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Impacts of different types of ramps on the traffic flow
ARTICLE IN PRESS
Physica A 352 (2005) 601–611 www.elsevier.com/locate/physa
Impacts of different types of ramps on the traffic flow K. Nassaba,b, M. Schreckenberga, S. Ouaskitb, A. Boulmakoulc, a
Theoretische Physik Fakulta¨t 4, Universita¨t Duisburg-Essen, D-47048 Duisburg, Germany Laboratoire de Physique de la Matie`re Condense´e, Faculte´ des Sciences ben M’sik, BP 7955 Ben M’Sik, Casablanca, Maroc c Laboratoire Informatique des Syste`mes de Transport, Faculte´ des Sciences et Techniques de Mohammedia, B.P. 146 Mohammedia, Maroc b
Received 10 November 2004 Available online 14 December 2004
Abstract The impact of the on- and off-ramps in a cellular automaton model for the traffic flow is studied. We include to the model the effect of spacing between the on- and the off-ramps on a same periodic road at a intersection (interchange) with another road. First, we use the Nagel–Schreckenberg (NaSch) model (J. Phys. I 2 (1992) 2221) without modifications to extract the basic phenomena of traffic flow, and in the following step we focus our investigation on the NaSch model with velocity-dependent randomization (VDR model) (Eur. Phys. J. B 5 (1998) 793) to examine the other system behaviors. Our results provide evidence that the metastable states and the phase separation can occur in the same way like in the models with local site defects. r 2004 Published by Elsevier B.V.
1. Introduction Increasing traffic jams have negative impacts on humans and environment. Especially in densely populated regions the capacity of the existing road networks is often exceeded. Therefore, the optimization of the traffic networks usage is an Corresponding author. Fax: +212 23 31 53 53.
E-mail address: [email protected] (A. Boulmakoul). 0378-4371/$ - see front matter r 2004 Published by Elsevier B.V. doi:10.1016/j.physa.2004.11.044
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important aspect of infra-structural planning. In many cases of experimental and theoretical investigations [1–4] of the traffic flow it is obvious that congested traffic states can be linked to a large variety of possible bottlenecks, e.g., on- and off-ramps, road constructions, and lane reductions. It has been provided [5–7] that the ramps reduce the capacity on highway networks, lead to the transitions from free to congested flow, and have similar effects as those of local perturbations. Usually, the on- and off-ramps are connected to the intersections between freeways and the other freeways or roads. The high demand and the traffic movement within interchanges can cause serious problems at the ramps. In fact, the short distance between the on- and the off-ramps on a same road and at the same interchange influences the movement of the vehicles entering and leaving the same road. To the best of our knowledge, the influence of the spacing between the on- and off-ramps on the traffic dynamics is not yet analyzed in detail by using the Nagel–Schreckenberg (NaSch) model [8]. The most important point that is focused on the present paper is the effect of the distance between the on- and the off-ramps on the same road and at the same interchange. This distance depends on the geometry of the road interchanges, and might change the behavior of the traffic flow. In order to find out and to analyze these changes of the traffic flow behavior, we use the NaSch model on a simple one-dimensional lattice with periodic boundary conditions and with an additional on- and off-ramp. As mentioned, the impact of the on-ramp in the NaSch model is similar to the effect of a local blockage which can separate the system into different phases depending on the global density and which leads to the formation of jams. But one of the wellknown experimental result [9,10], which cannot be obtained by the NaSch model, is the occurrence of the metastable states near the density of the maximum flow. These states can be explained by the fundamental diagram with a form similar to the mirror image of the Greek letter ‘‘l’’. Recently, this form of the fundamental diagram is found by using modifications in the rules of the NaSch model. In this article, we focus a part of our investigations on the NaSch model with velocity-dependent randomization (VDR model) [11]. It turns out that the VDR model can extract very interessant properties (e.g., the wide moving jams and the stop-and-go) of the traffic flow near the on- and off-ramps which are localized on a same road interchange. The article is organized as follows. First, the definition of the model will be considered. Second, the effect of two different distances between the on- and offramps by using the NaSch model will be discussed. Third, the existence of the wide moving jams and the stop-and-go waves will be researched using the VDR model for a chosen distance between the on- and off-ramps. Finally, conclusion and discussions will be presented.
2. Definition of the model In this section, we will briefly present the definition of the NaSch model which is applied to a periodic single lane (a one-dimensional lattice) with ramps. The lattice is
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divided into cells of length 7.5 m. Each cell can either be empty or occupied by just one vehicle (particle). The speed of each particle can take an integer value v ðv ¼ 0; 1; . . . ; vmax Þ: The state of the lattice at time t þ 1 can be obtained from that of the time step t by applying the following rules to all particles at the same time (parallel dynamics): Step Step Step Step
1: 2: 3: 4:
Acceleration: vn ! minðvn þ 1; vmax Þ: Braking: vn ! minðvn ; d n 1Þ: Randomization with probability PðvÞ: vn ! maxðvn 1; 0Þ: Movement: xn ! xn þ vn :
Here, d n ¼ xn þ 1 xn denotes the distance to the next particle ahead, and p controls the velocity fluctuations of moving particles. The length of the on- and off-ramps is chosen to Lramp ¼ 25 sites, and the particles can leave (enter) the system at the off-ramp (on-ramp) with the probability pout (pin ). The spacing between the on- and the off-ramps is noted Dx ¼ xon xoff (see Fig. 1 left). The schematic representations of the on- and off-ramps are shown (see Fig. 1, right). Because we focus our interest on the effect of the spacing between the on- and off-ramps on the same road and at the same interchange, we consider that these ramps separate the road into two sections (these sections are connected between them by an interchange and form a one-dimensional lattice). Using a simple strategy of the activity of the on- and off-ramps, the lattice will be searched in the region of the on-ramp (xon to xon þ Lramp ) until an empty site is found. Then a particle will be inserted into this site, and will take the maximal velocity vmax (see Fig. 1, right). In the same way, an occupied site will be searched in the region of the off-ramp (xoff to xoff þ Lramp ), and its particle will be removed (see Fig. 1, right). Note that an inserted particle into the system can lead to strong fluctuations which depend on the global density.
Fig. 1. Schematic representation of a road interchange (left), an off-ramp (right), and an on-ramp (right). Dx ¼ xon xoff denotes the spacing between the on- and off- ramps. With respect to the forward direction of the traffic flow, the first site on the off-ramp can be located before (Dx40) or after (Dxo0) that of the on-ramp.
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3. Simulations results In this section, we discuss the effects of the ramps and the distance separating them. The activity of the ramps is characterized by the rate of particles entering (pin ) and leaving (pout ) the system per unit time. By using the NaSch model we set p to 0.05, and we keep the other parameters of the model fixed (L ¼ 2000; vmax ¼ 5; and pin ¼ pout ¼ 0:1). 3.1. Fundamental diagram The fundamental diagrams of the NaSch model for three values of the spacing Dx between the on- and off-ramps are shown in Fig. 2. Without effects of the ramps and with only the defects due to the particles (p ¼ 0:05) the system can be driven to the state of the maximum flow near the global density r ¼ 0:16: For comparison, the fundamental diagram in this case is shown as a solid line. The dependence of the flow q on the global density r changes significantly when the ramps are activated. Therefore, a density regime rlow ororhigh is observed where the flow qðrÞ is independent of the density. For the same density regime, the value of the flow is lower than that found in the model without ramps, and changes by changing the value of spacing Dx between the on- and the off-ramps. An increase of the probabilities pin and pout causes a decrease in the plateau value. This result is qualitatively in accord with the results in Refs. [5,12], and can be explained by the activity of the on-ramp as a local defect. If one compares the fundamental diagrams obtained for different values of the distance Dx; it is clearly seen that the plateau value for Dx ¼ 70 is lower than that for Dxo0: The case of
Fig. 2. Fundamental diagrams (q ¼ f ðrÞ) obtained by the NaSch model for two different values of the distance Dx between the on- and off-ramps. The plateau value for Dxo0 is lower than that for Dx ¼ 70: The other parameters are given by L ¼ 2000; p ¼ 0:05; pin ¼ Pout ¼ 0:1; and vmax ¼ 5:
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Dxo0 explains that the position xon of the first site on the on-ramp is localized before the position xoff of the first site on the off-ramp in the forward direction of the main traffic flow. Note that these geometries are motivated by the dimensions found on German highways. The decrease of the plateau value induced by the change of the distance Dx can be explained by the probability that the site (on the on-ramp) in front of the inserted particle is already occupied. This probability is higher for Dxo0 than that for Dx ¼ 70; and leads to the slowing down of the particles. Up to a certain density rlow ; the flow in the system is obviously free, and increases with the increase of the density. Above the density rhigh the system is characterized by the congested states, and the flow decreases by increasing the density. For the regime between the densities rlow and rhigh ; the flow is constant and is limited by the capacity of the ramps and by the impact of the spacing between the on- and offramps. 3.2. Density profiles and space-time plots The behavior of the average flow observed on the fundamental diagram can be understood by looking at the density profiles. Fig. 3 shows the density profile for the system with the distance Dx ¼ 70: The system self-organizes into a macroscopic high density pinned at the on-ramp. The width of this region increases if the global density increases, and is limited by the effect of the off-ramp. So far the macroscopic properties are comparable to results obtained by the NaSch model with a distributed defect on a region of lenght Ld 41 site [13]. In contrast, the density profiles of the model with the distance Dxo0 (see Fig. 4) show another behavior. In high- and low-density phases, one can only observe local deviations from a constant profile. For intermediate densities, a separation into macroscopic high- and low-density regions appear, and the length of these regions
Fig. 3. Density profiles in three different phases for the model with the distance Dx ¼ 70: In the low- and high-density phases only local inhomogeneities occur on the region between the ramps. But, for intermediate densities, the system self-organizes into two regions of different densities. The model parameters are given by L ¼ 2000; p ¼ 0:05; pin ¼ Pout ¼ 0:1; and vmax ¼ 5:
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changes by increasing the global density. These results are analogous to those obtained by the NaSch model with a local defect of lenght Ld ¼ 1 site [14]. If we look at the microscopic structure (see Fig. 5) of the interchange region in both cases of the distance Dx (for intermediate densities), we see that all jams emerging near the on-ramp grow by moving backwards for Dxo0: But a part of these dissipates near the off-ramp for Dx ¼ 70; because the off-ramp is localized before the on-ramp in the forward direction of the main traffic flow. 3.3. Wide moving traffic jams and stop-and-go waves Obviously, the NaSch model has extracted the phenomenon of the emergence of jams near the ramps for both cases of the spacing Dx; and has demonstrated that the
Fig. 4. Density profiles in three different phases for the model with the distance Dxo0: In the low- and high-density phases only local inhomogeneities occur on the region between the ramps, but for intermediate densities one observes a phase separation. The model parameters are the same as that for Dx ¼ 70:
Fig. 5. Space–time plots of the system near the ramps show how the system behaves near the density of the maximum flow for both cases of the distance Dx around the position x ¼ 1000: The high-density region is compact for Dxo0 (left), and is mixed with low-density regions if the jams move far from the ramps for Dx ¼ 70 (right). The NaSch model parameters are given by L ¼ 2000; vmax ¼ 5; p ¼ 0:05; pin ¼ pout ¼ 0:1; and rstart ¼ 0:2:
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ramps at the same road interchange act as a local defect which influences the traffic dynamics (particularly) near the global density of the maximum flow. These results are qualitatively analogous to those found in Refs. [5,12]. But various theoretical and experimental studies [11,15] have found out that the existence of localized perturbations leads to the occurrence of the metastable states and the moving jams. Very recently, such states have been reproduced in a modified version of the NaSch model (the VDR model). It turns out that the VDR model can extract these properties in our simulated system with on- and off-ramps on the same road and at the same interchange. Due to the considerable change in the macroscopic behaviors of the traffic flow caused by the distance Dxo0 in the NaSch model, we restrict ourselves to this case (Dxo0) in the further simulations. Note that the randomization parameter PðvÞ depends on the velocity, whereas it is constant in the NaSch model. This parameter will be to determined before the acceleration ‘‘step 1’’ of the NaSch model. For simplicity we use the following rule: PðvÞ ¼ p0 for v ¼ 0;
and
PðvÞ ¼ p for v40 .
Starting from the global density rstart ¼ 0:2; setting the braking probability of the moving particles to p ¼ 0:01; and independently of the braking probability p0 of the stopping particles (this case corresponds to the NaSch model without modifications) we can clearly see that a sequence of small jams (see Fig. 6) occurs near the ramps. The downstream front of the emerged jams which are separated by very negligible regions of the free flow is fixed at the region of ramps. In contrast to the NaSch model, the VDR model where p0 ¼ 0:4 and p ¼ 0:01 has extracted two interesting phenomena. We can see these properties in the space–time plot for rstart ¼ 0:2 (see Fig. 6): (1) wide moving jams which can be formed out of the region of ramps, and which moves through the system, and (2) a jams pattern which consists of jams alternating with the free-flow regions. This form of jams is similar to the so-called ‘‘stop-and-go’’ phenomenon, and the downstream front of each jam of the stop-and-go waves is fixed at the region of ramps.
Fig. 6. Space–time plots of the system with Dxo0 at the ramps and near the global density of the maximum flow. In contrast to the NaSch model (left), the VDR model (right) leads to the formation of the wide moving jams and the jams waves which alternate with the free flow regions. This results are obtained by starting from the global density rstart ¼ 0:2: The other model parameters are given by L ¼ 2000; vmax ¼ 5; p ¼ 0:01; pin ¼ pout ¼ 0:1; and p0 ¼ 0:4:
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The formation of wide moving jams can be explained by the effect of the probability p0 which leads to the growing of spontaneously emerging jams. In fact, the particles arrive to the spontaneous jam (occurring due to the velocity fluctuations) and must be stopped inside the jam during a certain time interval. This phenomenon can be obtained in the same manner by the VDR model without ramps, but only one large moving jam can occur. The stop-and-go waves cannot be reproduced by the VDR model on a system without local defects, e.g. the on- and off-ramps. The term ‘‘small jams’’ is related to the width of the jams. A look at the distributions of the jams width and of the gaps in Figs. 7 and 8 reveals some interesting remarks. In the NaSch model without modifications, the width of the jams is only small, and the spacing between a jam and another jam or a single particle can take one value of the large interval of the gaps. In contrast, in the VDR model the jams of different high values of the width can be formed. Because a considerable number of particles are blocked within the width jams, and the spacing between a jam and another jam or a single particle can be very large compared to those obtained by the NaSch model. The pattern of the stop-and-go waves change by varying the values of the parameters p; p0 ; and pin (see Fig. 9). In fact, the jams can be mostly very small if the probability p is decreased, and two types of jams (jams of short and long lifetimes) can occur and can be separated by the growing regions of the free flow. Qualitatively, by varying the parameter p0 the jams of the stop-and-go waves have the analogous behavior to the case of the probability p; and the large jams moving through the system grow considerably if p0 is increased. For smaller rates of the entering and leaving particles at the ramps (in the case where pin ¼ pout ), the jams are rare. This occurs because after a jam is emerged, it is then rapidly dissipated. And due to the velocity fluctuations the large jams occur and move through the ramps
Fig. 7. Comparison between the jams widths distributions found in the both models (the NaSch and VDR models). The NaSch model (VDR model) is represented by the symbol ‘‘o’’ (‘‘’’). For rstart ¼ 0:2; L ¼ 2000; p ¼ 0:01; p0 ¼ 0:4 and pin ¼ pout ¼ 0:1; the jams can be only small in the NaSch model, and can be small or very large in the VDR model.
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Fig. 8. Comparison between the distributions of gaps found in the both models (the NaSch and VDR models). The NaSch model (VDR model) is represented by the symbol ‘‘’’ (‘‘’’). For the same parameters of the Fig. 7, the gap can take one of the large interval value of the spacings, but only in the case of the VDR this spacing is large enough.
Fig. 9. Overview of the phenomena induced by varying the model parameters, e.g., the variation of the width of the jams and of free flow regions.
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region without modifications. For higher in- and out-flow rates at the ramps, the width of the jams becomes large and can collide with the small jams (if these are not rapidly dissipated).
4. Conclusion and discussion In this paper, we have analyzed the effect of ramps and the spacing Dx between them on the traffic dynamics in microscopic models. The basic model used here is the well-known NaSch model. We have compared the two types of the distance DxðDx40 and Dxo0Þ between the on- and off-ramps on a same road and at the same road interchange. By using the NaSch model without modifications we have found the basic phenomena of the traffic flow analogous to those found for the systems with local defects, e.g., the plateau formation in the fundamental diagram, the reduction of the road capacity, and the phase transition from the free flow to the congested states. Qualitatively, the effects of both types of the distance Dx are similar. But only the plateau value for Dxo0 is lower than that for Dx40: This difference can be explained by the difference of the probability that the inserted vehicles slow down, because the sites in front of them are already occupied. Other important phenomena in the traffic flow which are observed in the VDR model are not observed here by using the NaSch model, e.g. the wide moving jams and the stop-and-go waves. That is the reason why we have used the VDR model to extract the other properties of the traffic in our simulated system. We have restricted our interest to the system with the spacing Dxo0 due to its remarkable effect on the road capacity. The VDR model shows that near the density of the maximum flow, large moving jams and stop-and-go waves occur. The wide moving jams can emerge out of the ramps region and be propagated through the system, and the stop-and-go waves are formed near the ramps. The width of the jams and the free-flow regions separating them is changed depending on the parameters of the model, e.g., the breaking probability of stopped and moved vehicles, and the rates of removed and inserted vehicles. Our results show that the system properties can change completely, if the change of spacing between the on- and off-ramps is taken into account. These results might be extended to other complex geometries of the road interchanges.
References [1] [2] [3] [4] [5] [6] [7] [8]
B.S. Kerner, H. Rehborn, Phys. Rev. E 53 (1996) R4275. B.S. Kerner, H. Rehborn, Phys. Rev. Lett. 79 (1997) 4030. L. Neubert, L. Santen, A. Schadschneider, M. Schreckenberg, Phys. Rev. E 60 (1999) 6480. H.Y. Lee, H.-W. Lee, D. Kim, Phys. Rev. Lett. 81 (1998) 1130. G. Diedrich, L. Santen, A. Schadschneider, J. Zittartz, Int. J. Mod. Phys. C 11 (2000) 335. H.Y. Lee, H.-W. Lee, D. Kim, Phy. Rev. E 59 (1999) 5101. D. Helbing, M. Treiber, Phys. Rev. Lett. 81 (1998) 3042. K. Nagel, M. Schreckenberg, J. Phys. I 2 (1992) 2221.
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[9] D.E. Wolf, M. Schreckenberg, A. Bachem (Eds.), Traffic and Granular Flow, World Scientific, Singapore, 1996. [10] B.S. Kerner, in: D. Helbing, H.J. Hermann, M. Schreckenberg, D.E. Wolf (Eds.), Traffic and Granular Flow, Springer, Berlin, 2000, p. 253. [11] R. Barlovic, L. Santen, A. Schadschneider, M. Schreckenberg, Eur. Phys. J. B 5 (1998) 793. [12] K. Nassab, R. Barlovic, M. Schreckenberg, S. Ouaskit, Physica A, submitted for publication. [13] L. Santen, Dissertation Thesis, Universita¨t zu Ko¨ln, 1999. [14] W. Knospe, L. Santen, A. Schadschneider, M. Schreckenberg, in: M. Schreckenberg, D.E. Wolf (Eds.), Traffic and Granular Flow, Springer, Singapore, 1998, p. 349. [15] S. KrauX, P. Wagner, C. Gawron, Phys. Rev. E 55 (1997) 5597.
Physica A 352 (2005) 601–611 www.elsevier.com/locate/physa
Impacts of different types of ramps on the traffic flow K. Nassaba,b, M. Schreckenberga, S. Ouaskitb, A. Boulmakoulc, a
Theoretische Physik Fakulta¨t 4, Universita¨t Duisburg-Essen, D-47048 Duisburg, Germany Laboratoire de Physique de la Matie`re Condense´e, Faculte´ des Sciences ben M’sik, BP 7955 Ben M’Sik, Casablanca, Maroc c Laboratoire Informatique des Syste`mes de Transport, Faculte´ des Sciences et Techniques de Mohammedia, B.P. 146 Mohammedia, Maroc b
Received 10 November 2004 Available online 14 December 2004
Abstract The impact of the on- and off-ramps in a cellular automaton model for the traffic flow is studied. We include to the model the effect of spacing between the on- and the off-ramps on a same periodic road at a intersection (interchange) with another road. First, we use the Nagel–Schreckenberg (NaSch) model (J. Phys. I 2 (1992) 2221) without modifications to extract the basic phenomena of traffic flow, and in the following step we focus our investigation on the NaSch model with velocity-dependent randomization (VDR model) (Eur. Phys. J. B 5 (1998) 793) to examine the other system behaviors. Our results provide evidence that the metastable states and the phase separation can occur in the same way like in the models with local site defects. r 2004 Published by Elsevier B.V.
1. Introduction Increasing traffic jams have negative impacts on humans and environment. Especially in densely populated regions the capacity of the existing road networks is often exceeded. Therefore, the optimization of the traffic networks usage is an Corresponding author. Fax: +212 23 31 53 53.
E-mail address: [email protected] (A. Boulmakoul). 0378-4371/$ - see front matter r 2004 Published by Elsevier B.V. doi:10.1016/j.physa.2004.11.044
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important aspect of infra-structural planning. In many cases of experimental and theoretical investigations [1–4] of the traffic flow it is obvious that congested traffic states can be linked to a large variety of possible bottlenecks, e.g., on- and off-ramps, road constructions, and lane reductions. It has been provided [5–7] that the ramps reduce the capacity on highway networks, lead to the transitions from free to congested flow, and have similar effects as those of local perturbations. Usually, the on- and off-ramps are connected to the intersections between freeways and the other freeways or roads. The high demand and the traffic movement within interchanges can cause serious problems at the ramps. In fact, the short distance between the on- and the off-ramps on a same road and at the same interchange influences the movement of the vehicles entering and leaving the same road. To the best of our knowledge, the influence of the spacing between the on- and off-ramps on the traffic dynamics is not yet analyzed in detail by using the Nagel–Schreckenberg (NaSch) model [8]. The most important point that is focused on the present paper is the effect of the distance between the on- and the off-ramps on the same road and at the same interchange. This distance depends on the geometry of the road interchanges, and might change the behavior of the traffic flow. In order to find out and to analyze these changes of the traffic flow behavior, we use the NaSch model on a simple one-dimensional lattice with periodic boundary conditions and with an additional on- and off-ramp. As mentioned, the impact of the on-ramp in the NaSch model is similar to the effect of a local blockage which can separate the system into different phases depending on the global density and which leads to the formation of jams. But one of the wellknown experimental result [9,10], which cannot be obtained by the NaSch model, is the occurrence of the metastable states near the density of the maximum flow. These states can be explained by the fundamental diagram with a form similar to the mirror image of the Greek letter ‘‘l’’. Recently, this form of the fundamental diagram is found by using modifications in the rules of the NaSch model. In this article, we focus a part of our investigations on the NaSch model with velocity-dependent randomization (VDR model) [11]. It turns out that the VDR model can extract very interessant properties (e.g., the wide moving jams and the stop-and-go) of the traffic flow near the on- and off-ramps which are localized on a same road interchange. The article is organized as follows. First, the definition of the model will be considered. Second, the effect of two different distances between the on- and offramps by using the NaSch model will be discussed. Third, the existence of the wide moving jams and the stop-and-go waves will be researched using the VDR model for a chosen distance between the on- and off-ramps. Finally, conclusion and discussions will be presented.
2. Definition of the model In this section, we will briefly present the definition of the NaSch model which is applied to a periodic single lane (a one-dimensional lattice) with ramps. The lattice is
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divided into cells of length 7.5 m. Each cell can either be empty or occupied by just one vehicle (particle). The speed of each particle can take an integer value v ðv ¼ 0; 1; . . . ; vmax Þ: The state of the lattice at time t þ 1 can be obtained from that of the time step t by applying the following rules to all particles at the same time (parallel dynamics): Step Step Step Step
1: 2: 3: 4:
Acceleration: vn ! minðvn þ 1; vmax Þ: Braking: vn ! minðvn ; d n 1Þ: Randomization with probability PðvÞ: vn ! maxðvn 1; 0Þ: Movement: xn ! xn þ vn :
Here, d n ¼ xn þ 1 xn denotes the distance to the next particle ahead, and p controls the velocity fluctuations of moving particles. The length of the on- and off-ramps is chosen to Lramp ¼ 25 sites, and the particles can leave (enter) the system at the off-ramp (on-ramp) with the probability pout (pin ). The spacing between the on- and the off-ramps is noted Dx ¼ xon xoff (see Fig. 1 left). The schematic representations of the on- and off-ramps are shown (see Fig. 1, right). Because we focus our interest on the effect of the spacing between the on- and off-ramps on the same road and at the same interchange, we consider that these ramps separate the road into two sections (these sections are connected between them by an interchange and form a one-dimensional lattice). Using a simple strategy of the activity of the on- and off-ramps, the lattice will be searched in the region of the on-ramp (xon to xon þ Lramp ) until an empty site is found. Then a particle will be inserted into this site, and will take the maximal velocity vmax (see Fig. 1, right). In the same way, an occupied site will be searched in the region of the off-ramp (xoff to xoff þ Lramp ), and its particle will be removed (see Fig. 1, right). Note that an inserted particle into the system can lead to strong fluctuations which depend on the global density.
Fig. 1. Schematic representation of a road interchange (left), an off-ramp (right), and an on-ramp (right). Dx ¼ xon xoff denotes the spacing between the on- and off- ramps. With respect to the forward direction of the traffic flow, the first site on the off-ramp can be located before (Dx40) or after (Dxo0) that of the on-ramp.
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3. Simulations results In this section, we discuss the effects of the ramps and the distance separating them. The activity of the ramps is characterized by the rate of particles entering (pin ) and leaving (pout ) the system per unit time. By using the NaSch model we set p to 0.05, and we keep the other parameters of the model fixed (L ¼ 2000; vmax ¼ 5; and pin ¼ pout ¼ 0:1). 3.1. Fundamental diagram The fundamental diagrams of the NaSch model for three values of the spacing Dx between the on- and off-ramps are shown in Fig. 2. Without effects of the ramps and with only the defects due to the particles (p ¼ 0:05) the system can be driven to the state of the maximum flow near the global density r ¼ 0:16: For comparison, the fundamental diagram in this case is shown as a solid line. The dependence of the flow q on the global density r changes significantly when the ramps are activated. Therefore, a density regime rlow ororhigh is observed where the flow qðrÞ is independent of the density. For the same density regime, the value of the flow is lower than that found in the model without ramps, and changes by changing the value of spacing Dx between the on- and the off-ramps. An increase of the probabilities pin and pout causes a decrease in the plateau value. This result is qualitatively in accord with the results in Refs. [5,12], and can be explained by the activity of the on-ramp as a local defect. If one compares the fundamental diagrams obtained for different values of the distance Dx; it is clearly seen that the plateau value for Dx ¼ 70 is lower than that for Dxo0: The case of
Fig. 2. Fundamental diagrams (q ¼ f ðrÞ) obtained by the NaSch model for two different values of the distance Dx between the on- and off-ramps. The plateau value for Dxo0 is lower than that for Dx ¼ 70: The other parameters are given by L ¼ 2000; p ¼ 0:05; pin ¼ Pout ¼ 0:1; and vmax ¼ 5:
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Dxo0 explains that the position xon of the first site on the on-ramp is localized before the position xoff of the first site on the off-ramp in the forward direction of the main traffic flow. Note that these geometries are motivated by the dimensions found on German highways. The decrease of the plateau value induced by the change of the distance Dx can be explained by the probability that the site (on the on-ramp) in front of the inserted particle is already occupied. This probability is higher for Dxo0 than that for Dx ¼ 70; and leads to the slowing down of the particles. Up to a certain density rlow ; the flow in the system is obviously free, and increases with the increase of the density. Above the density rhigh the system is characterized by the congested states, and the flow decreases by increasing the density. For the regime between the densities rlow and rhigh ; the flow is constant and is limited by the capacity of the ramps and by the impact of the spacing between the on- and offramps. 3.2. Density profiles and space-time plots The behavior of the average flow observed on the fundamental diagram can be understood by looking at the density profiles. Fig. 3 shows the density profile for the system with the distance Dx ¼ 70: The system self-organizes into a macroscopic high density pinned at the on-ramp. The width of this region increases if the global density increases, and is limited by the effect of the off-ramp. So far the macroscopic properties are comparable to results obtained by the NaSch model with a distributed defect on a region of lenght Ld 41 site [13]. In contrast, the density profiles of the model with the distance Dxo0 (see Fig. 4) show another behavior. In high- and low-density phases, one can only observe local deviations from a constant profile. For intermediate densities, a separation into macroscopic high- and low-density regions appear, and the length of these regions
Fig. 3. Density profiles in three different phases for the model with the distance Dx ¼ 70: In the low- and high-density phases only local inhomogeneities occur on the region between the ramps. But, for intermediate densities, the system self-organizes into two regions of different densities. The model parameters are given by L ¼ 2000; p ¼ 0:05; pin ¼ Pout ¼ 0:1; and vmax ¼ 5:
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changes by increasing the global density. These results are analogous to those obtained by the NaSch model with a local defect of lenght Ld ¼ 1 site [14]. If we look at the microscopic structure (see Fig. 5) of the interchange region in both cases of the distance Dx (for intermediate densities), we see that all jams emerging near the on-ramp grow by moving backwards for Dxo0: But a part of these dissipates near the off-ramp for Dx ¼ 70; because the off-ramp is localized before the on-ramp in the forward direction of the main traffic flow. 3.3. Wide moving traffic jams and stop-and-go waves Obviously, the NaSch model has extracted the phenomenon of the emergence of jams near the ramps for both cases of the spacing Dx; and has demonstrated that the
Fig. 4. Density profiles in three different phases for the model with the distance Dxo0: In the low- and high-density phases only local inhomogeneities occur on the region between the ramps, but for intermediate densities one observes a phase separation. The model parameters are the same as that for Dx ¼ 70:
Fig. 5. Space–time plots of the system near the ramps show how the system behaves near the density of the maximum flow for both cases of the distance Dx around the position x ¼ 1000: The high-density region is compact for Dxo0 (left), and is mixed with low-density regions if the jams move far from the ramps for Dx ¼ 70 (right). The NaSch model parameters are given by L ¼ 2000; vmax ¼ 5; p ¼ 0:05; pin ¼ pout ¼ 0:1; and rstart ¼ 0:2:
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ramps at the same road interchange act as a local defect which influences the traffic dynamics (particularly) near the global density of the maximum flow. These results are qualitatively analogous to those found in Refs. [5,12]. But various theoretical and experimental studies [11,15] have found out that the existence of localized perturbations leads to the occurrence of the metastable states and the moving jams. Very recently, such states have been reproduced in a modified version of the NaSch model (the VDR model). It turns out that the VDR model can extract these properties in our simulated system with on- and off-ramps on the same road and at the same interchange. Due to the considerable change in the macroscopic behaviors of the traffic flow caused by the distance Dxo0 in the NaSch model, we restrict ourselves to this case (Dxo0) in the further simulations. Note that the randomization parameter PðvÞ depends on the velocity, whereas it is constant in the NaSch model. This parameter will be to determined before the acceleration ‘‘step 1’’ of the NaSch model. For simplicity we use the following rule: PðvÞ ¼ p0 for v ¼ 0;
and
PðvÞ ¼ p for v40 .
Starting from the global density rstart ¼ 0:2; setting the braking probability of the moving particles to p ¼ 0:01; and independently of the braking probability p0 of the stopping particles (this case corresponds to the NaSch model without modifications) we can clearly see that a sequence of small jams (see Fig. 6) occurs near the ramps. The downstream front of the emerged jams which are separated by very negligible regions of the free flow is fixed at the region of ramps. In contrast to the NaSch model, the VDR model where p0 ¼ 0:4 and p ¼ 0:01 has extracted two interesting phenomena. We can see these properties in the space–time plot for rstart ¼ 0:2 (see Fig. 6): (1) wide moving jams which can be formed out of the region of ramps, and which moves through the system, and (2) a jams pattern which consists of jams alternating with the free-flow regions. This form of jams is similar to the so-called ‘‘stop-and-go’’ phenomenon, and the downstream front of each jam of the stop-and-go waves is fixed at the region of ramps.
Fig. 6. Space–time plots of the system with Dxo0 at the ramps and near the global density of the maximum flow. In contrast to the NaSch model (left), the VDR model (right) leads to the formation of the wide moving jams and the jams waves which alternate with the free flow regions. This results are obtained by starting from the global density rstart ¼ 0:2: The other model parameters are given by L ¼ 2000; vmax ¼ 5; p ¼ 0:01; pin ¼ pout ¼ 0:1; and p0 ¼ 0:4:
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The formation of wide moving jams can be explained by the effect of the probability p0 which leads to the growing of spontaneously emerging jams. In fact, the particles arrive to the spontaneous jam (occurring due to the velocity fluctuations) and must be stopped inside the jam during a certain time interval. This phenomenon can be obtained in the same manner by the VDR model without ramps, but only one large moving jam can occur. The stop-and-go waves cannot be reproduced by the VDR model on a system without local defects, e.g. the on- and off-ramps. The term ‘‘small jams’’ is related to the width of the jams. A look at the distributions of the jams width and of the gaps in Figs. 7 and 8 reveals some interesting remarks. In the NaSch model without modifications, the width of the jams is only small, and the spacing between a jam and another jam or a single particle can take one value of the large interval of the gaps. In contrast, in the VDR model the jams of different high values of the width can be formed. Because a considerable number of particles are blocked within the width jams, and the spacing between a jam and another jam or a single particle can be very large compared to those obtained by the NaSch model. The pattern of the stop-and-go waves change by varying the values of the parameters p; p0 ; and pin (see Fig. 9). In fact, the jams can be mostly very small if the probability p is decreased, and two types of jams (jams of short and long lifetimes) can occur and can be separated by the growing regions of the free flow. Qualitatively, by varying the parameter p0 the jams of the stop-and-go waves have the analogous behavior to the case of the probability p; and the large jams moving through the system grow considerably if p0 is increased. For smaller rates of the entering and leaving particles at the ramps (in the case where pin ¼ pout ), the jams are rare. This occurs because after a jam is emerged, it is then rapidly dissipated. And due to the velocity fluctuations the large jams occur and move through the ramps
Fig. 7. Comparison between the jams widths distributions found in the both models (the NaSch and VDR models). The NaSch model (VDR model) is represented by the symbol ‘‘o’’ (‘‘’’). For rstart ¼ 0:2; L ¼ 2000; p ¼ 0:01; p0 ¼ 0:4 and pin ¼ pout ¼ 0:1; the jams can be only small in the NaSch model, and can be small or very large in the VDR model.
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Fig. 8. Comparison between the distributions of gaps found in the both models (the NaSch and VDR models). The NaSch model (VDR model) is represented by the symbol ‘‘’’ (‘‘’’). For the same parameters of the Fig. 7, the gap can take one of the large interval value of the spacings, but only in the case of the VDR this spacing is large enough.
Fig. 9. Overview of the phenomena induced by varying the model parameters, e.g., the variation of the width of the jams and of free flow regions.
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region without modifications. For higher in- and out-flow rates at the ramps, the width of the jams becomes large and can collide with the small jams (if these are not rapidly dissipated).
4. Conclusion and discussion In this paper, we have analyzed the effect of ramps and the spacing Dx between them on the traffic dynamics in microscopic models. The basic model used here is the well-known NaSch model. We have compared the two types of the distance DxðDx40 and Dxo0Þ between the on- and off-ramps on a same road and at the same road interchange. By using the NaSch model without modifications we have found the basic phenomena of the traffic flow analogous to those found for the systems with local defects, e.g., the plateau formation in the fundamental diagram, the reduction of the road capacity, and the phase transition from the free flow to the congested states. Qualitatively, the effects of both types of the distance Dx are similar. But only the plateau value for Dxo0 is lower than that for Dx40: This difference can be explained by the difference of the probability that the inserted vehicles slow down, because the sites in front of them are already occupied. Other important phenomena in the traffic flow which are observed in the VDR model are not observed here by using the NaSch model, e.g. the wide moving jams and the stop-and-go waves. That is the reason why we have used the VDR model to extract the other properties of the traffic in our simulated system. We have restricted our interest to the system with the spacing Dxo0 due to its remarkable effect on the road capacity. The VDR model shows that near the density of the maximum flow, large moving jams and stop-and-go waves occur. The wide moving jams can emerge out of the ramps region and be propagated through the system, and the stop-and-go waves are formed near the ramps. The width of the jams and the free-flow regions separating them is changed depending on the parameters of the model, e.g., the breaking probability of stopped and moved vehicles, and the rates of removed and inserted vehicles. Our results show that the system properties can change completely, if the change of spacing between the on- and off-ramps is taken into account. These results might be extended to other complex geometries of the road interchanges.
References [1] [2] [3] [4] [5] [6] [7] [8]
B.S. Kerner, H. Rehborn, Phys. Rev. E 53 (1996) R4275. B.S. Kerner, H. Rehborn, Phys. Rev. Lett. 79 (1997) 4030. L. Neubert, L. Santen, A. Schadschneider, M. Schreckenberg, Phys. Rev. E 60 (1999) 6480. H.Y. Lee, H.-W. Lee, D. Kim, Phys. Rev. Lett. 81 (1998) 1130. G. Diedrich, L. Santen, A. Schadschneider, J. Zittartz, Int. J. Mod. Phys. C 11 (2000) 335. H.Y. Lee, H.-W. Lee, D. Kim, Phy. Rev. E 59 (1999) 5101. D. Helbing, M. Treiber, Phys. Rev. Lett. 81 (1998) 3042. K. Nagel, M. Schreckenberg, J. Phys. I 2 (1992) 2221.
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[9] D.E. Wolf, M. Schreckenberg, A. Bachem (Eds.), Traffic and Granular Flow, World Scientific, Singapore, 1996. [10] B.S. Kerner, in: D. Helbing, H.J. Hermann, M. Schreckenberg, D.E. Wolf (Eds.), Traffic and Granular Flow, Springer, Berlin, 2000, p. 253. [11] R. Barlovic, L. Santen, A. Schadschneider, M. Schreckenberg, Eur. Phys. J. B 5 (1998) 793. [12] K. Nassab, R. Barlovic, M. Schreckenberg, S. Ouaskit, Physica A, submitted for publication. [13] L. Santen, Dissertation Thesis, Universita¨t zu Ko¨ln, 1999. [14] W. Knospe, L. Santen, A. Schadschneider, M. Schreckenberg, in: M. Schreckenberg, D.E. Wolf (Eds.), Traffic and Granular Flow, Springer, Singapore, 1998, p. 349. [15] S. KrauX, P. Wagner, C. Gawron, Phys. Rev. E 55 (1997) 5597.