Xiaoling Cui, Xiaoyu Chen and Song Xiao*

(1. School of Mechanical & Vehicle Engineering, Linyi University, Linyi 276000, Shandong China;2. Laboratory of Electromagnetic Processes of Materials, Northeastern University, Shenyang 110004, China)

Abstract: The effect of the exit control feedback policy on traffic flow was investigated in this paper.Here, the exit rate(β)can be defined as a function of the hopping rate(p), the current(J)and the bulk density(ρbulk), which can be rewritten as β= p-J/ρbulk.A model based on normal totally asymmetric simple exclusion process(TASEP)has been analyzed by mean field approach.It is found that a phase transformation point exists in the phase diagram, which is determined by p. In addition, the traffic flow of the system achieves maximum current when the exit rate maintains itself atβ= p/2 for all other phases except the low density(LD)phase.The result implies that we can use the control feedback policy to make the traffic flow reach the maximum value when the traffic system is in the traffic jam status.

Keywords: TASEP; exit control feedback; phase diagram

0 Introduction

Recently, many people investigated a model for transport phenomena called the totally asymmetric simple exclusion process(TASEP)with applications to study surface growth[1], protein synthesis[2], molecular motors[3], vehicular traffic[4], etc.The TASEP model is a simplified version of ASEP, where particles are allowed to move stochastically and independently along a particular direction(generally, from left to right).Because of its conceptual simplicity, TASEP and its extension models are applied in vehicular traffic modeling, such as various updates and lattice geometries applied on a one-dimensional single lane TASEP[5-12].

Kolomeisky[5]and Liu et al.[6]studied TASEP with open boundary conditions with one defect site by different updated rules, random rule, and updated rule, respectively.Based on Ref.[5], Shaw et al.[2]investigated arbitrary size particles in TASEP with a single defect site, indicating the steady phase diagrams are identical to that in standard TASEP with shifts of phase boundaries.Subsequently, Liu et al.[7]and Xiao et al.[8]explored TASEP models with a range of consecutive defect sites by random and parallel updated rules.The results show that the steady state diagrams are more complex.More recently, two refined mean field methods have been employed to examine the influence of a single defect site near the lattice boundary on the steady state properties[9].Both methods can describe TASEP with stronger molecular interactions better, and the obtained results agree with simulations.

TASEP with ramps can be used to depict vehicular traffic.At present, many studies are focused on the TASEP with ramps.Some important results have been obtained[10-12], such as zone inhomogeneity and on-ramp on TASEP.The existence of the MC/MC, MC/LD and MC/HD phases in the steady phase diagram resulted from the different relationship between hopping ratepand on-ramp rateq[10].

More recently, many researchers have focused on traffic management strategies because these policies are vital for lightening traffic congestions and improving traffic flows at bottlenecks.A junction can be defined as a place where traffic alters different lines, directions, or joints.However, it always becomes system’s bottleneck due to the low efficiency of traffic flow in all directions.A junction of two parallel lanes merged into a single lane has been studied by TASEP model[13].An improved domain wall method has been introduced to discuss the first-order phase boundaries due to the density correlations.Afterward, a multi-input multi-output junction was investigated by TASEP with the parallel updated rule[14], displaying the potential applications of a junction.

Nevertheless, in reality, vehicles’ different road priority in traffic(or bandwidth requirements in internet traffic)results in unequal injection rates.Therefore, the influence of unequal entrance rates with junction[15]on the traffic has been explored, implying that the stationary-state phases depended on the entrance rate of the first sub-lane.Based on this model, the influences of hopping rates on the system have also been obtained[16], namely, the phase diagram structure qualitatively varied with different hopping rate.Consequently, they were extensively investigated by TASEP[17-25].Traffic management policies always control the entrance(α)or exit(β)rate of particles(vehicles)according to the lane state, for instance, the lane density[18-22].In the light of the lattice density, the variation of input rate was introduced by Woelki[22]with the TASEP approach.This method is understood as a type of ramp metering in vehicular traffic[26-27].

However, in reality, there are many road network nodes in the traffic system, which will become bottlenecks if the control strategy is weak.We explore the system through the TASEP model with the exit feedback control strategy to improve traffic flows at road network nodes.This paper considers a feedback control strategy for a single lane TASEP model and investigates how the exit control strategy affects the traffic flow rate.All possible input rates at which the system achieves maximum possible current under feedback will be shown.

1 Model Description

In this section, TASEP with open boundaries under exit control feedback policy will be described, as illustrated in Fig.1.

Fig.1 Graphic view of TASEP with open boundaries under the exit control feedback policy

The system containsLsites in one dimensional lattice, and there are only two states, occupied and empty, for each site.A particle which is at siteiwill attempt to go to sitei+1 with ratep, only the sitei+1 is vacant.Note that the particle will remain at sitei, if sitei+1 is occupied.In addition, particles can enter and exit the system from the left boundary(empty)and the right boundary(occupied)with ratesαandβ, respectively[28].Importantly, the leaving rateβis defined as the average over the occupancy probability of allLsites(ρi)as follows:

(1)

Obviously, the system will reduce to the normal TASEP when the exit control feedback policy is canceled.All kinetic properties can be given by exact solutions[29], and the system characteristics are as follows.

For low density(LD)phase, the entrance rate dominates its kinetic whenαandβsatisfyα<β,α

(2)

For the high density(HD)phase, the kinetics depends on the exiting process asβ<α,β

(3)

At last, the maximum current(MC)phase is related to the large entrance and exit ratesα>p/2,β>p/2.Therefore, we can describe the particle current and densities as

(4)

2 Theoretical Analysis

The assumption that the sensors on various sections of the road are able to measure the average occupancy of sites is applied in our study.Clearly, the control feedback policy was used to lead to the system achieving maximum flow rate whenever possible.According to Eqs.(2)-(4), the recurrence relationship for the average occupancies of each steady phase can be obtained by using mean field approximation as follows:

(5)

Therefore, Eq.(5)can be used to simplify Eq.(1).We note that the grand totalρiexists in any phase,ρbulkdominates over the values at the endpoints, and the limit average densities converge toρbulk.AsLis large enough, it yields

(6)

According to Eq.(2),J/ρbulk=p-αis obtained when the system is in the LD phase, implying that

β=α

(7)

While according to Eq.(3), we can obtainJ/ρbulk=βwhen the system lies in the HD phase, indicating that

β=p/2

(8)

At last, when the system is in the MC phase,J/ρbulk=p/2 is obtained according to Eq.(4), which again leads to

β=p/2

(9)

Therefore, the effect of exit control policy is a collapse of the phase transition diagram Fig.2(a)onto the phase transition line as shown in Fig.2(b).Obviously,Jis independent ofβand adjusted based onα.The current of the system will be in one of the two possible phases under the control policy as follows:

(10)

(11)

According to Eqs.(10)and(11), the exit control feedback strategy leads to the system operating at the MC phase even in the HD phase.This behavior is summarized in Table 1.In addition, the coordinate of the PointAin Fig.2(b)is defined as the phase transformation point, and it depends on the hopping ratep, namely,A(α=p/2,J=p/4), as illustrated in Fig.3.Evidently, the left boundary of the MC phase will move towards right with the growth ofp; meanwhile,Jincreases.The current and the entrance rate for PointAsatisfyJ=α/2.

Fig. 2 Exit control feedback policy changes the phase diagram by collapsing the phase diagram on the red line

Fig. 3 Relationship of Point A coordinate with different hopping rates p

Table 1 Effect of exit control feedback policy on one dimensional TASEP

When the density of the system is obtained and equal to 0.6, according toJ/ρbulk=β,β=p/2.4 is obtained.At this time, the parameterspis given as 0.6, and the exit rate can be calculated andβ= 0.25.According to Eq.(3), the density and current of the system can be obtained as 0.67 and 0.134, respectively.Clearly, the theoretical results are approximately equal to our real value.The error is caused by the fact that the correlation between neighboring particles was ignored.

3 Conclusions

The investigation in this paper presents the effect of the exit control feedback policy for TASEP on one-dimensional single lane.According to the exit control feedback strategy, the leaving rateβcan be rewritten asβ=p-J/ρbulk.When the system is in the free flow(the LD phase), the entrance rate satisfiesα

Additionally, the phase transformation PointAcan be described by the coordinate(α=p/2,J=p/4).It is decided by the hopping ratep, and the current and the entrance rate for PointAsatisfyJ=α/2.Note thatJgrows with the increase ofp.

The results indicate that our study model can be employed to control the traffic flow of the highway system.When the density of the highway system reaches high density, the traffic flow decreases sharply.At this time, the speed of vehicles can be adjusted to control the exit rate, thus leading to the traffic flow being in the maximum current and reducing the clogging time.