop time during the passing process of vehicles in one group.
MOSIM’10 - May 10-12, 2010 - Hammamet - Tunisia
The completion time of a vehicle group is defined as the maximum completion time of vehicles in this group, i.e., CG = max{C(i,l,j) }, where v(i,l,j) ∈ G. It is easy to notice that vehicles pass intersection in the sequence of vehicle group. Then we can describe the order of vehicles as a group sequence GS: GS = (G1 , G2 , . . . , Gb ), b > m. The problem of finding an optimal passing order of vehicles changes to the problem of partitioning vehicles into different groups and finding an optimal group sequence to minimize the vehicle evacuation time. Since there are lanes in each vehicle class and vehicles on each lane should pass the intersection in a FIFOway, we need to partition vehicles into groups, taking into account the vehicle sequence on different lanes. We can then have the following lemma. Lemma 3.1 Consider there are two vehicles v(i,l,j) , v(i,l,j 0 ) on lane l(i,l) with v(i,l,j 0 ) arriving at the stop line after v(i,l,j) , i.e., j < j 0 . Then we have:
time of all ‘un-partitioned’ vehicles on this lane after Gr . Then we can have the following property: Property 3.2 There is an optimal vehicle group sequence, in which any group Gx from VC i (1 6 x 6 b) has at least one vehicle belonging to lane l(i,1) . Proof. Suppose there are only two lanes in VC i , the last group before Gx is Gr and its completion time is CGr . l(i,1) is the lane that has maximum P(i,l) of these two lanes after time CGr , we denote another lane as l(i,2) . Let v(i,1,y) and v(i,2,y0 ) denote the first ‘un-partitioned’ vehicle on lane l(i,1) and l(i,2) after time CGr , respectively. An example is presented in figure. 2. In the figure, each rectangle indicates a vehicle in control range, the left side of rectangle shows the arrival time of this vehicle and the length of rectangle represents its passing time. The rectangle order from left to right denotes the vehicle arriving sequence on same lane. Suppose
1. a(i,l,j) < a(i,l,j 0 ) , and C(i,l,j) < C(i,l,j 0 ) . 2. In the required group sequence, either v(i,l,j) and v(i,l,j 0 ) (j < j 0 ) are included in same group, or included in two different groups: (a) if v(i,l,j) and v(i,l,j 0 ) in group G, any {v(i,l,k) , j < k < j 0 } ∈ G. (b) if v(i,l,j) and v(i,l,j 0 ) are contained by two different group G and G0 , respectively, there must be G passing the intersection before G0 in the group sequence. Suppose there is a partial group sequence, in which some groups are already made, but no decision has been taken yet on partitioning the remaining ‘unpartitioned’ vehicles. Some extra notations are given: • CGr , the completion time of the partial vehicle group sequence, i.e., the completion time of the last group Gr of the partial sequence, where 1 6 r < b. • v(i,l,y) , the first ‘un-partitioned’ vehicle on lane l(i,l) of VC i after Gr . • P(i,l) , the sum of passing time of all ‘unpartitioned’ vehicles on lane l(i,l) of VC i , i.e., ∑n(i,l) P(i,l) = j=y {p(i,l,j) }. Each time after a group is formed, we reindex the lane that has the maximum P(i,l) of all lanes in VC i as lane 1, i.e., l(i,1) and P(i,1) is the sum of passing
Figure 2: Example of property 3.2
that Gx only contains vehicle v(i,2,y0 ) without v(i,1,y) in the required group sequence GS. It’s easy to see that after the pass of group Gx , the lower bound of evacuating all ‘un-partitioned’ vehicles in VC i is the maximum P(i,l) of these two lanes plus the integral stop time of VC i , i.e., P(i,1) + si . We can obtain a new vehicle group sequence GS 0 by deleting group Gx and inserting v(i,2,y0 ) into the group that contains v(i,1,y) . One can notice that this change will reduce the overall evacuation time. Thus, GS 0 is also optimal. Continuing this procedure, we can eventually have an optimal group sequence with this property. This property can be easily applied to the vehicle class with more than two lanes. Consider we are going to partition vehicles in class VC i after a partial group sequence, and the completion time of this partial sequence is CGr . We can have the property follows: Property 3.3 Suppose Gx (1 6 x 6 b) is a group containing vehicles in VC i , If 1. C(i,l,y0 ) 6 C(i,1,y) or
MOSIM’10 - May 10-12, 2010 - Hammamet - Tunisia 2. 0 < C(i,l,y0 ) − C(i,1,y) 6 P(i,l) − (P(i,1) − p(i,1,y) ), there is an optimal group sequence, in which vehicle v(i,1,y) and v(i,l,y0 ) are both contained in Gx . Proof. Suppose there are only two lanes in VC i : l(i,1) and l(i,2) . Under the assumptions above, we can deduce that the completion time of these two vehicles v(i,1,y) and v(i,2,y0 ) are C(i,1,y) = max{CGr + si , a(i,1,y) } + p(i,1,y) and C(i,2,y0 ) = max{CGr + si , a(i,2,y0 ) } + p(i,2,y0 ) . By property 3.2 we know that there is an optimal group sequence in which Gx contains vehicle v(i,1,y) . If C(i,2,y0 ) 6 C(i,1,y) (figure. 3. for example), it is
is an optimal group sequence in which vehicle v(i,1,y) and v(i,2,y0 ) are contained in same vehicle group. Properties 3.2, 3.3 are mainly about the first ‘unpartitioned’ vehicle on each lane of same vehicle class after the completion time CGr of a partial sequence. we now give a property with regard to all the ‘unpartitioned’ vehicles of same vehicle class. Property 3.4 There is an optimal sequence, in which a vehicle group Gx contains all ‘un-partitioned’ vehicles in VC i after time CGr if we have 2si > CGx − CGr − P(i,1) . Proof. Suppose there are only two lanes in VC i , the last group before Gx is Gr and its completion time is CGr . An example is given in figure. 5. In the
Figure 3: Example of property 3.3, 1)
Figure 5: Example of property 3.4
clearly to see that the containing of vehicle v(i,2,y0 ) in group Gx does not delay rest vehicles. Otherwise, it will probably cause a time duration to make vehicle v(i,2,y0 ) pass the intersection in a group after Gx . The first item above then stands. Since from time CGr , the lower bound of evacuating all ‘un-partitioned’ vehicles in VC i is P(i,1) + si . Then, if C(i,1,y) < C(i,2,y0 ) (see figure. 4. as an example), and group Gx contains both v(i,1,y) and v(i,2,y0 ) , this lower bound will be increased from
example, there are three ‘un-partitioned’ vehicles on lane 1 and two on lane 2 after time CGr . By property 3.2, there is an optimal sequence in which v(i,1,y) is contained in Gx . Since P(i,1) + si means the lower bound of evacuating all ‘un-partitioned’ vehicles in VC i , it’s easy to see that CGx − (CGr + si + P(i,1) ) denotes that the extra time caused by making all ‘un-partitioned’ vehicles in VC i pass intersection at one time if we group all these vehicles into Gx , and si > CGx − (CGr + si + P(i,1) ) indicates that the integral stop time of VC i is equal to or bigger than this extra time. Thus, the overall evacuation time will be increased if we do not put all these five vehicles in group Gx because it will cost at least one extra integral stop time for the rest ‘un-partitioned’ vehicles.
Figure 4: Example of property 3.3, 2)
Suppose Gr is from vehicle class VC j and Gx is from VC i , i 6= j. Set qi = CGx − (si + CGr ) − PGx , where PGx is the sum of passing time of vehicles in l(i,1) . Then qi denotes the extra time used in the group Gx if we partition Gx after Gr . Similarly, if we add new vehicles into the last group Gr of partial sequence, define qj as the extra time caused by the new added vehicles. We can have the following proposition to predict the minimum time that will be wasted by the coming vehicles for each partial sequence.
P(i,1) + si to P(i,1) + si + (C(i,2,y0 ) − C(i,1,y) ) because of the containing of v(i,2,y0 ) . However, if Gx does not contain v(i,2,y0 ) , a time duration at least P(i,2) − (P(i,1) − p(i,1,y) ) will be added to make rest vehicles in VC i pass. If 0 < C(i,2,y0 ) − C(i,1,y) 6 P(i,2) − (P(i,1) − p(i,1,y) ), one can deduce that there
Proposition 3.5 In a partial group sequence, where some groups have been formed, but no decision
MOSIM’10 - May 10-12, 2010 - Hammamet - Tunisia
has been taken yet on grouping the remaining ‘unpartitioned’ vehicles, a group penalty, noted as ∆, attaches to each partial group sequence to estimate the minimum time that will be wasted by the coming vehicles. The algorithm for obtaining ∆ is given as follows. Algorithm 1: Group penalty 1
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begin /* initialization, ∆0 ← ∞, ∆ ← ∞ */ while there are still ‘un-partitioned’ vehicles in VC j do for Vehicle class VC i , i ∈ [1, m] and i 6= j do Form a group Gx from the ‘un-partitioned’ vehicles in VC i , count qi ; if there are ‘un-partitioned’ vehicles in VC r then ∆i = qi + qj + sj ; else ∆i = qi + qj ; 0
∆ ← min {∆ , ∆i };
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∆ ← min {∆, ∆0 }; Add vehicles to Gr from VC j by virtue of property 3.2, 3.3 and 3.4. Compute the CGr ;
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4.1
Fathoming and backtracking
In this branch-and-bound algorithm, a node is fathomed if: 1. It is a leaf node, i.e., a complete solution finished. 2. The lower bound exceeds or equals the incumbent upper bound. In the process of branch and bound algorithm, if a complete solution that has smaller evacuation time than the current upper bound is found, the new evacuation time should be regarded as a new upper bound. Fathoming initiates backtracking to the first node that still not fathomed. If no such node is found, the search terminates. 4.2
Lower bound
Finding a tight lower bound is crucial to the Branch and Bound algorithm. We can obtain the lower bound of each node by three parts. 1. LBS: time used by the vehicles already grouped. 2. LBR: the lower bound of the time that will be used by the rest vehicles. 3. LBP = ∆: the group penalty of the partial group sequence.
For each partial group sequence, the group penalty is the estimation of the minimum extra time wasted by the coming vehicles, this parameter will be used to form the lower bound of the following Branch and Bound Algorithm. 4
BRANCH AND BOUND ALGORITHM
In this section, we present a Branch-and-Bound algorithm for finding an optimal passing sequence of the approaching vehicles. At beginning, the P(i,l) of each lane of all vehicle classes is computed { } and we index the lane that has max16l6li P(i,l) as l(i,1) of vehicle class VC i , where l ∈ [1, li ] and i ∈ [1, m] . This computation and reindex procedure should be done each time after a group is formed during the searching process. In this branch-and-bound scheme, each node is partitioned into k branches: one branch indicates the group just formed should add more vehicles in same vehicle class if there are still vehicles in it, other k − 1 branches indicate that we give the authorization of using intersection to vehicles in other k − 1 vehicle classes, where k 6 m. The search tree is constructed in a depth-first fashion. Other components of the branch-and-bound scheme are presented as follows.
The LBS can be easily obtained by the data of the vehicles already partitioned. For calculating the LBR, we have the following proposition. Proposition 4.1 In a partial group schedule, where some vehicles have been formed in groups, but no decision has been taken yet on grouping the remaining ‘un-partitioned’ vehicles, a lower bound of the time that ∑m will be used by all ‘un-partitioned’ vehicles is i=1 (si + P(i,1) ) − si0 , where si0 is the integral stop time of vehicle class that contains the last vehicle in the partial schedule. Proof. For each vehicle class VC i , the time used by all the vehicles in it after time CGr should be at least P(i,1) . Integral stop time for switching authorization between different vehicle classes are also necessary. ∑m Thus, the i=1 (si + P(i,1) )−si0 can be seemed as the minimum time that will be used by the rest vehicles. Thus, the lower bound of each node in the searching process will be LB = LBS + LBR + LBP.
MOSIM’10 - May 10-12, 2010 - Hammamet - Tunisia
Initial upper bound (heuristic) Lane 1
Vehicle class 1
Providing a low initial upper bound is important for enhancing the exclusion rate of the branch-andbound, i.e., the rate with which nodes are fathomed. Thus, it is worth expending some computational effort to achieve that end. Moreover, such a upper bound can be used as an stand-alone heuristic for solving the problem. The algorithm of finding initial upper bound is given as follows.
Vehicle class 3 Vehicle class 2
4.3
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Algorithm 2: Initial upper bound (IUB) 1 2
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begin while there are ‘un-partitioned’ vehicles do /* from time CGr (the completion time of the last group in partial group), CGr ← 0 at beginning */ for vehicle class VC i , i ∈ [1, m] do Reindex the lane l(i,1) in each VC i ; Form a group Gx in VC i by virtue of property 3.2, 3.3 and 3.4; Count qi ; /* qi is the extra time caused by group Gx */ q ← min {qi }; Add Gx with q to the end of the partial sequence; CGr ← CGx ;
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IU B ← CGr ; /* the final CGr is the evacuation time of an approximate solution. */ end
4.4
Figure 6: Arrival time and passing time of example
the first vehicle group. Note that the vehicles enclosed by || are the vehicles partitioned in one group. Thus, the initial lower bound is LB = LBS+LBR+LBP = 27s. Initial upper bound: Calculated by the algorithm stated earlier, IU B = 32s, which gives the following vehicle group sequence: |v(1,1,1) |; |v(2,1,1) , v(2,2,1) |; |v(1,1,2) , v(1,2,1) , v(1,3,1) |; |v(2,1,2) , v(2,2,2) |; |v(3,1,1) , v(3,1,2) , v(3,1,3) , v(3,2,1) , v(3,2,2) |; |v(1,1,3) , v(1,2,2) |. Branch and bound: The search tree is presented in figure. 7 and the index of the node corresponds to the sequence of the search procedure. Some important nodes are listed as follows, and an explanation is provided whenever necessary. root
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Numerical example
To make the algorithm more understandable, we give the following example.
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Consider from a start time, there are 15 vehicles approaching an isolated intersection and vehicles are partitioned in three vehicle classes. The integral stop time of each vehicle class is: s1 = 1s, s2 = 2s, s3 = 3s. The arrival time and the passing time of vehicles are given in the figure. 6. In this figure, each rectangle represents a vehicle. Three colors are used to distinguish vehicle classes. The length of each rectangle shows the passing time of the corresponding vehicle. Initial lower bound: At the beginning, lower bound of ‘un-partitioned’ vehicles can be computed by proposition 4.1: LBS + LBR = 0 + 24 = 24s. Group penalty can be computed by proposition 3.5. Since there is no partitioned group at beginning, we suppose that the previous groups is from one of three vehicle classes and choose the smallest group penalty: ∆ = LBP = 1 + 2 = 3s. This penalty is obtained by scheduling the vehicles as |v(1,1,1) |, |v(1,1,2) , v(1,2,2) | as
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end
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Figure 7: Search tree of example Node 1: Vehicle group obtained: |v(1,1,1) |, LBS = 4, LBR = 20, LBP = 3; LB = 27. Node 2: Vehicle group obtained: |v(1,1,1) , v(1,1,2) , v(1,2,1) |, LBS = 8, LBR = 17, LBP = 3; LB = 28. Node 5: Vehicle group obtained: |v(1,1,1) , v(1,1,2) , v(1,2,1) |; |v(3,1,1) , v(3,2,1) |, LBS = 13, LBR = 14,
MOSIM’10 - May 10-12, 2010 - Hammamet - Tunisia
LBP = 4; LB = 31.
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Node 7: Vehicle group obtained: |v(1,1,1) , v(1,1,2) , v(1,2,1) |; |v(3,1,1) , v(3,2,1) |; |v(2,1,1) , v(2,1,2) , v(2,2,1) , v(2,2,2) |, LBS = 20, LBR = 11, LBP = 0; LB = 31. Node 9: Vehicle group obtained: |v(1,1,1) , v(1,1,2) , v(1,2,1) |; |v(3,1,1) , v(3,2,1) |; |v(2,1,1) , v(2,1,2) , v(2,2,1) , v(2,2,2) |; |v(3,1,2) , v(3,1,3) , v(3,2,2) |, LBS = 26, LBR = 5, LBP = 0; LB = 31. Node 10: First leaf node: |v(1,1,1) , v(1,1,2) , v(1,2,1) |; |v(3,1,1) , v(3,2,1) |; |v(2,1,1) , v(2,1,2) , v(2,2,1) , v(2,2,2) |; |v(3,1,2) , v(3,1,3) , v(3,2,2) |; |v(1,1,2) , v(1,2,2) , v(1,3,1) |. Time used: 31s.
Vehicle lass 1
After the first leaf node is found, the incumbent upper bound is changed to 31s. Other nodes are fathomed by this upper bound in the following search procedure. An final optimal solution is obtained by Node 10. This passing sequence is illustrated in figure. 8.
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Figure 8: Obtained optimal passing sequence 4.5
Algorithm application
The proposed Branch and Bound algorithm aims at minimizing the evacuation time of a set of vehicles in the control range. Since at the real-world intersection, vehicles keep entering the control range of the center controller, the algorithm should be executed when new vehicles are detected. However, if a group of vehicles are authorized to pass intersection, the recalculation process is delayed until all vehicles in this group have passed through intersection. Consider the numerical example above. At time 0s, 15 vehicles are detected in the control range, algorithm is executed to get an passing sequence. If at time 12s, a new vehicle of vehicle class 2 enters the control range. Since at t = 12s, the vehicles in class 3 have the authorization to use the intersection (v(3,1,1) , v(3,2,1) ), the algorithm should be re-executed without the vehicles already passed and v(3,1,1) , v(3,2,1) , but the new detected vehicle should be considered. Then, the new obtained vehicle sequence will be applied after v(3,1,1) , v(3,2,1) pass the intersection.
SIMULATION AND COMPUTATIONAL EXPERIMENTS
The fundamental measures for evaluating the performance of a traffic control algorithm at isolated intersection include the evacuation time, average queue size and average vehicle waiting time. The average queue size indicates the number of vehicles on each lane waiting to cross the intersection at same time. Average vehicle waiting time measures how long a vehicle has to wait before traversing the intersection. All the three measures are frequently used to evaluate the performance of a control algorithm. In this section, We will analyze the three measures with different relative traffic loads. Comparisons are done among the Branch and Bound algorithm, the heuristic and an optimized traditional fix-cycle traffic light control scheme. The simulation is implemented at an isolated fourapproach intersection, each approach has two lanes for incoming vehicles. Vehicles approaching the intersection are partitioned into four vehicle classes. According to statistics, we consider that the maximum traffic load for each of the four approach is 1800 vehicles/h (one vehicle every 2 seconds) and the traffic load for each incoming lane is quarter of the maximum load of one approach. In addition, each data point is obtained by taking the average over several separate simulation. Each simulation run 10 minutes of traffic flow. Besides, the integral stop time for each vehicle class is randomly generated integers varied from 3 to 8 seconds. Passing time of vehicles are varied from 2 to 8 seconds. Vehicles are equally distributed among the compatible lanes of one vehicle class. All approaches are coded in C++ and run on a desktop computer with Linux system (kernel 2.6.28). Since the algorithm aims at evacuating approaching vehicles as soon as possible, we first simulate the overall evacuation time of all vehicles entering the control range during the 10 minutes of traffic flow. The result is presented in figure. 9. We observe that the Branch and Bound algorithm and the heuristic reduce significantly the overall evacuation time for about 20 seconds, then the throughput can be improved in consequence. The comparisons of average queue size and average vehicle waiting time are given in figure. 10 and figure. 11. One can notice that the performance of the Branch and Bound algorithm and heuristic are almost the same in reducing the average queue length and average vehicle waiting time. However, the Branch and Bound algorithm shows a little better than the heuristic at high relative traffic load, i.e., > 0.4. For some data points, the heuristic performs a little better than the Branch and Bound which is an exact searching method, this is because our objective function is to
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6 Branch and Bound Heuristic Fix Cycle Time
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In this paper, we presented a new approach to sequence the vehicles passing an isolated intersection via Vehicle-to-Infrastructure communications. Vehicles were treated as discrete individuals in the proposed control strategy and our objective was to evacuate detected vehicles as soon as possible. A Branch and Bound algorithm and a heuristic were developed based on the carefully analysis of structural properties of the problem. Both methods were compared with a traditional fix-cycle time traffic light technique. The results showed that on the one hand, the proposed algorithms can significantly improve the traffic situation; On the other hand, the average running time of the proposed algorithms can satisfy the need of an real-time control system. In the future, differences between normal vehicles and special used vehicles such as ambulances, police cars will be considered. Special used vehicles should have the privileges to pass through intersection. Besides, pedestrian crossings will be taken into consideration. Further more, several neighbor intersections will be controlled together by one controller to save the computational resources as well as guarantee a more global optimization.
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Dresner K. and P. Stone, 2004. Multiagent traffic management: a reservation-based intersection control mechanism. Autonomous Agents and Multiagent Systems (AAMAS’04), New York, USA, pp.530-537.
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Contet J.-M., F. Gechter, P. Gruer, and A. Koukam, 2007. Application of reactive multiagent system to linear vehicle platoon.The 19th IEEE International Conference on Tools with Artificial Intelligence, vol. 2, pp. 67–70. Tan C.-W., S. Park, H.C. Liu, Q. Xu, P. Lau, 2008. Prediction of transit vehicle arrival time for signal priority control: Algorithm and performance. IEEE Transactions on Intelligent Transportation Systems, vol.9, No. 4, pp. 688-696.
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minimize the overall evacuation time, not the average queue size or average vehicle waiting time.
Dresner K. and P. Stone, 2006. Traffic intersections of the future. The 21th National Conference on Artificial Intelligence, NECTARTrack (AAAI 06), pp. 1593–1596. WU J., A. Abbas-Turki and A. El Moudni, 2009. Discrete Methods for Urban Intersection Traffic Controlling. The IEEE 69th Vehicular Technology Conference, (VTC Spring’09), Barcelona. Huang Q.-F. and R. Miller, 2003. The design of reliable protocols for wireless traffic signal systems.
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Department of Computer Science and Engineering, Washington University, Tech. Rep. Wunderlich R., C.-B. Liu, I. Elhanany, and T. Urbanik, II, 2008. A novel signal-scheduling algorithm with quality-of-service provisioning for an isolated intersection. IEEE Transactions on Intelligent Transportation Systems, Vol.9, No.3, pp.536-547. Nadeem T., S. Dashtinezhad, C.-Y. Liao and L. Iftode, 2004. TrafficView: A Scalable Traffic Monitoring System. The 2004 IEEE International Conference on Mobile Data Management (MDM’04), Berkeley, California, USA, pp.13-26. Yu X.-H. and A. R. Stubberud, 1997. Markovian decision control for traffic signal systems. The 36th IEEE Conference Decision Control, San Diego, vol. 5, pp. 4782–4787. Pan Y.-J., 2008. Decentralized control of vehicles in platoons with robust nonlinear state estimation. The 4th IEEE Conference on Automation Science and Engineering, Key Bridge Marriott, Washington DC, USA, pp.145-150.
A NOVEL VEHICLE SEQUENCING ALGORITHM WITH VEHICULAR INFRASTRUCTURE INTEGRATION FOR AN ISOLATED INTERSECTION F. YAN, M. DRIDI and A. EL MOUDNI Laboratoire Syst`emes et Transports Universit´e de Technologie de Belfort-Montb´eliard 90010 Belfort cedex, France {fei.yan, mahjoub.dridi, abdellah.el-moudni}@utbm.fr
ABSTRACT: In this paper, we study a novel vehicle sequencing algorithm for an isolated intersection with a new control method. The control method is based on the utilization of Vehicular Infrastructure Integration and the consideration that each vehicle arrives individually. We consider knowledge of each vehicle’s arrival time and the time each vehicle needs to traverse the intersection. Our objective is to schedule all approaching vehicles to pass this intersection in the shortest duration, the throughput of the intersection will then be increased. A Branch and Bound algorithm and a heuristic are proposed to evacuate the approaching vehicles as soon as possible. Structural properties of the problem are carefully investigated to simplify the search procedure of an optimal passing sequence. Simulation results in evacuation time, average waiting time and average queue size demonstrate the performance gain obtained when using the proposed schemes. KEYWORDS: Isolated intersection, Vehicular Infrastructure Integration, Vehicle arrival time, Branch and bound.
1
INTRODUCTION
Traffic congestion at intersections is one of the main issues to be addressed by today’s traffic management schemes. Countless efforts have been made directed toward efficiently improving the traffic situation at intersections. However, although there are well-planned road management schemes, sufficient infrastructure for transportation, we still face the congestion of traffic, especially at the intersections. The construction of additional infrastructure may be considered as one of the solution for the problem, but it is less feasible owing to political and environmental concerns. This urges researchers to find other methods to improve the efficiency of traffic control at intersections. The enhancement of high-tech information and communication technology such as the low cost wireless connectivity, miniaturization of computing devices, and availability of Global Positioning System (GPS) provoked the emergence of intelligent vehicles equipped with In-vehicle Information System (IVIS). The Vehicular Infrastructure Integration (VII) technology came forth under this background. Many realtime message dissemination systems and reliable information exchange protocols were proposed to cater to the development of new traffic control systems. For example, a wireless traffic light system is pro-
posed, where information about current lights status, location of intersection, and a reference point are broadcasted periodically (Huang Q.-F. and R. Miller, 2003). Intelligent devices that can be embedded in next generation vehicles were invented to provide the drivers with a real-time view of the road traffic far beyond what they can physically see (Nadeem T. et al., 2004). Vehicles equipped with this device can disseminate traffic information using short-range wireless communication. Besides, approaches and algorithms were proposed focusing on scheduling the traffic signal to maximize the traffic throughput while minimizing the average latency with various theories. For instance, based on new embedded devices and communication technologies, car platooning investigations aiming at increasing the capacity of roads by reducing temporal distances between vehicles were extensively discussed (Pan Y.-J., 2008), (Contet J.-M. et al., 2007). A reservation based system for alleviating traffic congestion at intersections is proposed in (Dresner K. and P. Stone, 2004) and (Dresner K. and P. Stone, 2006). Vehicles are controlled by agents under the assumption. Queuing theory is implemented in traffic control, and the problem of scheduling traffic at an intersection was addressed by structuring the problem as a Markov decision process (Yu X.-H. and A. R. Stub-
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berud, 1997). A new approach of traffic control at intersection is proposed that requires communication with a center controller at intersection (Wunderlich R. et al., 2008). This approach needs the information about the intended route of each vehicle that reaches the intersection. There are also researches to predict vehicle’s accurate arrival time at the intersection with both the historical and real-time GPS vehicle location data (Tan C.-W. et al., 2008). Based on the knowledge of each vehicle’s arrival time at intersection, algorithms in (Wu J. et al., 2009) aim at scheduling approaching vehicles to pass the intersection taking into account each vehicle arrives individually, but authors only considered the control strategy for a 2-way intersection. All these efforts in wireless communication and schedule algorithms share a common goal: to improve the traffic situation at controlled intersection while guaranteeing the driver safety. We study in this paper a more general problem to guide vehicles through the intersection with the consideration of Vehicular-to-Infrastructure communications. We assume that there is a controller in the center of the intersection and all vehicles equipped with IVIS are capable of communicating with the center controller in some fashion. The vital telemetry data of each vehicle can be obtained by the controller once the vehicle enters the controller’s control range. Basically, we assume knowledge of the vehicle’s accurate arrival time at the intersection, which is possible in very near future with development of Intelligent Transportation System. Decision of vehicle passing sequence will then be made and broadcasted to all vehicles. A Branch and Bound algorithm and a heuristic are proposed for finding an optimal vehicle passing sequence. Structural properties of the problem are carefully studied to simplify the search procedure of the Branch and Bound algorithm. The rest of this paper is structured as follows: In Section 2, we specify the studied problem, along with some useful notations that will be used in the sequel. In Section 3, the structural properties of the problem are carefully investigated. Algorithms are presented in the section follows. In Section 5, various traffic scenarios are simulated to compare the proposed control strategies with a traditional fix-cycle time traffic lights technique. Conclusions are drawn in the last section. 2 2.1
PROBLEM DEFINITION System model
To specify the problem we study, an example of the intersection under consideration is presented in figure. 1. This intersection appears frequently in realword traffic networks. It is a four-approach intersection with through lanes (some also serve as right-turn
lanes) and exclusive left-turn lanes. The controller is located in the center of intersection with same control range for the four directions. Vehicles approaching the intersection should inform the center controller their data after enter the control range, and vehicle overtaking on same lane is not allowed. According to frequently used traffic convention, lanes on which vehicles can pass through intersection without conflicts are compatible. For example, lane 1 and lane 2 in figure. 1 are compatible lanes. We define all the vehicles on compatible lanes as one Vehicle Class (VC). Thus, all the vehicles in the control range can be divided into several vehicle classes according to the lanes they are running on. Vehicles on incompatible lanes (or vehicles in different vehicle classes) can not traverse the intersection at same time. For the reason of safety, there is usually a time duration between two vehicles from different vehicle classes during which no vehicles behind the stop lines can pass intersection. We define this time as the integral stop time and we assume that this time duration is a constant only decided by the vehicle class that will get the authorization of using the intersection. For instance, the integral stop time when changing authorization from vehicles on lane 1 to vehicles to lane 3 is only decided by the vehicles class that contains lane 3. Suppose at time t0 = 0, there are n vehicles in the control range approaching the intersection from different directions, and the data of all vehicles have been received by the controller in a very short time. The information of each vehicle contains four parts: • Vehicle identification (ID): used to identify the records belonging to different vehicles. • Lane number: which lane it is moving on. • Vehicle arrival time: the precise time it arrives at the stop line from time t0 . • Vehicle passing time: the time it needs to accelerate from stop line until it reaches a safe distance to its follower on same lane (only related to the type of vehicles). The vehicle passing time depends on the type of vehicles, for example, cars and trucks. Normally, trucks are slower to change speed and, therefore, use more time to accelerate until it can reach a safe distance for the vehicle following it on same lane to start. Our objective is to decide an vehicle passing sequence to minimize the evacuation time (the time all vehicles passing the intersection) of all vehicles. The minimization of the evacuation time will naturally increase the throughput of the intersection in consequence. It should be noted that the control technique may be applied to any intersection layout.
MOSIM’10 - May 10-12, 2010 - Hammamet - Tunisia • li , the number of lanes in VC i . • l(i,l) , the lth lane in VC i , where 1 6 l 6 li . • n(i,l) , the number of vehicles on lane l(i,l) . • v(i,l,j) , the j th vehicle on lane l(i,l) , the j is indexed according the arrival precedence of vehicles on lane l(i,l) . • a(i,l,j) , the arrival time of v(i,l,j) , i.e., the time v(i,l,j) needs to arrive at the stop lane (or the waiting queue) from time t0 . • s(i,l,j) , the time vehicle v(i,l,j) starts to pass the intersection. Figure 1: Schematic of intersection control with Control Device
• p(i,l,j) , the passing time of v(i,l,j) , i.e., the time v(i,l,j) needs to accelerate from stop line until it reaches a safe space to the vehicle following it on same lane.
As a conclusion, the constraints of the problem can be described as:
• C(i,l,j) , the time that vehicle v(i,l,j) have a safe space to the vehicle following it on the same lane.
1. All vehicles approaching the intersection are partitioned into several vehicle classes.
Thus, for n vehicles detected at time t0 , the objective function to minimize can be described as
4. Before a vehicle pass the intersection, the vehicle after it on same lane cannot start to pass even it has arrived.
max{C(i,l,j) } 1 6 i 6 m, 1 6 l 6 li , 1 6 j 6 n(i,l) s > a(i,l,j) (i,l,j) C(i,l,j) = s(i,l,j) + p(i,l,j) s.t. s > C(i,l,j) (i,l,j+1) C − C(i0 ,l0 ,j 0 ) > si0 , if i 6= i0 and (i,l,j) C(i,l,j) > C(i0 ,l0 ,j 0 )
5. Vehicles in same class but not the same lane may pass the intersection simultaneously.
In order to design an efficient algorithm, we need to study the structural properties of the problem first.
2. There are several lanes in each vehicle class. 3. Vehicles on each lane need to pass the intersection in First In First Out (FIFO)-way.
6. There is a time duration (integral stop time) between two vehicles from different vehicle classes during which no vehicles behind the stop line can pass intersection. 2.2
Formulation
In the following, the formal description of the problem is given. At first, some notations that will be used are defined. Suppose there are n vehicles approaching the intersection and all vehicles are partitioned into m vehicle classes according to the lanes where they are running on. In each class, vehicles are separated on different lanes.
3
PROPERTIES OF THE PROBLEM
In order to derive an algorithm that can find an optimal vehicle passing order from time t0 = 0, we need study the structural properties of vehicle data. These properties will be used to simplify the search procedure of the algorithms we are going to present. First, we need some definition about the vehicle group. A vehicle group G is defined as a set of vehicles from same VC that pass intersection without the interruption of vehicles in other VC. Specifically, a vehicle group will have the following properties: 1. A vehicle class has at least one group of vehicles. 2. Each vehicle should be in one and only one group.
• VC i , the i
th
vehicle class, where 1 6 i 6 m.
• ni , the number of vehicles in VC i . • si , the integral stop time of VC i .
3. There may be idle time waiting for some vehicles to arrive in a group, but there is no integral stop time during the passing process of vehicles in one group.
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The completion time of a vehicle group is defined as the maximum completion time of vehicles in this group, i.e., CG = max{C(i,l,j) }, where v(i,l,j) ∈ G. It is easy to notice that vehicles pass intersection in the sequence of vehicle group. Then we can describe the order of vehicles as a group sequence GS: GS = (G1 , G2 , . . . , Gb ), b > m. The problem of finding an optimal passing order of vehicles changes to the problem of partitioning vehicles into different groups and finding an optimal group sequence to minimize the vehicle evacuation time. Since there are lanes in each vehicle class and vehicles on each lane should pass the intersection in a FIFOway, we need to partition vehicles into groups, taking into account the vehicle sequence on different lanes. We can then have the following lemma. Lemma 3.1 Consider there are two vehicles v(i,l,j) , v(i,l,j 0 ) on lane l(i,l) with v(i,l,j 0 ) arriving at the stop line after v(i,l,j) , i.e., j < j 0 . Then we have:
time of all ‘un-partitioned’ vehicles on this lane after Gr . Then we can have the following property: Property 3.2 There is an optimal vehicle group sequence, in which any group Gx from VC i (1 6 x 6 b) has at least one vehicle belonging to lane l(i,1) . Proof. Suppose there are only two lanes in VC i , the last group before Gx is Gr and its completion time is CGr . l(i,1) is the lane that has maximum P(i,l) of these two lanes after time CGr , we denote another lane as l(i,2) . Let v(i,1,y) and v(i,2,y0 ) denote the first ‘un-partitioned’ vehicle on lane l(i,1) and l(i,2) after time CGr , respectively. An example is presented in figure. 2. In the figure, each rectangle indicates a vehicle in control range, the left side of rectangle shows the arrival time of this vehicle and the length of rectangle represents its passing time. The rectangle order from left to right denotes the vehicle arriving sequence on same lane. Suppose
1. a(i,l,j) < a(i,l,j 0 ) , and C(i,l,j) < C(i,l,j 0 ) . 2. In the required group sequence, either v(i,l,j) and v(i,l,j 0 ) (j < j 0 ) are included in same group, or included in two different groups: (a) if v(i,l,j) and v(i,l,j 0 ) in group G, any {v(i,l,k) , j < k < j 0 } ∈ G. (b) if v(i,l,j) and v(i,l,j 0 ) are contained by two different group G and G0 , respectively, there must be G passing the intersection before G0 in the group sequence. Suppose there is a partial group sequence, in which some groups are already made, but no decision has been taken yet on partitioning the remaining ‘unpartitioned’ vehicles. Some extra notations are given: • CGr , the completion time of the partial vehicle group sequence, i.e., the completion time of the last group Gr of the partial sequence, where 1 6 r < b. • v(i,l,y) , the first ‘un-partitioned’ vehicle on lane l(i,l) of VC i after Gr . • P(i,l) , the sum of passing time of all ‘unpartitioned’ vehicles on lane l(i,l) of VC i , i.e., ∑n(i,l) P(i,l) = j=y {p(i,l,j) }. Each time after a group is formed, we reindex the lane that has the maximum P(i,l) of all lanes in VC i as lane 1, i.e., l(i,1) and P(i,1) is the sum of passing
Figure 2: Example of property 3.2
that Gx only contains vehicle v(i,2,y0 ) without v(i,1,y) in the required group sequence GS. It’s easy to see that after the pass of group Gx , the lower bound of evacuating all ‘un-partitioned’ vehicles in VC i is the maximum P(i,l) of these two lanes plus the integral stop time of VC i , i.e., P(i,1) + si . We can obtain a new vehicle group sequence GS 0 by deleting group Gx and inserting v(i,2,y0 ) into the group that contains v(i,1,y) . One can notice that this change will reduce the overall evacuation time. Thus, GS 0 is also optimal. Continuing this procedure, we can eventually have an optimal group sequence with this property. This property can be easily applied to the vehicle class with more than two lanes. Consider we are going to partition vehicles in class VC i after a partial group sequence, and the completion time of this partial sequence is CGr . We can have the property follows: Property 3.3 Suppose Gx (1 6 x 6 b) is a group containing vehicles in VC i , If 1. C(i,l,y0 ) 6 C(i,1,y) or
MOSIM’10 - May 10-12, 2010 - Hammamet - Tunisia 2. 0 < C(i,l,y0 ) − C(i,1,y) 6 P(i,l) − (P(i,1) − p(i,1,y) ), there is an optimal group sequence, in which vehicle v(i,1,y) and v(i,l,y0 ) are both contained in Gx . Proof. Suppose there are only two lanes in VC i : l(i,1) and l(i,2) . Under the assumptions above, we can deduce that the completion time of these two vehicles v(i,1,y) and v(i,2,y0 ) are C(i,1,y) = max{CGr + si , a(i,1,y) } + p(i,1,y) and C(i,2,y0 ) = max{CGr + si , a(i,2,y0 ) } + p(i,2,y0 ) . By property 3.2 we know that there is an optimal group sequence in which Gx contains vehicle v(i,1,y) . If C(i,2,y0 ) 6 C(i,1,y) (figure. 3. for example), it is
is an optimal group sequence in which vehicle v(i,1,y) and v(i,2,y0 ) are contained in same vehicle group. Properties 3.2, 3.3 are mainly about the first ‘unpartitioned’ vehicle on each lane of same vehicle class after the completion time CGr of a partial sequence. we now give a property with regard to all the ‘unpartitioned’ vehicles of same vehicle class. Property 3.4 There is an optimal sequence, in which a vehicle group Gx contains all ‘un-partitioned’ vehicles in VC i after time CGr if we have 2si > CGx − CGr − P(i,1) . Proof. Suppose there are only two lanes in VC i , the last group before Gx is Gr and its completion time is CGr . An example is given in figure. 5. In the
Figure 3: Example of property 3.3, 1)
Figure 5: Example of property 3.4
clearly to see that the containing of vehicle v(i,2,y0 ) in group Gx does not delay rest vehicles. Otherwise, it will probably cause a time duration to make vehicle v(i,2,y0 ) pass the intersection in a group after Gx . The first item above then stands. Since from time CGr , the lower bound of evacuating all ‘un-partitioned’ vehicles in VC i is P(i,1) + si . Then, if C(i,1,y) < C(i,2,y0 ) (see figure. 4. as an example), and group Gx contains both v(i,1,y) and v(i,2,y0 ) , this lower bound will be increased from
example, there are three ‘un-partitioned’ vehicles on lane 1 and two on lane 2 after time CGr . By property 3.2, there is an optimal sequence in which v(i,1,y) is contained in Gx . Since P(i,1) + si means the lower bound of evacuating all ‘un-partitioned’ vehicles in VC i , it’s easy to see that CGx − (CGr + si + P(i,1) ) denotes that the extra time caused by making all ‘un-partitioned’ vehicles in VC i pass intersection at one time if we group all these vehicles into Gx , and si > CGx − (CGr + si + P(i,1) ) indicates that the integral stop time of VC i is equal to or bigger than this extra time. Thus, the overall evacuation time will be increased if we do not put all these five vehicles in group Gx because it will cost at least one extra integral stop time for the rest ‘un-partitioned’ vehicles.
Figure 4: Example of property 3.3, 2)
Suppose Gr is from vehicle class VC j and Gx is from VC i , i 6= j. Set qi = CGx − (si + CGr ) − PGx , where PGx is the sum of passing time of vehicles in l(i,1) . Then qi denotes the extra time used in the group Gx if we partition Gx after Gr . Similarly, if we add new vehicles into the last group Gr of partial sequence, define qj as the extra time caused by the new added vehicles. We can have the following proposition to predict the minimum time that will be wasted by the coming vehicles for each partial sequence.
P(i,1) + si to P(i,1) + si + (C(i,2,y0 ) − C(i,1,y) ) because of the containing of v(i,2,y0 ) . However, if Gx does not contain v(i,2,y0 ) , a time duration at least P(i,2) − (P(i,1) − p(i,1,y) ) will be added to make rest vehicles in VC i pass. If 0 < C(i,2,y0 ) − C(i,1,y) 6 P(i,2) − (P(i,1) − p(i,1,y) ), one can deduce that there
Proposition 3.5 In a partial group sequence, where some groups have been formed, but no decision
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has been taken yet on grouping the remaining ‘unpartitioned’ vehicles, a group penalty, noted as ∆, attaches to each partial group sequence to estimate the minimum time that will be wasted by the coming vehicles. The algorithm for obtaining ∆ is given as follows. Algorithm 1: Group penalty 1
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begin /* initialization, ∆0 ← ∞, ∆ ← ∞ */ while there are still ‘un-partitioned’ vehicles in VC j do for Vehicle class VC i , i ∈ [1, m] and i 6= j do Form a group Gx from the ‘un-partitioned’ vehicles in VC i , count qi ; if there are ‘un-partitioned’ vehicles in VC r then ∆i = qi + qj + sj ; else ∆i = qi + qj ; 0
∆ ← min {∆ , ∆i };
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∆ ← min {∆, ∆0 }; Add vehicles to Gr from VC j by virtue of property 3.2, 3.3 and 3.4. Compute the CGr ;
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4.1
Fathoming and backtracking
In this branch-and-bound algorithm, a node is fathomed if: 1. It is a leaf node, i.e., a complete solution finished. 2. The lower bound exceeds or equals the incumbent upper bound. In the process of branch and bound algorithm, if a complete solution that has smaller evacuation time than the current upper bound is found, the new evacuation time should be regarded as a new upper bound. Fathoming initiates backtracking to the first node that still not fathomed. If no such node is found, the search terminates. 4.2
Lower bound
Finding a tight lower bound is crucial to the Branch and Bound algorithm. We can obtain the lower bound of each node by three parts. 1. LBS: time used by the vehicles already grouped. 2. LBR: the lower bound of the time that will be used by the rest vehicles. 3. LBP = ∆: the group penalty of the partial group sequence.
For each partial group sequence, the group penalty is the estimation of the minimum extra time wasted by the coming vehicles, this parameter will be used to form the lower bound of the following Branch and Bound Algorithm. 4
BRANCH AND BOUND ALGORITHM
In this section, we present a Branch-and-Bound algorithm for finding an optimal passing sequence of the approaching vehicles. At beginning, the P(i,l) of each lane of all vehicle classes is computed { } and we index the lane that has max16l6li P(i,l) as l(i,1) of vehicle class VC i , where l ∈ [1, li ] and i ∈ [1, m] . This computation and reindex procedure should be done each time after a group is formed during the searching process. In this branch-and-bound scheme, each node is partitioned into k branches: one branch indicates the group just formed should add more vehicles in same vehicle class if there are still vehicles in it, other k − 1 branches indicate that we give the authorization of using intersection to vehicles in other k − 1 vehicle classes, where k 6 m. The search tree is constructed in a depth-first fashion. Other components of the branch-and-bound scheme are presented as follows.
The LBS can be easily obtained by the data of the vehicles already partitioned. For calculating the LBR, we have the following proposition. Proposition 4.1 In a partial group schedule, where some vehicles have been formed in groups, but no decision has been taken yet on grouping the remaining ‘un-partitioned’ vehicles, a lower bound of the time that ∑m will be used by all ‘un-partitioned’ vehicles is i=1 (si + P(i,1) ) − si0 , where si0 is the integral stop time of vehicle class that contains the last vehicle in the partial schedule. Proof. For each vehicle class VC i , the time used by all the vehicles in it after time CGr should be at least P(i,1) . Integral stop time for switching authorization between different vehicle classes are also necessary. ∑m Thus, the i=1 (si + P(i,1) )−si0 can be seemed as the minimum time that will be used by the rest vehicles. Thus, the lower bound of each node in the searching process will be LB = LBS + LBR + LBP.
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Initial upper bound (heuristic) Lane 1
Vehicle class 1
Providing a low initial upper bound is important for enhancing the exclusion rate of the branch-andbound, i.e., the rate with which nodes are fathomed. Thus, it is worth expending some computational effort to achieve that end. Moreover, such a upper bound can be used as an stand-alone heuristic for solving the problem. The algorithm of finding initial upper bound is given as follows.
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Algorithm 2: Initial upper bound (IUB) 1 2
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begin while there are ‘un-partitioned’ vehicles do /* from time CGr (the completion time of the last group in partial group), CGr ← 0 at beginning */ for vehicle class VC i , i ∈ [1, m] do Reindex the lane l(i,1) in each VC i ; Form a group Gx in VC i by virtue of property 3.2, 3.3 and 3.4; Count qi ; /* qi is the extra time caused by group Gx */ q ← min {qi }; Add Gx with q to the end of the partial sequence; CGr ← CGx ;
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IU B ← CGr ; /* the final CGr is the evacuation time of an approximate solution. */ end
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Figure 6: Arrival time and passing time of example
the first vehicle group. Note that the vehicles enclosed by || are the vehicles partitioned in one group. Thus, the initial lower bound is LB = LBS+LBR+LBP = 27s. Initial upper bound: Calculated by the algorithm stated earlier, IU B = 32s, which gives the following vehicle group sequence: |v(1,1,1) |; |v(2,1,1) , v(2,2,1) |; |v(1,1,2) , v(1,2,1) , v(1,3,1) |; |v(2,1,2) , v(2,2,2) |; |v(3,1,1) , v(3,1,2) , v(3,1,3) , v(3,2,1) , v(3,2,2) |; |v(1,1,3) , v(1,2,2) |. Branch and bound: The search tree is presented in figure. 7 and the index of the node corresponds to the sequence of the search procedure. Some important nodes are listed as follows, and an explanation is provided whenever necessary. root
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Numerical example
To make the algorithm more understandable, we give the following example.
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Consider from a start time, there are 15 vehicles approaching an isolated intersection and vehicles are partitioned in three vehicle classes. The integral stop time of each vehicle class is: s1 = 1s, s2 = 2s, s3 = 3s. The arrival time and the passing time of vehicles are given in the figure. 6. In this figure, each rectangle represents a vehicle. Three colors are used to distinguish vehicle classes. The length of each rectangle shows the passing time of the corresponding vehicle. Initial lower bound: At the beginning, lower bound of ‘un-partitioned’ vehicles can be computed by proposition 4.1: LBS + LBR = 0 + 24 = 24s. Group penalty can be computed by proposition 3.5. Since there is no partitioned group at beginning, we suppose that the previous groups is from one of three vehicle classes and choose the smallest group penalty: ∆ = LBP = 1 + 2 = 3s. This penalty is obtained by scheduling the vehicles as |v(1,1,1) |, |v(1,1,2) , v(1,2,2) | as
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end
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Figure 7: Search tree of example Node 1: Vehicle group obtained: |v(1,1,1) |, LBS = 4, LBR = 20, LBP = 3; LB = 27. Node 2: Vehicle group obtained: |v(1,1,1) , v(1,1,2) , v(1,2,1) |, LBS = 8, LBR = 17, LBP = 3; LB = 28. Node 5: Vehicle group obtained: |v(1,1,1) , v(1,1,2) , v(1,2,1) |; |v(3,1,1) , v(3,2,1) |, LBS = 13, LBR = 14,
MOSIM’10 - May 10-12, 2010 - Hammamet - Tunisia
LBP = 4; LB = 31.
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Node 7: Vehicle group obtained: |v(1,1,1) , v(1,1,2) , v(1,2,1) |; |v(3,1,1) , v(3,2,1) |; |v(2,1,1) , v(2,1,2) , v(2,2,1) , v(2,2,2) |, LBS = 20, LBR = 11, LBP = 0; LB = 31. Node 9: Vehicle group obtained: |v(1,1,1) , v(1,1,2) , v(1,2,1) |; |v(3,1,1) , v(3,2,1) |; |v(2,1,1) , v(2,1,2) , v(2,2,1) , v(2,2,2) |; |v(3,1,2) , v(3,1,3) , v(3,2,2) |, LBS = 26, LBR = 5, LBP = 0; LB = 31. Node 10: First leaf node: |v(1,1,1) , v(1,1,2) , v(1,2,1) |; |v(3,1,1) , v(3,2,1) |; |v(2,1,1) , v(2,1,2) , v(2,2,1) , v(2,2,2) |; |v(3,1,2) , v(3,1,3) , v(3,2,2) |; |v(1,1,2) , v(1,2,2) , v(1,3,1) |. Time used: 31s.
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After the first leaf node is found, the incumbent upper bound is changed to 31s. Other nodes are fathomed by this upper bound in the following search procedure. An final optimal solution is obtained by Node 10. This passing sequence is illustrated in figure. 8.
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Figure 8: Obtained optimal passing sequence 4.5
Algorithm application
The proposed Branch and Bound algorithm aims at minimizing the evacuation time of a set of vehicles in the control range. Since at the real-world intersection, vehicles keep entering the control range of the center controller, the algorithm should be executed when new vehicles are detected. However, if a group of vehicles are authorized to pass intersection, the recalculation process is delayed until all vehicles in this group have passed through intersection. Consider the numerical example above. At time 0s, 15 vehicles are detected in the control range, algorithm is executed to get an passing sequence. If at time 12s, a new vehicle of vehicle class 2 enters the control range. Since at t = 12s, the vehicles in class 3 have the authorization to use the intersection (v(3,1,1) , v(3,2,1) ), the algorithm should be re-executed without the vehicles already passed and v(3,1,1) , v(3,2,1) , but the new detected vehicle should be considered. Then, the new obtained vehicle sequence will be applied after v(3,1,1) , v(3,2,1) pass the intersection.
SIMULATION AND COMPUTATIONAL EXPERIMENTS
The fundamental measures for evaluating the performance of a traffic control algorithm at isolated intersection include the evacuation time, average queue size and average vehicle waiting time. The average queue size indicates the number of vehicles on each lane waiting to cross the intersection at same time. Average vehicle waiting time measures how long a vehicle has to wait before traversing the intersection. All the three measures are frequently used to evaluate the performance of a control algorithm. In this section, We will analyze the three measures with different relative traffic loads. Comparisons are done among the Branch and Bound algorithm, the heuristic and an optimized traditional fix-cycle traffic light control scheme. The simulation is implemented at an isolated fourapproach intersection, each approach has two lanes for incoming vehicles. Vehicles approaching the intersection are partitioned into four vehicle classes. According to statistics, we consider that the maximum traffic load for each of the four approach is 1800 vehicles/h (one vehicle every 2 seconds) and the traffic load for each incoming lane is quarter of the maximum load of one approach. In addition, each data point is obtained by taking the average over several separate simulation. Each simulation run 10 minutes of traffic flow. Besides, the integral stop time for each vehicle class is randomly generated integers varied from 3 to 8 seconds. Passing time of vehicles are varied from 2 to 8 seconds. Vehicles are equally distributed among the compatible lanes of one vehicle class. All approaches are coded in C++ and run on a desktop computer with Linux system (kernel 2.6.28). Since the algorithm aims at evacuating approaching vehicles as soon as possible, we first simulate the overall evacuation time of all vehicles entering the control range during the 10 minutes of traffic flow. The result is presented in figure. 9. We observe that the Branch and Bound algorithm and the heuristic reduce significantly the overall evacuation time for about 20 seconds, then the throughput can be improved in consequence. The comparisons of average queue size and average vehicle waiting time are given in figure. 10 and figure. 11. One can notice that the performance of the Branch and Bound algorithm and heuristic are almost the same in reducing the average queue length and average vehicle waiting time. However, the Branch and Bound algorithm shows a little better than the heuristic at high relative traffic load, i.e., > 0.4. For some data points, the heuristic performs a little better than the Branch and Bound which is an exact searching method, this is because our objective function is to
MOSIM’10 - May 10-12, 2010 - Hammamet - Tunisia
6 Branch and Bound Heuristic Fix Cycle Time
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Figure 9: Simulation results of evacuation time for 10 min of traffic flow
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In this paper, we presented a new approach to sequence the vehicles passing an isolated intersection via Vehicle-to-Infrastructure communications. Vehicles were treated as discrete individuals in the proposed control strategy and our objective was to evacuate detected vehicles as soon as possible. A Branch and Bound algorithm and a heuristic were developed based on the carefully analysis of structural properties of the problem. Both methods were compared with a traditional fix-cycle time traffic light technique. The results showed that on the one hand, the proposed algorithms can significantly improve the traffic situation; On the other hand, the average running time of the proposed algorithms can satisfy the need of an real-time control system. In the future, differences between normal vehicles and special used vehicles such as ambulances, police cars will be considered. Special used vehicles should have the privileges to pass through intersection. Besides, pedestrian crossings will be taken into consideration. Further more, several neighbor intersections will be controlled together by one controller to save the computational resources as well as guarantee a more global optimization.
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Figure 10: Simulation results of average queue length
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Dresner K. and P. Stone, 2004. Multiagent traffic management: a reservation-based intersection control mechanism. Autonomous Agents and Multiagent Systems (AAMAS’04), New York, USA, pp.530-537.
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Contet J.-M., F. Gechter, P. Gruer, and A. Koukam, 2007. Application of reactive multiagent system to linear vehicle platoon.The 19th IEEE International Conference on Tools with Artificial Intelligence, vol. 2, pp. 67–70. Tan C.-W., S. Park, H.C. Liu, Q. Xu, P. Lau, 2008. Prediction of transit vehicle arrival time for signal priority control: Algorithm and performance. IEEE Transactions on Intelligent Transportation Systems, vol.9, No. 4, pp. 688-696.
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CONCLUSION ADN FUTURE WORK
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Figure 11: Simulation results of average waiting time
minimize the overall evacuation time, not the average queue size or average vehicle waiting time.
Dresner K. and P. Stone, 2006. Traffic intersections of the future. The 21th National Conference on Artificial Intelligence, NECTARTrack (AAAI 06), pp. 1593–1596. WU J., A. Abbas-Turki and A. El Moudni, 2009. Discrete Methods for Urban Intersection Traffic Controlling. The IEEE 69th Vehicular Technology Conference, (VTC Spring’09), Barcelona. Huang Q.-F. and R. Miller, 2003. The design of reliable protocols for wireless traffic signal systems.
MOSIM’10 - May 10-12, 2010 - Hammamet - Tunisia
Department of Computer Science and Engineering, Washington University, Tech. Rep. Wunderlich R., C.-B. Liu, I. Elhanany, and T. Urbanik, II, 2008. A novel signal-scheduling algorithm with quality-of-service provisioning for an isolated intersection. IEEE Transactions on Intelligent Transportation Systems, Vol.9, No.3, pp.536-547. Nadeem T., S. Dashtinezhad, C.-Y. Liao and L. Iftode, 2004. TrafficView: A Scalable Traffic Monitoring System. The 2004 IEEE International Conference on Mobile Data Management (MDM’04), Berkeley, California, USA, pp.13-26. Yu X.-H. and A. R. Stubberud, 1997. Markovian decision control for traffic signal systems. The 36th IEEE Conference Decision Control, San Diego, vol. 5, pp. 4782–4787. Pan Y.-J., 2008. Decentralized control of vehicles in platoons with robust nonlinear state estimation. The 4th IEEE Conference on Automation Science and Engineering, Key Bridge Marriott, Washington DC, USA, pp.145-150.
