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Physica A 369 (2006) 841–852

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Effect of the lane reduction in the cellular automata modelsapplied to the two-lane traffic

K. Nassaba,b,�, M. Schreckenberga, A. Boulmakoulc, S. Ouaskitb

aFachbereich Physik, Universitat Duisburg-Essen, D-47048 Duisburg, GermanybFaculte des Sciences ben M’sik, Universite Hassan II Casablanca, Casablanca, Maroc

cLaboratoire Informatique des Systemes de Transport, Faculte des Sciences et Techniques de Mohammedia, B.P. 146 Mohammedia, Maroc

Received 18 November 2005; received in revised form 23 January 2006

Available online 21 February 2006

Abstract

More investigated situations in the field of traffic modelling are those of traffic bottlenecks caused by slow vehicles or

road defects. The new aspect of this paper is the simulation of vehicular dynamics near a partial reduction in a road from

two lanes to one lane. In order to reduce the bad impact of waiting vehicles behind the defect region, a strategy regulating

the vehicle movement in the vicinity of the reduced lane is taken into account. The simulation model is based on the cellular

automata model of Nagel–Schreckenberg with additional rules of lane change. The partial lane reduction strongly reduces

the road capacity, and the added regulation strategy leads to a more interesting shape of the fundamental diagram, which

depends on different constraints on the model parameters, e.g., the length of the reduced lane, the maximal speed, and the

length of the connection sites near the entry of the reduced lane.

r 2006 Elsevier B.V. All rights reserved.

Keywords: Cellular automata; Stochastic process; Transport; Phase transitions; Formation of congestions

1. Introduction

Several simple microscopic models were proposed during the last decades to simulate traffic flow on a largescale of road networks and to describe and reproduce its instabilities. Due to their simplicity, discretemicroscopic models based on stochastic cellular automata (CA) model of NaSch [1–5] are frequently used forthe simulation of road traffic flow. It is obviously more pronounced in one-lane traffic models that slowvehicles or geometrical defects of roads influence the fluidity of the traffic and cause a phase transition fromfree to congested flow. A part of traffic flow models treated the effect of an additional lane in a generalmanner, i.e., the passage from a one-lane to a multi-lane road traffic model. These additional lanes lead to anincrease in road capacity due to the possibilities of lane change. Note that taking into account the hindrances,which reduce the number of lanes, a two-lane traffic model is not an easy task. The most usual situation ofsuch a road defect can be caused by road works that partially reduce the width of a motorway to one lane. The

e front matter r 2006 Elsevier B.V. All rights reserved.

ysa.2006.01.073

ing author. Fachbereich Physik, Universitat Duisburg-Essen, D-47048 Duisburg, Germany.

ess: nbkd2@yahoo.fr (K. Nassab).

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vehicle dynamics in such a situation seem to be rather complicated. In our knowledge, no theoretical studybased on the approach of CA was made within the framework of a traffic flow simulation in the vicinity of areduced lane.

Recently, it has been shown in two-lane traffic models that slow vehicles can dominate vehicle dynamics atlow densities [6]. These models are based on the model of NaSch with additional rules of lane change proposedby Rickert et al. [7] and Nagatani [8]. The vehicle movement can be done by dividing the update step into twosuccessive sub-steps: either the vehicle movement in the first sub-step and the lane change in the second sub-step [7], or the left lane in a sub-step and the right lane in the next sub-step [8]. The defects caused by slowvehicles in these last models were represented by vehicles with very low speed [8]. The lane-change rules can besymmetric or asymmetric with respect to the lanes and to the vehicles. These sets of rules can reproduce thelane usage inversion, which was observed experimentally and studied in Refs [7,9–11].

If one wants to study the road traffic flow in the vicinity of a partial reduction of lanes (PRL), one must askoneself how the vehicles will change from lane to lane in order to exceed the region of a PRL. In particular,how can the so-called ‘‘procedure of zipper circulation’’ (PZC), which is well known in countries likeGermany, help to improve traffic flow near the defect region?

The objective of this paper is to take a useful step towards the traffic flow description in the vicinity of aPRL on a two-lane road. Generally, the road capacity is reduced in the presence of vehicular defects (like slowvehicles [6,12]) and geometrical road defects (like ramps [13–16] and open boundaries [17,18]).

The simulation results presented in this paper will show that a PRL has an important influence on thecapacity of a two-lane road. In order to simulate a good approximated model of the real application of thePZC, some modifications in the lane-change rules near the region of a PRL will be taken into account. In thispaper, it will be proved that the approximated model of the PZC can help to extract the important features oftwo-lane traffic flow in the presence of a PRL.

2. Simulation model and model of a reduced lane

In this paper, the simulated system is composed of a section of a two-lane motorway with periodicboundaries and the simulation model of CA will be based on that of NaSch [1] with additional rules of lanechange. In the discrete NaSch model, each lane is represented by a one-dimensional chain of L cells (sites) oflength 7.5m, each cell can be empty or occupied by just one vehicle. The speed of each vehicle can take one ofthe vmax þ 1-allowed integer values v ¼ 0, 1y, vmax. At each step of discrete time, from t to tþ 1, the state ofthe road can be obtained by applying the following rules to all vehicles at the same time (parallel dynamics):

Rule 1: Acceleration: vn ! min (vn þ 1, vmax)Rule 2: Braking: vn ! min (vn, dn � 1)Rule 3: Randomization with probability p: vn-max (vn � 1, 0)Rule 4: Driving: xn ! xn þ vn

Here dn ¼ xnþ1 � xn denotes the distance to the next vehicle ahead and vmax represents the maximum speed.The lane-change rules in discrete microscopic models of CA were discussed in several scientific works [6–11].

Rickert et al. [7] have assumed a symmetric rule set where vehicles change lanes if the following criteria arefulfilled.

Incentive criterion:

(1) vmove4gapsame, avec vmove ¼ min(vn+1, vmax).

Criteria of safety:

(2) gaptarget4gapsame.(3) gapbackXvmax.

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Here gapsame, gaptarget, and gapback denote the number of unoccupied sites between the vehicle and itspredecessor on the actual lane, and between the same vehicle and its two neighbour vehicles on the desired

lane, respectively.

The symmetric rules quoted above have been used to change from right to left and conversely because theovertaking manoeuvre is allowed on both lanes. However, asymmetric rule sets have been used for thedescription of road traffic in countries like Germany where overtaking on the right lane is forbidden.

In this work, the symmetric rule set defined above will be used to change from right to left. In order to describethe lane change from left to right, the following simple set of asymmetric rules will be applied to vehicles:((vmovepgapsame) or (gaptargetpgapsame)) and (gapbackXvmax), with vmove ¼ min(v+1, vmax). In addition, thecriterion (gaptargetXvmove) must be achieved. The first criterion concerning the comparisons between vmove,gapsame, and gaptarget is simply the negation of the criterion ((vmove4gapsame) and (gaptarget4gapsame)) used inthe symmetric rule set of lane change. The usage of asymmetric rules for the lane change from left to right canhelp to describe the traffic flow in countries like Germany where drivers are also advised to apply the PZC near aPRL. Moreover, it should be noted that the capacity of the simulated system becomes strongly reduced in thepresence of a PRL if the overtaking manoeuvre is allowed on both lanes.

Various implementations of a PRL are possible. But the most usual case of these possibilities, which allowsone to keep the set of parameters as small as possible, is the presence of an obstacle of a finite length on theright lane. This simple scenario can extract the qualitative and important aspects of the application of the PZCnear a reduced lane.

2.1. Strategies A and B of lane change near the obstacle

The obstacle causing the PRL is of length Lr and is placed between the positions xr1 ¼ L=2 andxr2 ¼ L=2þ Lr. Obviously, the parameter Lr controls the strength of the defect. For the sake of simplicity, thisparameter is kept fixed (Lr ¼ 16). Two different strategies of lane change before the beginning of the reducedlane will be presented in this paper. (i) Within the strategy A, vehicles can change on all positions (except in thedefect region) of the simulated system from right to left and conversely, if the lane-change rules are fulfilled.Then, this strategy does not contain modifications in the lane-change rules, or in other words, the PZC is notapplied in this strategy. (ii) Within the strategy B, the lane change from right to left and conversely isforbidden in a region located behind the defect region (Fig. 1). This region is called ‘‘region C’’ and is of lengthLc. The vehicles can change the lane from right to left only at the exit of this region through two sites calledconnection sites (CS; Fig. 1). In the vicinity of CS, a change in the value of ‘‘gapback’’ will be taken intoaccount. This implementation of strategy B is closer to a good approximation of a real case of PZCapplication. In real traffic, the PZC is usually done by applying the two following principal steps: (i) in the firststep, vehicles must use the control lane up to the beginning of the PRL and (ii) in the second step, vehicledrivers who drive straight make the place turn by turn (alternatively) for the inserting vehicles which try toexceed the obstacle (from the right lane) before the beginning of the PRL.

In order to increase the number of vehicles changing towards the left lane through the CS, it becomesnecessary to change the constraint concerning the safety spacing gapback in strategy B. Fig. 2 shows that thelane-change rate n(x) from right to left through the CS is high if the value of gapback is very low (gapback ¼ 1site). Note that the minimal spacing ‘‘gapback ¼ 1’’ does not cause accidents, which are also avoidable by thesecond rule of the NaSch model. Moreover, it is often observed on motorways that before the arrival at aPRL, vehicles move slowly forwards with small safety distances. Then, the choice of a small gapback is closer toa good approximation of the real case of two-lane traffic in the vicinity of a PRL. In order to emphasize onlythe effects of the PRL to the traffic flow, only a low p ¼ 0:01 will be taken into account (because the internaldynamics influenced by p are not out pointed in this work. The other parameters of the simulation model willbe given by L ¼ 1000 sites and vmax ¼ 5.

3. Results of simulation

The effect of lane reduction and also effects of lane-change strategies A and B introduced in Section 2 will bepresented in this section on the basis of numerical results.

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Fig. 1. Schematic representation of a PRL. The defect of length Lr is located on the right lane. Bottom: the lane change is not allowed in

region C if strategy B is applied to vehicles. However, vehicles on the right lane can change from right to left only through the CS to exceed

the obstacle. Top: within strategy A, vehicles do not follow any ordered rule to exceed the obstacle.

Fig. 2. Comparison between the lane-change rates n(x) near the CS obtained for two different values of gapback (vmax and 1). The number

of vehicles changing the lane increases if gapback is minimal. This result is found for density r ¼ 0:2, which corresponds to the so-called

plateau regime where the traffic flow is maximal and constant (Fig. 3). The system size is given by L ¼ 1000 sites.

K. Nassab et al. / Physica A 369 (2006) 841–852844

3.1. Application of strategy A of lane change near the PRL

As mentioned in Section 2, no additional constraints leading to a controlled movement of vehicles nearthe PRL are added to the rules of lane change within strategy A. In this case, simulation results show that theaverage flow q varies with respect to the lanes and to the density r, and that the PRL of length Lr reduces thecapacity of the simulated system (Fig. 3) and leads to different forms of its corresponding fundamentaldiagrams. At this point, it should be mentioned that two critical values of the density r exist, for which theaverage flow in the simulated system changes: r1 � 0:1 and r2 � 0:6.

Up to the density r1 the system is in a free flow state, i.e., no jams exist. Above the density r2 the systemresides in the so-called jammed state. However, the most interesting regime lies between the two densities r1and r2, which can be also called rlow and rhigh, respectively. In this density phase, the average flow on the rightlane decreases and that on the left lane increases. Near the density r ¼ 0:31, values of the average flow on bothlanes are the same. The resulting average flow of the global system (two lanes) becomes constant and forms

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Fig. 3. Fundamental diagram of the global simulated system (two lanes) is compared to that of a two-lane road without a PRL (solid line)

and to that of right and left lanes. The defect causes a strong reduction in the capacity of the simulated system. This capacity is identified to

the plateau value which is formed in the fundamental diagram between densities r1 and r2. Left: the fundamental diagram of the right (left)

lane is obtained by computing the average flow and also the density on the right (left) lane. Right: the average flow computed on each lane

varies with respect to the density of the global simulated system.

K. Nassab et al. / Physica A 369 (2006) 841–852 845

a plateau in the fundamental diagram (Fig. 3). This plateau identifies the intermediate density regime(r1prpr2), and its value corresponds to the capacity of the simulated system. The capacity of the reducedlane of length Lr limits the value of the indicated plateau. If Lr increases, this value decreases further.

3.1.1. Density profiles

In order to understand the variations of the average flow shown in Fig. 3 and already described above(Section 3.1), it is useful to look at the density profile r(x) and at the rate n(x) of lane change on each lane withrespect to the traffic flow regimes. In the free-flow phase (r ¼ 0:03), only small and local deviations of aconstant density profile can be observed for each lane in Fig. 4 (top). At the intermediate density phase(r ¼ 0:27 and r ¼ 0:4), a separation in macroscopic regions of different values of the density (high and low)can be seen for each lane of the simulated system in Fig. 4 (bottom).

The congested region on the right lane is denser and less wide than that on the left lane. This result, which isfound in the plateau regime, can be explained by a long waiting time of right-lane vehicles behind the obstacleto be inserted in the reduced lane. This waiting time is due to the intense passage of left-lane vehicles straighttowards the reduced lane without allowing those on the right lane to have more insertion chances for insertion.These facts can be confirmed by the difference between the density values of the parallel congestions in Fig. 4(bottom). In addition, other vehicles arriving towards the entry of the compact congestion on the right lane tryto change the lane early from right to left leading to the formation of a parallel congestion.

At a high-density regime (r ¼ 0:78), the density of congestion on the right lane increases more quickly thanthat on the left lane (Fig. 4, bottom). It is to stress that the density in the defect region on the right laneremains zero, because it is not usable by the vehicles.

The results in Figs. 3 and 4 show also that the presence of an obstacle of a finite length on the right lane andthe formation of a parallel jam on the left lane act like a blockage which limit the capacity of the simulatedsystem in the density regime r1prpr2.

3.1.2. Rates of lane change

As a further explanation of the impact of the PRL on the average flow in the simulated system, plots of lanechange rate n(x) are given with respect to traffic flow regimes in Fig. 5. In the free-flow regime, the rate n(x) ofthe lane change from right to left is higher near the beginning of the PRL than in other regions on the rightlane (Fig. 5, right). This is due to the fact that vehicles are more obliged to exceed the hindrance of the defectregion. But the lane change in other regions on the right lane is possible if overtaking is necessary and if thelane change conditions are fulfilled. Vehicles passing through the reduced lane tend to accelerate at its exit(the exit of the reduced lane) and change the lane frequently from left to right, where they find more space afterthe exceeding of the obstacle (Fig. 5, left).

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Fig. 4. Density profiles of each lane of the simulated system obtained for different values of the density r. At low densities (r ¼ 0:03), onlya local and low disturbance in each lane can be observed near the defect (top). By increasing the value of the density r (r ¼ 0:27, 0.40, and0.78), the obstacle leads to the formation of parallel congestions, which grow in size on both lanes of the simulated system. But the

congestion on the right lane remains more compact than that on the left lane (bottom).

Fig. 5. Lane-change rate n(x) obtained for the left and right lanes in the free-flow regime (r ¼ 0:03). Left: the vehicles change the lane

frequently from left to right after the exceeding of the obstacle. Right: the vehicles change frequently from right to left near the entry on the

reduced lane.

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In the plateau regime (r ¼ 0:27), the rate of lane change from right to left decreases strongly near thebeginning of the reduced lane and increases before the entry of the congestion, which is formed behind theregion of the PRL (Fig. 6). This result confirms what is explained for this regime in Section 3.1.1.

The results presented up to now show that only the already existing vehicles in the non-defective lane, beforetheir arrival near the beginning of the PRL, can enter the reduced lane with a short waiting time in acongestion. This can be as a consequence of not taking into account the constraints, which help to control thetraffic movement in the vicinity of the PRL. The aim of the following subsections is to show if the applicationof the PZC (strategy B) can improve the traffic movement near the region of the PRL and especially the rate oflane change from right to left near the entry of the reduced lane.

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Fig. 6. Lane-change rate n(x) obtained for the right and left lanes in the plateau regime (r ¼ 0:27). Left: the vehicles frequently change thelane from left to right after the exceeding of the obstacle and also before the entry on the congested region (n(x) begins to increase). Right:

the rate n(x) of lane change from right to left increases before the entry on the congested region (which is formed behind the obstacle) and

decreases at the entry of the reduced lane.

Fig. 7. Comparison between the fundamental diagrams of the simulated system obtained for strategies B (dotted line) and A (symbol o).

Left: the plateau regime is sub-divided into three so-called ‘‘sub-phases’’ of the density due to the effect of strategy B. Right: strategy B also

increases the density interval, in which the maximum flow on the right lane remains constant and higher than that on the left lane.

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3.2. Application of strategy B of lane change in the vicinity of the PRL

Now, we discuss the effect of the PRL by taking into account the PZC, which is an important aspect ofstrategy B. Fig. 7 (left) shows a comparison between two fundamental diagrams corresponding to strategies Aand B. In the regimes of free (rprlow) and jammed (rXrhigh) flow, the values of average flow of the globalsimulated system (two lanes) do not change remarkably with respect to the strategy of lane change. However,the more interesting density regime lies in the plateau regime between rlow and rhigh, where the plateau issubdivided into two so-called ‘‘sub-plateaus’’ of different values due to the application of strategy B. The firstsub-plateau is formed in the density phase rlowprpr3 and the second sub-plateau corresponds to the regimeof the densities between r4 and rhigh. Within the density regime r3prpr4, the average flow is not constant(it increases). In addition, a decrease in the maximal average value can be observed in Fig. 7 (left) in the case ofapplication of strategy B. This is evident and can be explained by the following three facts: (i) lane changefrom right to left and conversely is forbidden in region C, (ii) vehicles on the right lane can be inserted only atthe entry of the reduced lane through two CS, and (iii) vehicles on both lanes decelerate alternatively in orderto make the insertion turn-by-turn a success in the reduced lane.

Despite the decrease in the capacity of the global simulated system, the comparison between the plots ofFigs. 7 (right) and 3 (left) leads to a more interesting result. It concerns the increase in the value interval inwhich the maximum flow on the right lane remains constant and higher than that on the left lane if strategy Bis applied to the vehicles. This can mean that the flow through the CS from right to left is improved, andconsequently this effect influences the flow in the left lane (at the entry of the reduced lane). The explanation of

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this result and also of the formation of two sub-plateaus will be more detailed in the following subsections.Furthermore, the result shown in Fig. 2 confirms the increase in the flow through the CS from right to left inthe plateau regime if the minimal gapback is taken into account in the simulation model. Also, it should benoted that the width of the first sub-plateau decreases and that of second sub-plateau increases if Lc decreasesor L increases. In our case, the length of region C is given by Lc ¼ 400 sites.

3.2.1. Density profiles and lane-change rates

In the following, we focus on the variations of density profiles and lane-change rates with respect to the sub-regimes located between rlow and rhigh for each lane of the simulated system to demonstrate the effects causedby the additional control process (PZC) of the traffic movement near the entry of the reduced lane. Inparticular, we discuss the impacts observed for the regimes of the two sub-plateaus. As one can see in Fig. 8(top), the regime of the first sub-plateau (rlowprpr3) is characterized by the formation of a congested regionbehind the region of the PRL in each lane of the simulated system. The congestion on the right lane is widerthan that on the left lane. While the small congested region on the left lane remains constant, the other on theright lane increases in length by keeping its density constant. But the length of the congestion on the right lanedoes not exceed that of region C. For these reasons, the average flow becomes constant and forms the first sub-plateau in the fundamental diagram. In the regime of the second sub-plateau (r4prprhigh), the sizes ofthe parallel congestions are the same and continue to increase in parallel, but their densities remain constant(Fig. 8, bottom). This explains the formation of the second sub-plateau in the fundamental diagram of theglobal simulated system. The existence of two sub-plateaus in the density regime rlowprprhigh can be tracedback to the behaviour of the congestions on the simulated system with respect to the length of region C. Infact, above the density r3, the size of the congested region on the right lane exceeds the length of region C andobliges the vehicles to change intensively to the left lane where the parallel congestion widens. Thus, thedensity phase r3prpr4 constitutes a bridge between the two sub-plateau regimes.

Fig. 8. Density profiles obtained by using strategy B are compared with respect to the lanes of the simulated system and to the sub-phases

of the plateau regime (between rlow and rhigh). Top: in the first sub-plateau phase (r ¼ 0:09 and r ¼ 0:12), while the left-lane congestionremains small and local, that on the right lane grows in size in region C where lane change is prohibited. Bottom: in the second sub-plateau

phase (r ¼ 0:31), after the size of the right-lane congestion has exceeded, the size of the region C and the left-lane congestion has widened,

the mentioned two congestions continue to increase in size in parallel by keeping the values of their densities constant. Note, in the case of

r ¼ 0:27, the length of the congested region on the left lane has already reached that on the right lane, but its density tends to reach the

second constant value which characterizes the regime of the second sub-plateau).

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To explain more clearly the dependence of the behaviour of the indicated congestions on the length ofregion C, let us consider region C of the simulated system to be sub-divided in two open and parallel sub-systems. Each sub-system is constituted of one lane and is connected to the rest of the global system at theboundaries (Fig. 9). This system sub-division is due to the lane-change prohibition in region C. The obstacle oflength Lr leads to the emerging of congestion in the first open-lane system and in the density regimerlowprpr3. Vehicles entering this congested region can only move slowly towards the CS to enter the reducedlane. This means that the capacity of the first open-lane system is limited by the capacities of the reduced laneand the CS. It is to stress here that the traffic flow through the CS has an effect similar to that of the trafficflow arriving from an on-ramp to a main road on a motorway. In our case, the flow of traffic crossing the CSacts on that in the second open-lane system. As it is well known in road traffic, the on-ramps are usually oflength Lramp ¼ 25 sites and act as a local blockage which reduces the capacity of motorway main roads [13,14].But in the case of the implemented scenario of this work, the length of the on-ramp representing the CS is verysmall (Lramp ¼ 2 sites) and the influenced traffic flow on motorway main roads represents that of the secondopen-lane system.

The increase in global system density in the regime of the densities between rlow and r3 increases thecongestion size only in the first open-lane system. Up to the density r3, the congestion on the first open-lanesystem reaches the length of region C and obliges the vehicles arriving at its entry to enter the second open-lane system. This occurs like an overflow of a container liquid. Thus, an important traffic flow from right toleft takes place and the usage of the second open-lane system increases. These results are confirmed in Fig. 10which show the comparison between the lane-change rates n(x) from right to left obtained for the first sub-plateau regime and for the density regime r3prpr4.

After the parallel congestions on both open-lane systems have reached the length equal to that of region C,they continue to increase simultaneously in size with respect to the density in the second sub-plateau regime.

Up to now, the simulation results have shown some interesting advantages of the application of thePZC. These are obtained without taking into account the constraints on maximum speed vmax. The aim of

FiG. 9. Schematic representation of the sub-division of region C into two open-lane systems.

FiG. 10. Lane-change rates n(x) obtained for the right lane by using strategy B. Left: in the first sub-plateau phase (r ¼ 0:09), a high rate

of lane change from right to left is observed only on the positions of the CS. Right: in the density regime r3prpr4 (r ¼ 0:22), the highrate of lane change from right to left is observed at both boundaries of region C.

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Section 3.3 is to present the effect of the limitation of vmax in region C during application of the PZC tovehicles in the vicinity of the PRL.

3.3. Reduction in the maximum speed in region C

Now, the maximal speed is reduced to vmax ¼ 3 in region C (in both open-lane systems). In the case of realtraffic, it is advised to the vehicles to respect the maximum speed limitation before their arrival at the entry ofthe reduced lane. As shown in Fig. 11, the values of the sub-plateaus formed in the fundamental diagram forvmax ¼ 3 do not change in comparison to the case of vmax ¼ 5. Also, a very low increase in the values of criticaldensities rlow and r4 can be observed in the fundamental diagram of the global simulated system if vmax ¼ 3 inregion C (Fig. 11). These low changes in the fundamental diagram are due to the slow movement of thevehicles in both open-lane systems of region C. This delays the arrival of a great number of the vehiclestowards the two parallel congestions, which become, consequently, less compact than those obtained withoutreducing vmax in region C. This result can be also seen near the density r4, where the form of the fundamentaldiagram is low modified due to the limitation of vmax in region C (see Fig. 12).

The additional constraint on maximum speed vmax is very important to apply the regulation procedure PZCin the vicinity of a PRL, because it can allow one to find more spacing at the entry of the reduced lane and toincrease the number of vehicles changing the lane from right to left through the CS. This result is confirmed inFig. 13 which shows an important improvement of lane change through the CS near the density r4, where alow modification in the fundamental diagram has taken place.

Fig. 11. Comparison between the fundamental diagrams obtained for two different values of vmax in region C (vmax ¼ 3 and 5). These

results are obtained by applying the PZC in the vicinity of the PRL.

Fig. 12. Comparison between the density profiles r(x) (on the right lane) obtained by using the PZC without and with the reduction of

vmax in region C of the simulated system. If vmax ¼ 3, then the congestion on the right lane widens and its density decreases (it becomes less

compact). These results are compared for the density r ¼ 0:22 (near r4 where a low modification in the fundamental diagram has taken

place (see Fig. 11)).

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Fig. 13. Comparison between the lane-change rates n(x) obtained without and with the reduction of vmax in region C of the simulated

system. The rate of lane change from right to left through the CS increases if vmax ¼ 3. These results are compared for the density r ¼ 0:22(near r4 where a low modification in the fundamental diagram has taken place (see Fig. 11)).

K. Nassab et al. / Physica A 369 (2006) 841–852 851

4. Conclusions and perspectives

This paper has shown the effect of a PRL on traffic flow in a discrete microscopic model of cellularautomata for a two-lane motorway. The simulation model is based on that of Nagel–Schreckenberg withadditional rules of lane change. The usage of asymmetric rules of lane change separates the simulated systeminto rapid (left lane) and slow (right lane) lanes and helps to simulate the traffic flow in two-lane roads likethose in Germany where the PZC is advised to vehicles near a reduced lane. The simulation results arecompared for two different strategies of lane change in the vicinity of the indicated PRL. A phase transition tothe congested states and an important reduction in the capacity of the simulated system occur. This capacityreduction becomes stronger if the length of the reduced lane is increased.

In order to simulate the PZC, strategy B of the lane change near the region of the PRL has been added tothe simulation model. Within this strategy, the vehicles cannot change the lane in a region (called region C) offinite length behind the obstacle. The vehicles must be inserted from the right lane, where the defect exists, tothe entry on the reduced lane through the CS. The application of the PZC gives more chances to the vehicleson the right lane to exceed the obstacle and to reduce their waiting time before entering the reduced lane. Butthe prohibition of the lane change in region C sub-divides the typical plateau of the fundamental diagram ofthe global simulated system into two sub-plateaus of two different values of the constant average flow. Thewidth of each sub-plateau depends on the length of region C and also on the system size.

The simulation results have shown that the properties of two-lane traffic can change completely due tooccurrence of a PRL, and they have also shed light on two important additional constraints which canimprove the efficiency of the PZC in the simulation model: (i) the minimal safety spacing gapback near the entryof the reduced lane and (ii) the limitation of the maximum speed vmax in region C of the simulated system.These constraints increase the rate of lane change (insertion of the vehicles) from right to left at the entry of thereduced lane through the CS.

Despite the above-mentioned improvement of the lane-change rate through the CS during application of thePZC with both additional constraints on gapback and vmax, the capacity of the global simulated system isdecreased. To increase this capacity, it is useful to take into account other additional constraints in thesimulation model. In particular, it is possible to reduce the length of region C and to increase the number ofthe CS. Finally, the results of this paper can be important and useful in the description of the traffic flow onmotorways where lane reduction occurs often due to road works.

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