Discrete Intersection Signal Control Jia Wu, Abdeljalil Abbas-Turki, Aur´elien Corr´e¨ıa and Abdellah El Moudni SeT, UTBM, 90010 Belfort Cedex

Abstract— In this paper, we propose a new control of traffic lights of a simple intersection, taking into account vehicle behaviour, integral red and orange phases. We consider the intersection as a resource shared between vehicles of two roads. Hence, the control of traffic light signal is performed to take into account each vehicle arrival individually. On the one hand, this allows us to distinguish different kinds of vehicles such as public transport vehicles and emergency vehicles. On the other hand, since the main objective is the minimization of the evacuation time, then the waiting time of each vehicle is dramatically reduced. Moreover, such a control can lead to autonomous vehicles, which would not need traffic lights anymore.

I. I NTRODUCTION As the rapid growth of urban traffic, the demand of transport becomes hard to meet. Furthermore, the transport is generally the principal source of air pollution in big cities. Several politicians stake on the reduction of the use of individual vehicles. Thus, they have to face the dilemma of the choice between pollution and the economic expansion of the city. This situation leads several researchers to look for new solutions for controlling the traffic in order to minimize stop times, queue lengths and situations of acceleration and deceleration. Traffic controls are mainly based on the estimation of the flow rate of vehicle arrivals. This is the case of both major kinds of techniques of the traffic light control, fixed cycle time and adaptive signal control. In the fixed signal control, historical data is used to set up control strategies. The fixed signal control is simple and does not require sensors to obtain information about real-time traffic condition. It is just based on the average flow rate of roads. The drawback of the fixed signal control is obvious: It can not respond to any change in traffic condition. The most widely known and used system is TRANSYT [1] which was developed originally at Transport and Road Research Laboratory (TRRL). In order to improve such a control, adaptive control strategies are proposed. They receive real-time data through sensors and create an optimal timing plan. Several approaches are proposed to develop successfully adaptive control at a single intersection, such as the ones based on fuzzy logic [2], [3], continuous or hybrid Petri Nets [4], [5], and Markov chains [6]. The basic principle of fuzzy signal control is to model the control based on human expert knowledge. The basic advantage of fuzzy control is that it fires many soft rules simultaneously and makes a decision which offers the compromise. The Petri Nets are widely used in traffic control. The safety requirements such as avoiding conflicting movements at the intersection can be easily ensured by PN model. Besides, the assumption of a continuous flow can be

modelled by continuous places and transitions. The Markov model provide a way of computing the queue dynamics and the probability of a queue to be observed at a certain time. Right now, various responsive systems of traffic control have been established. For example, SCOOT [7], one of the first major real-time traffic control system, based on feedforward strategy, makes continual incremental adjustments in real-time of cycle lengths, splits, and offsets. SCAT [8] uses voting algorithms to find the best signal plan. OPAC [9], PRODYN [10], operate in a different manner. OPAC uses a rolling horizon approach to operate a dynamic optimisation. PRODYN uses automatic control methods like: Bayesian estimation, Dynamic Programming, Decentralized methods. One can note that there are several approaches for the traffic light control. However, we draw the reader attention to the fact that all these approaches are mainly based on the estimation of the flow rate for minimizing vertical queue length. It is widely known that the estimated flow rate is not an effective measure for controlling traffic lights because it leads to a confusion about which actual traffic situation occurs. Moreover, there is a gap between the estimated traffic flow of the next time and the actual one. Hence, such systems are still requiring a human intervention when the traffic fluctuates dramatically. Another limitation of such systems is that their operation rules cannot be used for new concepts of traffic control such as the one of autonomous vehicles. Indeed, flow rate cannot be applied to vehicles that communicate their situation to each other in order to define their priority for passing through intersection. Such a new concept requires taking into consideration each vehicle individually. Besides, flow rate estimation can not make use of each second of green time efficiently. Because the portion of the signal timing cycle is allocated to each phase, there is the possibility that when the green light is on, no vehicle gets through the intersection. The intersection as a resource is not utilized in the maximum degree. Since approaches we find in the literature meet the limitation of efficiency and of flexibility for facing new technologies of communication, naturally, we are forced to look for new approaches. In this case, it is interesting (i) to consider only present vehicles according to their availability for passing through intersection and (ii) to try to vacate them as soon as possible. If we could reduce the evacuation time, the crossroad would absorb efficiently the new vehicle arrivals without the need of estimation of the future flow rate. Since we treat individually each vehicle arrival, we need a discrete event model. We use the model of street proposed

in [11]. The model is based on Timed Event Graphs (TEG) and allows us to evaluate microscopic and macroscopic performances. These evaluations meet the fundamental diagram properties. Besides, we can compute sojourn time of each vehicle in order to evaluate the time at which it will be ready to pass through intersection. By this way, we present in this paper an approach that attempt to expend each second efficiently. The remain of this paper is organized as follows. Section II presents the model that our control strategy is based on. Section III briefly describes traffic light control algorithm that we use. The simulation results are given in Section IV.

fi

init

fo Fi

Fo

li

lo P

d

n x

v c

Fig. 2.

TEG model of a non-empty street at simulation-starting date

II. T HE MODEL FOR TRAFFIC LIGHT CONTROL We consider an isolated three-phase intersection with controllable traffic lights on each corner, see Figure 1. It is supposed that there are two traffic flows of vehicles to be served in the intersection. Each road has one lane. We assume that there is a minimum waiting time s between two successive vehicles coming from different roads. This corresponds to the integral red time plus the wasted time for acceleration. Likewise, two successive vehicles from the same road are spaced at least by a time space p. Obviously, p < s.

A. Model of a segment of road

4

sensor

Road 2 3 2

one segment of Road 2

one segment

1

of Road 1

Road 1 5

4

3

2

1 shared surface

sensor

a vehicle

Fig. 1.

Event Graphs. Consequently, the behaviour of the system can be described by (M ax, +)-linear equations [12]. By this model, we can treat the vehicles individually. In the following, we present a modular model of simple intersection. Indeed, we cut this system into elementary subsystems that are either segments of one road or the shared surface. Sub-systems of the former type allow to describe roads and their heterogeneous properties, while the one of the latter represents the surface shared by the roads, as the hatched area of figure 1.

a simple intersection

For each movement of traffic flow, the signals are given in the following order: Red, Green and Yellow. A green interval is followed by a yellow change interval indicating that a vehicle must stop if it can ensure safety. One can note that in the described system, both movements of the traffic flow obey the same rule. Therefore, we are interested only on the modelling of one movement of flow. Models of traffic are important. They may support offline and on-line planning of traffic lights. Besides, they may be used for lane speed control. Hence, the traffic control approach depends strongly on the chosen model. For taking into consideration each vehicle individually for the traffic control, we need a model that considers each vehicle arrival as a discrete event. Thus, we use a model based on Timed

In this part, we briefly review the model of a segment of road [11]. For simplicity, we just give the elementary road traffic system modelling. Each direction of the road is treated separately and discretized into segments where the characteristic variables of flow depend only on the time but not on the position of vehicles into the segment. Each segment is considered as an elementary road traffic system. In the simple intersection depicted by figure 1, there are two main traffic flows. Both upstream roads are split into segments, of which behaviours are modelled by TEG in figure 2. The meaning and the computation of parameters are detailed in table I. Each token in place P represents a vehicle in the segment. Usually in the literature, the maximum flow rate at the entrance and at the exit of the segment are 0.5 veh/s, so fi1 = fo1 = 2 s. Note that the simulation must have a starting date, denoted t0 = 0. A virtual input init is connected to each output of segment. This virtual input prevents v1 from being fired before the date t0 and “freeze” the initial marking until this date. Given a transition y, we can evaluate the dater function y(k) [12], which gives the date of its k th firing. So the arrival time of each vehicle at the entrance of segment is obtained by dater function of transition x. Similarly, the arrival time at the exit of segment can be obtained easily by the dater function of transition v. If the segment is the last one of the road, dater function v gives also the arrival time at the entrance of the shared surface. B. Model of a shared surface endowed with traffic lights We present the model of a surface shared by two roads. In [13], authors propose a model which describes the behaviour

TABLE I PARAMETERS OF THE MODEL Parameter

2

2

lo1

lc1

Meaning

li

Number of lanes at entrance of segment

lo

Number of lanes at exit of segment

fi

Invert of max flow rate per lane at entrance of segment

fo

Invert of max flow rate per lane at exit of segment

n

Number of vehicles in segment initially

c

Maximum number of vehicles that can be held by seglength of segment × lanes ment, mean length of vehicles

d

Time to pass through segment,

P1

v1

x1

p1

R1 G1 x2−1

v1−2

length of segment average speed of vehicles

s2−1

Traffic light management

of vehicles at intersection and the three-phrase traffic light signal cycle, see figure 3. We assume that this shared area is occupied by at most one vehicle at a time. This model can be divided into three parts. R1 and R2 present, respectively, the behaviour of vehicles from road 1 and vehicles from road 2 when crossing the shared surface. lo1 denotes the number of lanes at exit of the last segment of road 1, and lc1 represents the number of lanes at the exit of the shared surface. The definition of lo2 and lc2 are similar for road 2. A token in place G1 means that a vehicle coming from road 1 would meet the green light. We assume that every vehicle crosses the shared surface at the same speed, so they will spend the same time units p1 to cross it. R2 works in the same way. The part of traffic light management deals with the threephrase traffic light signal cycle. In figure 3, a token in place G1 means that the green light for vehicles coming from road 1 is on. When we need to switch traffic lights, the token moves to place S1−2 and remains there for a fixed time s1−2 including the integral red time plus the wasted time for acceleration of vehicles. Then, the token moves in place G2 and the green light is turned on for vehicles coming from road 2. By deciding to fire transtions v1−2 or v2−1 , it is possible to control the signals of intersection. Our control strategy is applied on the model of figure 3. III. T RAFFIC C ONTROL S TRATEGY As we have mentioned, we consider the intersection as a resource shared between vehicles of two roads. Hence, the control strategy is proposed to allocate the authority of using the intersection to different roads according to the arrival time of vehicles. The objective is to free the resource as soon as possible. This means that we aim at vacating both roads and intersection as earlier as we can. For example, see figure 1, there are two directions of traffic flow. In each direction, the arrival times are the following: 5 vehicles in road 1, the arrival times are 0s, 4s, 10s, 15s, 21s; 4 vehicles in road 2, the arrival times are 2s, 8s, 15s, 22s, see figure 4. For simplicity reasons, we suppose vehicles of two roads use identical time units p1 = p2 = 2s when crossing the shared surface. We assume that the integral red time plus the wasted time for acceleration are s1−2 = s2−1 = 5s. The optimal solution in this case is known and is given in figure 5. The minimum evacuation time of all vehicles in this

s1−2

S1−2

S1−2

v2−1

x1−2 G2

R2 v2

Fig. 3.

P2

p2

x2

lo2

lc2

2

2

TEG model of a shared surface endowed with traffic lights

Road

2

1 1

1

4

3

2 3

2

5

4

t (s) 0

2

4

6

8

10

Fig. 4.

12

14

16

18

20

22

the arrival time of vehicles

example is 33s. For sake of generality, the question is how can we allocate the resource of intersection to each vehicle to carry out our objective? The Petri net model informs us about vehicle arrivals and conflicting situations. However, a control strategy is still required. A. Problem Specification Before introducing the control strategy, we need data presented in table II that are derived from the intrinsic infrastructure parameters such as the length, the maximum allowed speed and flow rate of each road. Besides, we need, at each second, information about traffic situation like the number of vehicles at each road, the arrival time of vehicles Road 1 2

1

Road 2

1

Fig. 5.

2

3

4

5

3

the optimal strategy for traffic lights control

4

and so on. TABLE II

G (q1 , q2 , g) = min {G(q1′ , q2′ , g ′ )+sg,g′ +p(qg ,g) +τ } (1) ′ g =1,2

ROAD PARAMETERS

qf′

variable g ng

definition the index of road, g = 1, 2 the number of vehicles in road g

(qg , g)

the qg th vehicle in road g, 1 ≤ qg ≤ ng

r(qg ,g)

the arrival time of vehicle (qg , g)

p(qg ,g)

the time which vehicle (qg , g) need to cross the shared surface

sg′ ,g

the integral red time plus the wasted time for acceleration when the traffic light changes between road g ′ and g, g ′ 6= g, g ′ = 1, 2.

G (q1 , q2 , g)

by traffic control strategy, the minimum total travelling time for vehicles (1, 1), . . . , (q1 , 1) and (1, 2), . . . , (q2 , 2). The last vehicle (qg , g)th in this strategy is from road g

We assume that vehicles of each road leave the intersection in a stringent order according to their arrival time. This means that we vacate each lane according to FIFO rule. Indeed, for sake of correctness, we can not admit vehicles to pass through intersection before a previous vehicle of the same road. Hence, we have: (1, g) ≺ (2, g) ≺ . . . ≺ (ng , g) Since our objective is to evacuate the vehicles in the road as quickly as possible, we have to reduce the travelling duration of each vehicle in the road. We have defined G (q1 , q2 , g) as the total travelling time of vehicle (1, 1), . . . , (q1 , 1) and (1, 2), . . . , (q2 , 2) (see Table II). It can be comprehended as the duration of the last vehicle departure time (time when it leaves the intersection) minus the arrival time of the first vehicle (when the first vehicle is detected). Hence, the optimal strategy is the arrangement of traffic light that minimizes this objective function: min {G(n1 , n2 , 1), G(n1 , n2 , 2)} For example, see figure 1, the objective is to find an optimal strategy that satisfies: min {G(5, 4, 1), G(5, 4, 2)} B. Forward Dynamic Programming One can note from the problem definition and the PN model Pthat the problem can be considered as 1|rj , pj , sP Cj [14] where rj is r(qg ,g) , pj is p(qg ,g) , sf is f| sg′ ,g and Cj is G (q1 , q2 , g). In [14], it proposes a forward dynamic programming for solving the problem in order to find the optimal resolution. Since the completion time of this algorithm is polynomial (O(n2 )), it can be implemented for a real traffic situation. Thus, right now, we use the forward dynamic programming that is described in the following. Now, we give the formula of recursion.

Where = qf for f 6= g, otherwise qf′ = qf − 1 for f = g, f = 1, 2. If the last vehicle of the previous strategy is also from road g, we have then sg′ ,g = 0, where g ′ = g, that means the traffic light does not change. Otherwise, if the last vehicle of the previous strategy is from another road g ′ , where g ′ 6= g, then the traffic light change time is involved. At this stage, we can give the formula for computing the waiting time τ of vehicle (qg , g). We have: τ = max



0, r(qg ,g) − G(q1′ , q2′ , g ′ ) 0, r(qg ,g) − G(q1′ , q2′ , g ′ ) − sg′ ,g

g′ = g g ′ 6= g

(2)

The formula of recursion selects the previous optimal strategy by which vehicle (qg , g) is the last vehicle to pass through the intersection. The recursion keeps on until it reaches the initialization condition. To illustrate formula 1, we cite again the example of figure 1 and figure 4, G(5, 4, 1) = min{G(4, 4, 1) + p(5,1) + τ1 , G(4, 4, 2) + p(5,1) + s1,2 + τ2 } τ1 = max{0, r(5,1) − G(4, 4, 1)} τ2 = max{0, r(5,1) − G(4, 4, 2) − s1,2 } And r(5,1) = 21 s. C. Initialization We are now ready to give initialization for forward dynamic programming. The recursion stops when there is one vehicle in the road. For our problem, there are four cases that can be considered as initializations: G(1, 0, 1), G(0, 1, 2), G(1, 0, 2), G(0, 1, 1). s0,g denotes the traffic light change time before the first vehicle (1, g) in road g arrives.  G(1, 0, 1) =    + G(0, 1, 2) =    +

s0,1 + max(r(1,1) − s0,1 , 0) p(1,1) s0,2 + max(r(1,2) − s0,2 , 0) p(1,2)

(3)

If s0,g = 0, the initialization can be simplified as follows:  G(1, 0, 1) = r(1,1) + p(1,1) (4) G(0, 1, 2) = r(1,2) + p(1,2) Indeed, the other two cases, which are G(0, 1, 1) and G(1, 0, 2), can not be considered. Because they are contradictory of reality. Taking G(1, 0, 2) as example, on the one hand it denotes that there is no vehicle on the road 2, but on the other hand it denotes at the same time that there is one

D. Time Complexity As widely known, dynamic programming has three properties: overlapping sub problems, optimal substructure, and memorization. The overlapping means that the sub problems are solved by dividing into similar sub-sub problems until we reach the initialization condition. During the process, the same sub problems maybe appear several times. Hence, we do not need to compute them every time. Instead, we save the solutions to sub problems we have already solved and use them to solve many different larger problems. Then, if we need to solve the same problem later, we can retrieve and reuse our already-computed solution. This is called memorization. So, in our method, the time complexity of the algorithm is O(n2 ), where n is the number of vehicles (proof can be found in [15]). E. Extension of the traffic control strategy We can consider other elements in real traffic situation. For example, there are some vehicles which have urgency, like the police, the ambulance. The traffic control system should give the high authority to these important vehicles, and minimize their waiting times. We make a change of the dynamic programming to solve this problem. Suppose that every vehicle has a weight for evaluating its degree of urgency. To reduce the waiting time of the high urgency vehicles, we should minimize their weighted total travelling time. w(pg ,g) denotes the weight of vehicle (qg , g), qg = 1, . . . , nf , g = 1, 2. Redefine G(q1 , q2 , g) as the weighted total travelling time of vehicles (1, 1), . . . , (q1 , 1) and (1, 2), . . . , (q2 , 2). Hence, we have: G (q1 , q2 , g) = min {G(q1′ , q2′ , g ′ ) ′ g =1,2

+w(qg ,g) (sg,g′ + p(qg ,g) + τ )}

(6)

Where the last vehicle (qg , g) in this strategy is from road g. Then qf′ = qf for f 6= g, otherwise qf′ = qf −1 for f = g. And sg′ ,g = 0 if g = g ′ , otherwise traffic light change time is involved. Definitions of other variables are the same as the precedent assumption, i.e. without given weights. The initializations for the extended traffic control strategy are given in the following: G(1, 0, 1) = w(1,1) (s0,1 +max(r(1,1) −s0,1 , 0)+p(1,1) ) (7)

G(0, 1, 2) = w(1,2) (s0,2 +max(r(1,2) −s0,2 , 0)+p(1,2) ) (8) If the traffic light change time s0,g = 0, then:

fixed cycle time forward dynamic programming

400 350 300 waiting time[s]

vehicle on this road that pass through intersection. G(0, 1, 1) in the same way can be considered as impossible. We denote these two impossible cases as:  G(0, 1, 1) = ∞ (5) G(1, 0, 2) = ∞

250 200 150 100 50 0

Fig. 6.

0

50

100

150 200 time[s]

250

300

The waiting time of vehicles in the roads(λ1 = 0.1, λ2 = 0.1)



G(1, 0, 1) = w(1,1) (r(1,1) + p(1,1) ) G(0, 1, 2) = w(1,2) (r(1,2) + p(1,2) )

(9)

The other two impossible cases, G(0, 1, 1), G(1, 0, 2), are considered equal to infinity as it is shown in (5). IV. S IMULATION R ESULTS In this section, we present the simulation for controlling the traffic lights of a simple intersection. We use VISSIM to simulate vehicle behaviour at the crossroad. The simulations are performed on a 2.40 Ghz Pentium IV processor based PC. The flow rate in road 1 is λ1 veh/s (number of vehicles per second), road 2 is λ2 veh/s. These two flow rates must satisfy the condition: 0 < λ1 + λ2 < 0.5 veh/s. To make a comparison, the traditional approach fixed cycle time method and adaptive signal control [16] are applied in the experiments. In the fixed cycle time method, the cycle time is 60s. Considering the intersection with the following data: λ1 = 0.1 veh/s, λ2 = 0.1 veh/s. Figure 6 shows the total waiting time of vehicles in two roads at each second. Figure 7 presents the result of the number of vehicles waiting in two roads at each second. Augmenting the traffic flow : λ1 = 0.2 veh/s, λ2 = 0.2 veh/s. Figure 8 are the results of the total waiting time of vehicles. The results of number of vehicles waiting in two roads are shown in figure 9. We can see that the waiting time and the number of waiting vehicles of our method are obviously smaller than those of the fixed cycle time method. At this moment, we have to show that our approach improves the intersection performance by comparing it with an adaptive signal control method. For this, we use the adaptive signal control method from [16] that minimzes average waiting time by describing the evolution of the queue lengths as countinuous variables. We consider λ1 = 0.22 veh/s, λ2 = 0.13 veh/s, w1 = 2, w2 = 1. The average waiting time in two roads is 182.472s. And the average waiting time of our method is 23.3392s.

These experiments prove that our new method can efficiently reduce the waiting time of vehicles in two roads and the number of vehicles in the queue. Hence, we could vacate the vehicles as soon as possible.

16 fixed cycle time forward dynamic programming

14

V. C ONCLUSION

the number of vehicles

12 10 8 6 4 2 0

0

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150 200 time[s]

250

300

Fig. 7. The number of vehicles waiting in the roads (λ1 = 0.1, λ2 = 0.1)

In this paper, a new control for traffic signal has been presented. This strategy is based on the model [13] which treats the vehicle individually. Thus, the arrival time of vehicle is obtained easily. We use forward dynamic programming to find the optimal traffic control strategy for vacating all vehicles in the roads as soon as possible. Then, we extend the control strategy to minimize the waiting time of vehicles which have urgency. The weight is added to each vehicle for evaluating the urgency of each vehicle. Vehicles behaviours are simulated by using VISSIM. By comparing the fixed cycle time and the adaptive signal control method, our method proves its ability of efficiently reducing the waiting time of vehicles and the number of vehicles in real traffic situation. R EFERENCES

700

fixed cycle time forward dynamic programming

600

waiting time[s]

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150 200 time[s]

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The waiting time of vehicles in the roads(λ1 = 0.2, λ2 = 0.2)

Fig. 8.

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fixed cycle time forward dynamic programming

18

the number of vehicles

16 14 12 10 8 6 4 2 0

0

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150 200 time[s]

250

300

Fig. 9. The number of vehicles waiting in the roads (λ1 = 0.2, λ2 = 0.2)

[1] C. B.M. and L. C.J., “Transyt-the latest developments,” Traffic engineering and control, vol. 28, pp. 387–390, 1987. [2] J. Niittymaki and M. Pursula, “Signal control using fuzzy logic,” Fuzzy sets and systems, vol. 116, pp. 11–22, 2000. [3] J.Lin and K.Y.Kuo, “Application of fuzzy set theory on the change intervals at a signalized intersection,” Applied Soft Computing, vol. 1, pp. 161–177, 2001. [4] G. F. List and M. Cetin, “Modeling traffic signal control using petri nets,” IEEE Transacations on Intelligent Transportation Systems, vol. 5, no. 3, pp. 177–187, September 2004. [5] A. D. Febbraro, D. Giglio, and N. Sacco, “Urban traffic control structure based on hybrid petri nets,” IEEE Transactions on Intelligent Transportation Systems, vol. 5, no. 4, December 2004. [6] L. Qiaoru, W. Lianyu, and M. Shoufeng, “The model analysis of vehicles situation and distribution in intersections based on markov process,” Intelligent Transportation Systems, 2003. Proceedings. 2003 IEEE, vol. 2, October 2003. [7] D. I.Robertson and R. Bretherton, “Optimizing networks of traffic signals in real time-the scoot method,” IEEE Transacations on Vehicular Technology, vol. 4, no. 1, pp. 11–15, February 1991. [8] A.G.Sims and K.W.DOBINSON, “The sydney coordinated adaptive traffic(scat) system philosophy and benefits,” IEEE Transactions on vehicular technology, vol. VT-29, pp. 130–137, May 1980. [9] ——, “Implementation of the opac adaptive control strategy in a traffic signal network,” IEEE Intelligent Transactions Systems Conference Proceedings, pp. 25–29, August 2001. [10] J.J.Henry, J.L.Farges, and J.Tuffal, “The prodyn real time traffic algorithm,” Proc. 4th IFAC/IFORS Conf. Control Transportation System, pp. 305–309, 1983. [11] A. Corr´e¨ıa, A. Abbas Turki, R. Bouyekhf, and A. El Moudni, “A (max,+)-linear model for the analysis of urban traffic networks,” in 10th IEEE international conference on Emerging Technologies and Factory Automation (ETFA), Catania, Italy, Sept, 2005. [12] F. Baccelli, G. Cohen, G.-J. Olsder, and J.-P. Quadrat, Synchronization and linearity, an algebra for discrete event systems. Wiley, 1992. [13] A. Corr´e¨ıa, A. Abbas Turki, R. Bouyekhf, and A. El Moudni, “Modelling urban intersections in dioid algebra,” WSEAS transactions on systems, vol. 5, pp. 1223–1228, May 2006. [14] C. N.Potts and M. Y.Kovalyov, “Scheduling with batching: A review,” European Journal of Operational Research, vol. 120, pp. 228–249, 2000. [15] B. Peter, Scheduling Algorithms. Springer, 2004. [16] B. Schutter, “Optimal traffic light control for a single intersection,” Proceeding of the 1999 American Control Conference(ACC’99),San Diego, California, pp. 2195–2199, June 1999.