Mean-field theory for the Nagel-Schreckenberg model with overtaking strategy
MMean-field theory for the Nagel-Schreckenbergmodel with overtaking strategy
Zhu Su ∗ , Weibing Deng , Jihui Han , Wei Li , Xu Cai National Engineering Laboratory for Technology of Big Data Applications inEducation, Central China Normal University, Wuhan, P. R. China Complexity Science Center, Institute of Particle Physics, Central China NormalUniversity, Wuhan, P. R. China School of Computer and Communication Engineering, Zhengzhou University ofLight Industry, Zhengzhou, P.R. ChinaCorresponding address: [email protected]
May 2018
Abstract.
Based on the Nagel-Schreckenberg (NS) model with periodic boundaryconditions, a modified model considered overtaking strategy (NSOS) has been proposed[1, 2]. In this paper, we focus on the theoretical analysis of traffic flow for NSOS modelby using mean-field method. In the special case of v max = 1 where vehicles cannot overtake preceding ones, the features of stationary state can be obtained exactly.However, in the case of v max > p .
1. Introduction
Various dynamical models [3, 4] have been proposed to explain the complexityphenomena generated by traffic flow. From the microscopic point of view, the vehiculartraffic system can be regarded as being composed of interacting particles driven farfrom equilibrium, each individual vehicle is represented by a particle, and the waythat they influence others’ movement is treated as the interactions among particles.Therefore, vehicular traffic offers the possibility to study various fundamental aspectsof the dynamics of non-equilibrium systems which are of interest in statistical physics.During the last two decades, cellular automata (CA) [5] have obtained popularitydue to their simplicity and their ability to simulate large networks. One of the earlyCA based on traffic models is the NS model [6] developed by Nagel and Schreckenberg,and then a large amount of improved versions have been proposed by imposing someconditions on NS model to make it more realistic. In the NS model, the road is dividedinto sites, each site can be either empty or occupied, and all the space, time and velocities a r X i v : . [ n li n . C G ] J un ean-field theory for the Nagel-Schreckenberg model with overtaking strategy v ( v = 0 , . . . , v max ), where v max is the maximum velocity. To obtain the system’s statein the next time, one could adopt the following rules to all vehicles at the same time(parallel dynamics): (1) The first step is an acceleration process, if a vehicle’s velocity( v ) is lower than the speed limit ( v max ), its velocity is advanced by one. (2) The secondstep is designed to avoid accidents, if two adjacent vehicles have h empty sites betweeneach other, and the following vehicle has a speed larger than h , then its velocity isreduced to h . (3) The third step is considered random braking, a noise with probability p to reduce the velocity of a moving vehicle ( v >
0) to v −
1. (4) The last step dealswith the vehicle’ movement, which enables the position of a vehicle to be advanced byits speed v .Considered the overtaking case, the NSOS model has been proposed [1, 2], whereevery vehicle could be an overtaking one with probability q at each time step. The modelexists due to the following facts: (1) Overtaking obviously happens a lot, especiallywhen the preceding vehicles move quite slowly. (2) The overtaking vehicles are used togo back to the original lane once overtake successfully. (3) When the preceding vehiclesare overtaken, they would slow down rather than accelerate.Several approaches for analytical descriptions of the NS model have been studied[7, 8, 9, 10, 11, 12, 13], and they yield the exact solution for the special case of v max = 1and are good approximations for higher values in the case of v max >
1. In this paper, weuse mean-field method [9] to analyze the NSOS model. The general express equationshave been obtained in the stationary state in the case of v max >
2. In the special caseof v max = 1, where overtaking vehicles can not overtake the preceding ones, the detailresults in the stationary state have been done. In order to find the qualitatively differentbehaviors in the case of v max ≥
2, we calculate the results for v max = 2 where overtakinghappens. Figure 1. (Color online) Schematic graph of updating and moving in the NSOSmodel. The numbers in the sites represent the velocities after moving. The red carsare overtaking vehicles (left). The second car is an overtaking vehicle, it has tried toovertake the preceding one but failed (right). The first car is an overtaking vehicle, ithas successfully overtook the preceding one.
The paper is organized as follows: First briefly introduce the update rules of theNSOS model in Section 2, then analyze this model by applying mean-field method inSection 3, and the summary is discussed in the final Section. ean-field theory for the Nagel-Schreckenberg model with overtaking strategy
2. NSOS model
NSOS model is based on the NS model which considering overtaking strategy withprobability q [1, 2]. In this model, overtaking vehicles are picked up randomly withprobability q at every time, and they overtake preceding ones depending on theirconfigurations in the next time. Some assumptions have been considered as followings:(1) Overtaking vehicles would brake if they reached the same location with theirpreceding ones in the next time to avoid collisions. (2) Each overtaking vehicle isonly able to overtake one per time step. (3) The overtaking vehicle locates in frontof preceding one once overtake successfully. Similar to NS model, all the overtakingvehicles decrease by one with braking probability p except for the successful overtakingones. For convenience, we name the vehicles which are not overtaking as ordinary ones,and update their velocities according to the NS model. The detailed updating rules ofthe NSOS model are as follows: At time t , the j th vehicle becomes an overtaking vehicle with probability q , otherwiseit is an ordinary one. Update the velocity: (I)
If the j th vehicle is an ordinary one: (1) Acceleration: v ( j, t ) → min ( v ( j, t ) + 1 , v max ). (2) Deceleration: v ( j, t ) → min ( v ( j, t ) , d ( j, t )). (3) Random braking: v ( j, t ) → max ( v ( j, t ) − ,
0) with the probability p . (II) If the j th vehicle is an overtaking one: (1) Acceleration: v ( j, t ) → min ( v ( j, t ) + 1 , v max ). (2) If v ( j, t ) > d ( j, t ) + v ( j + 1 , t + 1), the position d ( j, t ) + v ( j + 1 , t + 1) + 1is empty and the ( j + 1)th vehicle does not overtake successfully, (i) Overtaking: v ( j, t ) → d ( j, t ) + v ( j + 1 , t + 1) + 1 . (3) Otherwise, (i)
Deceleration: v ( j, t ) → min ( d ( j, t ) + v ( j + 1 , t + 1) − a, v ( j, t )). (ii) Random braking with probability p : v ( j, t ) → min ( v ( j, t ) − , Movement: x ( j, t + 1) = x ( j, t ) + v ( j, t ).Here, v ( j, t ) denotes the velocity of the j th vehicle at time t and x ( j, t ) denotes itscorresponding position. The number of empty sites in front of the j th vehicle is denotedby d ( j, t ) = x ( j + 1 , t ) − x ( j, t ) −
1. To avoid collisions, we assume a = 2 if the ( j + 1)th ean-field theory for the Nagel-Schreckenberg model with overtaking strategy a = 1. An illustration of theNSOS model can be found in Fig. 1. The numbers in the sites stand for their velocitiesafter moving. In the left graph, the second car is an overtaking vehicle, it has tried toovertake the preceding one but failed. In the right graph, the first car is an overtakingvehicle, which has successfully overtook the preceding one.Since the velocity of overtaking vehicle at time t + 1 is relative to its precedingone, we should know the preceding vehicle’s velocity at time t + 1 first. Fortunately,the velocity of ordinary vehicle is independent of the preceding one, so we could pickup ordinary vehicles and update their velocity first, and then the rear vehicles. In oursimulations, we use parallel update and periodic boundary conditions, assume the firstand last vehicles as ordinary ones all the time, update their velocities first, and thenupdate others’ velocities from ( N − th car to the second one.
3. Mean-field theorem
The simplest analytical approach to the NS model is a microscopic mean-field (MF)theory [9]. Here one considers the probability c α ( i, t ) of vehicles with velocity α atsite i and time t , and we denote the probability that there is no vehicle at site i ( i = 1 , , , ..., L ) at time t by d ( i, t ). In the MF approach, correlations between sites arecompletely neglected. Therefore one has the normalization condition for all sites andall time steps, d ( i, t ) + v max (cid:88) α =0 c α ( i, t ) = 1 . (1)Denoting with c ( i, t ) the total probability for site i to be occupied at time step t ,i.e. (cid:80) v max α =0 c α ( i, t ), one simply has d ( i, t ) + c ( i, t ) = 1. In our model each car could be anordinary one with probability ¯ q (here, ¯ q = 1 − q ) or overtaking one with probability q .Therefore we have the equation c = c ∗ + c † , where c ∗ = qc denotes the probability of beingovertaking vehicles and c † = ¯ qc denotes the probability of being ordinary vehicles. Sincethe update rules of overtaking vehicles relate to the next time positions of precedingones, the configurations of them should be obtained first. Here we adopt C ( i + j, t )( D ( i + j, t ) = 1 − C ( i + j, t )) as the probability that there is (not) a vehicle at site i + j ( j = 1 , , , ..., v max ) at the next time before one updates the site i at time t , need tosay that, here, we just consider the vehicles in front of site i . We update the ordinaryvehicles first, and choose the site of an ordinary vehicle as a starting site to update therear vehicles.According to the update rules, the time evolution of these probability distributionscan be described by the following sets of equations:(1) For the ordinary vehicles, the update rules are the same with the ones in theoriginal NS model, therefore the MF equations for the stationary state ( t → ∞ ) read[9] : ean-field theory for the Nagel-Schreckenberg model with overtaking strategy c † = ( c + pd ) c † + (1 + pd ) c v max (cid:88) β =1 c † β ,c † α = d α [¯ pc † α − + (¯ pc + pd ) c † α + (¯ p + pd ) c v max (cid:88) β = α +1 c † β ] , < α < v max c † v max = ¯ pd v max ( c † v max − + c † v max ) . (2)(2) For the overtaking vehicles ( v max > c ∗ ( i, t ) = 0 ,c ∗ α ( i, t ) = c ∗ α − ( i, t ) , < α < v max c ∗ v max ( i, t ) = c ∗ v max ( i, t ) + c ∗ v max − ( i, t ) . (3)(ii) The deceleration stage: c ∗ ( i, t ) = c ∗ ( i, t ) ,c ∗ ( i, t ) = D ( i + 1 , t ) c ∗ ( i, t ) + D ( i + 1 , t ) C ( i + 2 , t ) c ∗ ( i, t ) ,c ∗ α ( i, t ) = c ∗ α ( i, t ) α (cid:89) j =1 D ( i + j, t ) (cid:124) (cid:123)(cid:122) (cid:125) I + c ∗ α +1 ( i, t ) α (cid:89) j =1 D ( i + j, t ) C ( i + α + 1 , t ) (cid:124) (cid:123)(cid:122) (cid:125) II + C ( i + α + 2 , t ) C ( i + α + 1 , t ) α (cid:89) j =1 D ( i + j, t ) v max (cid:88) β = α +2 c ∗ β ( i, t ) (cid:124) (cid:123)(cid:122) (cid:125) III + C ( i + α − , t ) D ( i + α, t ) α − (cid:89) j =1 D ( i + j, t ) v max (cid:88) β = α c ∗ β ( i, t ) (cid:124) (cid:123)(cid:122) (cid:125) IV , < α < v max c ∗ v max ( i, t ) = c ∗ v max ( i, t ) v max (cid:89) j =1 D ( i + j, t ) + C ( i + v max − , t ) × D ( i + v max , t ) v max − (cid:89) j =1 D ( i + j, t ) c ∗ v max ( i, t ) . (4)(iii) The braking stage: ean-field theory for the Nagel-Schreckenberg model with overtaking strategy c ∗ ( i, t ) = c ∗ ( i, t ) + pc ∗ ( i, t ) ,c ∗ ( i, t ) = ¯ pc ∗ ( i, t ) + p [ c ∗ ( i, t ) − C ( i + 1 , t ) D ( i + 2 , t ) v max (cid:88) β =2 c ∗ β ( i, t )] ,c ∗ α ( i, t ) = p [ c ∗ α +1 ( i, t ) − C ( i + α, t ) D ( i + α + 1 , t ) α − (cid:89) j =1 D ( i + j, t ) v max (cid:88) β = α +1 c ∗ β ( i, t )] (cid:124) (cid:123)(cid:122) (cid:125) I + ¯ p [ c ∗ α ( i, t ) − C ( i + α − , t ) D ( i + α, t ) α − (cid:89) j =1 D ( i + j, t ) v max (cid:88) β = α c ∗ β ( i, t )] (cid:124) (cid:123)(cid:122) (cid:125) II + C ( i + α − , t ) D ( i + α, t ) α − (cid:89) j =1 D ( i + j, t ) v max (cid:88) β = α c ∗ β ( i, t ) (cid:124) (cid:123)(cid:122) (cid:125) III , < α < v max c ∗ v max ( i, t ) = ¯ p [ c ∗ v max ( i, t ) − C ( i + v max − , t ) D ( i + v max , t ) v max − (cid:89) j =1 D ( i + j, t ) c ∗ v max ( i, t )]+ C ( i + v max − , t ) D ( i + v max , t ) v max − (cid:89) j =1 D ( i + j, t ) c ∗ v max ( i, t ) . (5)(iv) The motion stage: c ∗ α ( i, t + 1) = c ∗ α ( i − α, t ) . ≤ α ≤ v max (6)Since the overtaking vehicle is able to overtake the preceding one, it has moreconfigurations than an ordinary vehicle. In the deceleration stage (ii), the item Idescribes that the overtaking vehicle at site i with velocity α will keep its velocity if thepreceding one do not locate in these α sites in the next time. If the preceding vehiclewill locate at the site i + α + 1 in the next time, the vehicle with velocity α + 1 decreasesone to avoid collision (item II). The NSOS model forbids to overtake more than one, sostage (ii) contains an item III. If an overtaking vehicle overtakes its preceding vehicle, itwill close to the overtaken one according to our model, this contributes a situation withvelocity α when a vehicle overtakes one with velocity α − p expect for successful overtaking vehicles, so one obtains an item I which the velocitywith α + 1 of an overtaking vehicle minuses one with probability p except for successfulovertaking ones. Similarly, item II can be obtained without braking and the successfulovertaking vehicle with velocity α also has a contribution to this stage (item III).Even though these time evolution equations are nonlinear, in the limit t → ∞ , the C and D distributions become homogeneous in space (for periodic boundary conditions). ean-field theory for the Nagel-Schreckenberg model with overtaking strategy C ’s values apart from the time and site dependences tocalculate the traffic flow. c ∗ = pDc ∗ + pDCc ∗ + pDC v max (cid:88) β =2 c ∗ β ,c ∗ = ¯ pDc ∗ + (¯ pDC + pD ) c ∗ + pD Cc ∗ + pD C v max (cid:88) β =3 c ∗ β ,c ∗ α = D α − [(¯ pD + C ) c ∗ α − + (¯ pDC + pD + C ) c ∗ α + (¯ pDC + pD + 1) Cc ∗ α +1 + (¯ pDC + pD C + 1) C v max (cid:88) β = α +2 c ∗ β ] , < α < v max c ∗ v max = D v max − (¯ pD + C )( c ∗ v max + c ∗ v max − ) . (7)These equations are linear when we apply the relation C = 1 − D . So the equations(2) and (7) can be recast in matrix form as M (cid:126)c = (cid:126)c . The matrix M can be read offfrom (2) and (7), (cid:126)c is the vector with elements c α , α = 0 , . . . , v max . For small v max onecan calculate the probability c α explicitly.Since c † = ¯ qc , c ∗ = qc and c = c † + c ∗ , using the equations (2) and (7), we couldobtain the specific form of c α (0 ≤ α ≤ v max ). Moreover, combined with the equationof flow f ( c, p, q ) = v max (cid:80) α =1 αc α , we can calculate the flow as a function of p , q and c . Next,we calculate the flow in the case of v max = 1 and v max = 2, respectively. v max = 1No overtaking vehicles could overtake the preceding one in the limit of v max = 1. Infact, it becomes the takeover case which is discussed in the paper [14], that is to say,overtaking vehicles could advance to the position that was occupied by their precedingones at the previous time step.According to the update steps of the NSOS model, the time evolutions of theseprobability distributions can be described by the following four sets of equations:(1) For the ordinary vehicles with v max = 1, the MF equations for the stationarystate ( t → ∞ ) are given by [9] c † = ( c + pd ) c † ,c † = ¯ pdc † . (8)with c † = c † + c † .(2) For the overtaking vehicles: ean-field theory for the Nagel-Schreckenberg model with overtaking strategy q = 0 q = 0 . 2 5 f c Figure 2. (Color online) Fundamental diagram flow f vs density c for maximumvelocity v max = 1 in the mean-field approximation in the case of p = 0 .
25. Black lineis in the case of q = 0, and red line is in the case of q = 0 . S i m u l a t i o n M F f c Figure 3. (Color online) Fundamental diagram for v max = 1, p = 0 .
25 and q = 0 . c ∗ = pDc ∗ ,c ∗ = ¯ pDc ∗ . (9)Since c † = ¯ qc , c ∗ = pc and c = c † + c ∗ , we have ean-field theory for the Nagel-Schreckenberg model with overtaking strategy c = ( c + pd )¯ qc + pqDc,c = ¯ p ¯ qdc + ¯ pqDc. (10)The flow f ( c, p, q ) in the case v max = 1 is f ( c, p, q ) = c = ¯ p ¯ qdc + ¯ pqDc. (11)According to the definition of D , which denotes the empty possibility in the nexttime, considered the mean-field approximation it equals to the sum of distance d oftwo neighborhood vehicles with the flow f , i.e. D ≈ d + f . So the equation becomes f ( c, p, q ) = ¯ pdc − ¯ pqc . (12)The first information from this equation is that the flow is dominated by brakingprobability p , while the overtaking probability q is not the important factor since q hasa factor ¯ pc . Specially, the flow becomes the form of NS model when q = 0. Anotherfinding is that the flow is enhanced mainly in the jammed phase. If we keep the p and q invariant, with the growth of density c , the denominator decreases monotonically, whilethe value of flow increases larger. This result can be explained in physical terms. Inthe free flow phase, all the vehicles move freely which means the preceding ones has noinfluence to overtaking vehicles, the impact of D on the flow is the same as the one of d . While in the jammed phase, overtaking vehicles have larger probabilities to movethan ordinary ones due to D > d , so the flow in this phase is larger than the one of NSmodel. Moreover, the larger q induces, the quicker flow increases.The mean-field result yields, compared with the simulation data shown in Fig. 3,much too small values of the flow. This can easily be understood since the reduction toa single vehicles problem ignores all spatial correlations of the vehicles [9]. v max = 2It is the simplest case that overtaking vehicles could overtake the preceding ones inthe NSOS model in the case of v max = 2. Moreover, it is natural without innatenesshypothesis that an overtaking vehicle is only able to overtake one each time and occupythe site just in front of its preceding one if it could overtake successfully. So it isnecessary to calculate the exact equations in the case of v max = 2. Using the (2)-(5)equations, we have:(1) For the ordinary vehicles: c † = (1 + pd ) c − pd c † ,c † = ¯ p (1 − ¯ pd ) d − pd c † ,c † = ¯ p d − pd c † . (13) ean-field theory for the Nagel-Schreckenberg model with overtaking strategy q = 0 q = 0 . 2 5 f c Figure 4. (Color online) Fundamental diagram flow f vs. density c for maximumvelocity v max = 2 in the mean-field approximation in the case of p = 0 .
25. Black lineis in the case of q = 0, and red line is in the case of q = 0 . S i m u l a t i o n M F f c Figure 5. (Color online) Fundamental diagram for v max = 2, p = 0 .
25 and q = 0 . (2) For the overtaking vehicles: c ∗ = pDC − pD c ∗ ,c ∗ = ¯ pC + pD (1 − pD )1 − pD Dc ∗ ,c ∗ = (¯ pD + C )(1 − pD )1 − pD Dc ∗ . (14) ean-field theory for the Nagel-Schreckenberg model with overtaking strategy C + D = 1 and c ∗ + c ∗ = c ∗ − c ∗ . Again, we could calculate c α ( α = 0 , ,
2) using c α = c † α + c ∗ α and c † = ¯ qc , c ∗ = qc . But this time we assume D ≈ d due to the value of D − d is smaller than d . The flow can be calculated using thefollowing equation f ( c, p, q ) = ( c † + c ∗ ) + 2( c † + c ∗ ). The result is shown in Fig. 4. Onecould also observe that the flow of the NSOS model enlarged in the jammed regime thanthat of original NS model, this would be due to overtaking mechanism is beneficial todevelop the traffic flow. Again, the mean-filed result is still less than simulation data(Fig. 5), and one could observe that the maximum flow density does not coincide withthe simulation result, this may be the result of our simplicity D .
4. Conclusions
In this paper theoretical analysis of the NSOS model is performed by using the mean-field method, the equations for v max = 1 can be obtain exactly, while for larger valuesof v max they are just approximations. Even though mean-field theory is insufficient dueto the important correlations between neighboring sites are neglected, the reason thatwhy the NSOS model can improve traffic flow in the area where the flow exceed themaximum flow density has been explained, and braking probability as a major factorthat influence transition density has been discovered.
5. Acknowledgments
This work was supported by China Postdoctoral Science Foundation (Grant Nos.30205010003), Fundamental Research Funds for the Central Universities (Grant Nos.20205170444), and National Natural Science Foundation of China (Grant Nos. 11505071and 61702207).
Appendix A: Takeover Case
In this appendix we show that the stationary state of the NSOS model with v max = 1for overtaking vehicles:(i) The acceleration stage: c ∗ ( i, t ) = 0 ,c ∗ ( i, t ) = c ∗ ( i, t ) + c ∗ ( i, t ) . (15)(ii) The deceleration stage: c ∗ ( i, t ) = c ∗ ( i, t ) ,c ∗ ( i, t ) = D ( i + 1 , t ) c ∗ ( i, t ) . (16)(iii) The braking stage: ean-field theory for the Nagel-Schreckenberg model with overtaking strategy c ∗ ( i, t ) = c ∗ ( i, t ) + pc ∗ ( i, t ) ,c ∗ ( i, t ) = ¯ pc ∗ ( i, t ) . (17)(iv) The motion stage: c ∗ α ( i, t + 1) = c ∗ α ( i − α, t ) , ≤ α ≤ . (18)In the stationary state, distributions become homogeneous in space for periodicboundary conditions, so the site dependence could be omitted. Using this and combiningthe four update steps one gets the set of equations (10). Appendix B: Overtaking Case
In this appendix we show that the stationary state of the NSOS model with v max = 2for overtaking vehicles:(i) The acceleration stage: c ∗ ( i, t ) = 0 ,c ∗ ( i, t ) = c ∗ ( i, t ) ,c ∗ ( i, t ) = c ∗ ( i, t ) + c ∗ ( i, t ) . (19)(ii) The deceleration stage: c ∗ ( i, t ) = c ∗ ( i, t ) ,c ∗ ( i, t ) = D ( i + 1 , t ) c ∗ ( i, t ) + D ( i + 1 , t ) C ( i + 2 , t ) c ∗ ( i, t ) ,c ∗ ( i, t ) = D ( i + 1 , t ) D ( i + 2 , t ) c ∗ ( i, t ) + C ( i + 1 , t ) D ( i + 2 , t ) c ∗ ( i, t ) . (20)(iii) The braking stage: c ∗ ( i, t ) = c ∗ ( i, t ) + pc ∗ ( i, t ) ,c ∗ ( i, t ) = ¯ pc ∗ ( i, t ) + p [ c ∗ ( i, t ) − C ( i + 1 , t ) D ( i + 2 , t ) c ∗ ( i, t )] ,c ∗ ( i, t ) = ¯ p [ c ∗ ( i, t ) − C ( i + 1 , t ) D ( i + 2 , t ) c ∗ ( i, t )] + C ( i + 1 , t ) D ( i + 2 , t ) c ∗ ( i, t ) . (21)(iv) The motion stage: c ∗ α ( i, t + 1) = c ∗ α ( i − α, t ) , ≤ α ≤ . (22)In the stationary state, distributions become homogeneous in space for periodicboundary conditions, so the site dependence could be omitted. Using this and combiningthe four update steps one gets the set of equations (14). ean-field theory for the Nagel-Schreckenberg model with overtaking strategy References [1] Zhu S, Weibing D, Jihui H, Wei L and Xu C 2016
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