Velocity control for improving flow through a bottleneck
aa r X i v : . [ n li n . C G ] M a r Velocity control for improving flow through abottleneck
Hiroki Yamamoto , Daichi Yanagisawa and KatsuhiroNishinari Department of Aeronautics and Astronautics, School of Engineering, TheUniversity of Tokyo,7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan Research Center for Advanced Science and Technology, The University of Tokyo,4-6-1 Komaba, Meguro-ku, Tokyo 153-8904, JapanE-mail: [email protected]
January 2016
Abstract.
A bottleneck can largely deteriorate the flow, such as a traffic light or anon-ramp at a road. To alleviate bottleneck situations, one of the important strategiesis to control the input rate to suit the state of the road. In this study, we propose aneffective velocity control of traveling particles, in which the particle velocity dependson the state of a bottleneck. To analyze our method, we modify the totally asymmetricsimple exclusion process (TASEP) and introduce a slow-to-start rule, which we refer toas controlled TASEP in the present paper. Flow improvement is verified in numericalsimulations and theoretical analyses by using controlled TASEP.
1. Introduction
Recently, researchers in non-equilibrium physics have become interested in systems ofactive matter. These systems, with their remarkable self-propulsion features, havebeen vigorously investigated in numerical simulations, experiments, and theories [1–3].Despite their simplicity, such systems describe a wide range of phenomena far fromthermal equilibrium. Among the various models of active matter, the so-calledasymmetric simple exclusion process (ASEP) has attracted much attention as a minimalmodel.The ASEP was pioneered by MacDonald and Gibbs [4, 5], who sought to describethe kinetics of protein synthesis. Since exactly solving the model with open boundaryconditions [6–8], researchers have applied the ASEP and its extensions to diversesystems such as vehicular traffic [9], molecular motor traffic [10–15], exclusive queuingprocesses [16–18], and market systems [19].A simplified version of ASEP called the totally asymmetric simple exclusion process(TASEP), where particles are allowed to hop unidirectionally (left to right in the present elocity control for improving flow through a bottleneck α ) or output ( β ) rate ofparticles (vehicles) according to the lane state, such as the lane density [21–25]. In theTASEP approach of Woelki, the flow is improved by changing the input rate accordingto the lattice density [25]. In vehicular traffic, this method can be interpreted as a formof ramp metering [34,35]. However, these studies assume a constant hopping probability(rate) of particles.Our investigation adopts a different approach. We manage bottleneck flows bychanging a particle velocity according to the state of a bottleneck. In the TASEP, ifthe right-neighboring site of a particle is empty, the particle moves to that site with aconstant hopping probability v . We alter this rule depending on the state of the rightboundary, i.e., the exit of the lattice. Specifically, v is decreased only when the exitbehaves as a bottleneck. This method can be interpreted as variable speed limit (VSL)control [36] or jam-absorption driving [37, 38] in actual vehicular traffic. We note thatrelated models, in which a hopping probability (rate) can differ among particles [39–41]or sites [42, 43], have already been investigated.Additionally, we introduce a slow-to-start (SlS) rule [44–49], which is absent in therelated works [21–25]. The SlS rule represents the delay of restarting a blocked particle,accounting for the law of inertia and the reaction delay in actual situations. Note thatthe TASEP with the SlS rule can reproduce a metastable state, which is observed inreal expressways [48]. Furthermore, by introducing additional rules, we can see a widescattering area near the phase transition region from free flow to jam flow phase in thefundamental diagram [49].A similar approach to our control method was developed by Nishi et al [50]. Intheir model, two open lanes merge into one lane. Particles, obeying the SlS rule, cannotchange their lane and their hopping probability is decreased by particles in the other lane.The decreased hopping probability generates a spontaneous zipper merging, improvingthe particle flux at the merging point.The remainder of the present paper is organized as follows. Sec. II defines our elocity control for improving flow through a bottleneck
2. Model description
The original TASEP with open-boundary conditions is defined as a one-dimensionallattice of L sites, which are labeled from left to right as i = 0 , , ......, L − i th site at time t is occupied by a particle, its state is represented as n i ( t ) = 1; otherwise, its state is n i ( t ) = 0. In the present paper, we adopt discrete time and parallel updating, whichis generally used in traffic contexts. In parallel updating, the states of particles on thelattice are simultaneously determined in the next time step. Particles enter the latticefrom the left boundary with probability α , and leave the lattice from the right boundarywith probability β . In the bulk, if the right-neighboring site is empty, particles hop tothat site with probability v ; otherwise they remain at their present site. If p articlescan hop to multiples sites with acceleration and deceleration under parallel updating inone time step, the TASEP becomes the Nagel-Schreckenberg (NS) model [51]. (cid:1) v v α β L sites (cid:1) (cid:1) Figure 1:
Schematic of the original TASEP with open-boundary conditions. T (cid:1) τ T – τ β = 1 β = 0 β = 1 β = 0 T τ T – τ (cid:1)(cid:1) Figure 2:
Schematic of β ( t ). Our controlled TASEP differs from the original TASEP in several ways. First, inour model, β is not constant, but varies with t as follows. β = β ( t ) = ( nT ≤ t < nT + τ )0 ( nT + τ ≤ t < ( n + 1) T ) , (1)where T ( T ∈ N ) is the length of one cycle, τ (0 ≤ τ ≤ T ) is the length of a periodwhen β = 1 (in the case of traffic lights, this denotes the “green” period), and n ∈ N . elocity control for improving flow through a bottleneck β = 1 ( β = 0) is referred to as an (a) opening(closing) period. A schematic of β ( t ) is presented in Fig. 2. Note that particles at theright boundary, i.e., at the ( L − β = 1, butcannot leave the lattice when β = 0. We define β ∗ as the time-averaged value of β . β ∗ = τT (2) & (cid:1) = 0v = 0 (cid:1) = 1 &Configuration& The value of (cid:1) Hoppingprobability v = p v = 1 (a) blocked blockedunblockedblocked unblockedunblockedat t-1 v = 0 at t unblocked blocked v = (cid:1) s ( β =1 ) sp ( β =0 ) v = (cid:1) ( β =1 ) p ( β =0 ) Particle’s states Hopping probabilitybetween t and t+1 (b)Table 1: (a) Hopping probability of the left particle (indicated by arrows). The particle movesfrom left to right under the control rule. The hopping probability v decreases from1 to p during a closing period ( β = 0). (b) Explanation of the SlS rule for agiven particle. The hopping probability of the particle between time t and t + 1 isdetermined by the particle’s state at time t − t . The labels “blocked” and“unblocked” denote the presence and absence of a particle in the right-neighboringsite, respectively. Next, we introduce a particle control rule. In this rule, hopping probability v depends on the value of β . Particles can move no further than one site ahead (from leftto right) with hopping probability v ∈ { , p, } , where 0 ≤ p ≤
1, as shown in Table 1a.The value of v is determined by the state of the exit and by the state of the particle’sright-neighboring site. First, if the particle’s right-neighboring site is occupied, hoppingprobability v is set to 0. Second, if β = 0 and the particle’s right-neighboring site isempty, hopping probability v is set to p . Finally, if β = 1 and the particle’s right-neighboring site is empty, hopping probability v is set to 1. Under these rules, particlesnot yet involved in a jam near the exit are expected to avoid the jam. In contrast tothe related works [21–25], in which hopping probability is fixed, we adopt a variablehopping law.We introduce a parameter p , representing the degree of the particles’ decelerationfrom 1. When p = 1, there is no control (no deceleration), whereas 0 ≤ p < elocity control for improving flow through a bottleneck t −
1, its hopping probability between time t andtime t + 1 is 0, s , or sp , depending on other conditions (see Table 1b). Here s is the SlScoefficient, which takes values from 0 to 1. A relatively small (large) s indicates large(small) inertia and slow (fast) reaction delay of particles. If s = 0, particles must rest forone time step after being blocked by other particles; if s = 1, there is no SlS effect. Notethat our model is generally difficult to analyze from a theoretical perspective because(i) the SlS rule increases the particle states and (ii) β is time-inhomogeneous.
3. Fundamental and Phase diagram without control
In this section, we investigate and depict the fundamental diagram and phase diagramof the model when p = 1 (indicating no control) to observe the control effects in thefollowing sections.The average bulk density ρ bulk and the average global density ¯ ρ are defined as theaverage number of occupancy over the space [0 . L, . L ] and [1 , L ] in one time step,respectively. Similarly, the average flow Q is defined as the mean number of particlesexiting the right boundary in one time step. As for the fundamental diagram, we observea thousand sets of ( ρ bulk , Q ) for every T × after T × steps. On the other hand,as for the phase diagram we observe ( ¯ ρ , Q ) for T × after T × steps with eachset of ( α , β ∗ ). Note that the simulation period is fixed as [ T × , T × ] from thenext section. By evolving the lattice through 0 ≤ t < T × , we obtain ρ bulk , ¯ ρ , and Q in the steady state. Finally, in both cases, L is fixed as L = 200. The results with L = 1000 are almost the same as those with L = 200. Thus, we choose L = 200 todecrease calculation time. The fundamental diagram of the TASEP with the SlS rule is mainly constituted offree flow phase and jam flow phase. In the free flow phase, where every particle is notaffected by the SlS rule, the average flow Q free is equal to the average bulk density ρ bulk .As ρ bulk exceeds the critical density ρ cr , the lattice is reduced to the jam flow phase,where it is divided into two kinds of regions; a cluster region and a non-cluster region.Note that the lattice is divided into many regions, which are either a cluster region ora non-cluster region. In the cluster region, the average density is equal to 1. In thenon-cluster region, the interval of particles can be two values by the SlS rule; 1 sitewith probability s and 2 site with probability 1 − s . Therefore, the average interval ofparticles is reduced to 2 − s site. As a result, the average density becomes 1 / (3 − s ), whichcorresponds to the critical density ρ cr . Assuming that the total length of non-clusterregions accounts for a ratio x of the lattice, we obtain Eq. (3) below; ρ bulk = 13 − s × x + 1 × (1 − x ) . (3) elocity control for improving flow through a bottleneck Q jam = x/ (3 − s ), we obtain Q jam = 12 − s (1 − ρ bulk ) . (4)Finally, even if ρ bulk exceeds ρ cr , metastable state sometimes appears in the casewhere the SlS rule never occurs, i.e., every particle maintains a gap more than or equalto one site between them. We see the metastable state until ρ bulk becomes 1/2, whereparticles maintain one-site gap between them. In the metastable state, the average flow Q meta is equal to ρ bulk similar to the free flow phase.Eventually, we can describe Q as follows; Q = Q free = ρ bulk for ρ bulk < − s Q jam = − s (1 − ρ bulk ) for ρ bulk > − s Q meta = ρ bulk for − s < ρ bulk < . (5)Figure 3 plots the simulated (red dots) and theoretical (green line) with (a) s = 0and (b) s = 0 .
5. In both figures, the simulated values favorably agree with thetheoretical line, Eq. (5). Note that some of the simulated values which take valuesin Q = ( Q jam , Q meta ] can be obtained only with the value of β ∗ very close to 1. (a) (b)Figure 3: Fundamental diagram for (a) s = 0 and (b) s = 0 .
5, plotting the simulated (reddots) and theoretical (green line) values. The other parameters are set as L = 200, T = 100, and p = 1. The simulated values favorably agree with the theoretical linein the both figures. The theoretical results are obtained by Eq. (5). In this subsection, we discuss and depict the phase diagram of the model. The resultsof the numerical simulation are shown in Fig. 4 by plotting the values of ¯ ρ (Fig. 4a)and Q (Fig. 4b). The other parameters are set as T = 100 and s = 0. elocity control for improving flow through a bottleneck α , β ∗ ). (a) (b)Figure 4: Phase diagram without the control. Color bar indicates the values of ¯ ρ in (a) ( Q in (b)). The other parameters are set as L = 200, T = 100, p = 1, and s = 0. t = nT + 7t = nTt = nT + 6t = nT + 1t = nT + 2t = nT + 3t = nT + 4t = nT + 5 exit exit Figure 5:
Schematic of three sites near the exit in the HD phase. The time series is t = nT to t = nT + 7 (left: SlS rule, right: no SlS rule). The right figure is shown forcomparison. In the left (right) panels, each particle exits the lattice in every 3 (2)time steps in the left (right) figure. In this schematic, three particles accumulatein front of the exit during the closing period. Note that the parameter s is set as s = 0 in the left panel. In the HD phase, Q is governed by β ∗ , the time-averaged value of β , which is definedin Eq. (1). In this case, a jam is often formed in front of the exit. Therefore, all particles elocity control for improving flow through a bottleneck ⌈ τ / ⌉ (= ⌈ β ∗ T / ⌉ ) per cycle. Eventually, Q settles to ⌈ β ∗ T / ⌉ /T . Notethat ⌈ x ⌉ denotes the smallest integer greater than or equal to x .On the other hand, the SlS rule has little effect on the flow in the LD phase, becausethe opening period is long enough to let all particles accumulating in front of the exitevacuate. Therefore, in this phase, Q is governed by the input probability of particlesfrom the left boundary. Consequently, Q is given by α/ (1 + α ), which describes theparticle flow in the original TASEP with parallel updating [8].At the boundary between the LD and HD phases, the flows must match [46].Therefore, we have: (cid:24) β ∗ T (cid:25) T = α α . (6)From Eq. (6), the boundary between the HD and LD phases is described by β ∗ =3 α/ (1 + α ) when τ (= β ∗ T ) is a multiple of 3. On the other hand, when τ = 3 m − τ + 1) / T , where m ∈ N . Similarly, when τ = 3 m − τ + 2) / T .Finally, when α exceeds 1 / β ∗ = 1, particles never stop atthe exit and are never affected by the SlS rule. Therefore, we observe metastable state,where the flow is equal to Q = α/ (1 + α ) exceeding the maximal current 1/3. (a) (b)Figure 6: (a) Simulated (red dots) and theoretical (green line) of Q as a function of α forvarious β ∗ ∈ { . , . , } . (b) Simulated (red dots) and theoretical (green line) of Q as a function of β ∗ for various α ∈ { . , . , } . The other parameters are set as L = 200, T = 100, p = 1, and s = 0. Note that in the case of s = 1 (no SlS effect) we can derive the boundary betweenHD and LD with similar discussion (see Fig. 5) as follows: (cid:24) β ∗ T (cid:25) T = α α . (7) elocity control for improving flow through a bottleneck s changes from s = 1 to s = 0.Figure 6 compares the simulated values and the theoretical line of Q with (a) afixed β ∗ =0.6 or (b) a fixed α =0.2. In Fig. 6, the simulated values favorably agreewith the theoretical line. We also confirm the metastable state whose flow is equal to Q = α/ (1 + α ) when β ∗ = 1 in Fig. 6a. Note that in Fig. 6b the simulated valuesfor β ∗ = 1 seem to be deviated from the theoretical line because those cases are in themetastable state.
4. Investigation of flow improvement with control
In this section, we investigate whether the average flow Q is improved by our control,where we vary the values of p according to the state of a bottleneck. Q is mainlya function of p ; that is, we write Q ( p ). We examine how Q ( p ) changes between thecases of p = 1 and 0 < p <
1. Note that when p = 1, particles never decelerate(i.e., no control), whereas when 0 < p <
1, particles decelerate during a closing period(indicating that the control works). Hereafter, we set β ∗ = 0 . α . We discuss thesimulation results by changing β ∗ ( τ ) in Appendix A. We also review the dependenceof the effect of the control in Appendix B. Q ( p ) for various α In this subsection, we investigate whether Q (0 < p <
1) is improved over the case of Q (1) for various α . Here, we set T = 20 and s = 0 to investigate the cases where wehave the largest SlS effect.Figure 7 shows space-time diagrams of the model for p = 1 and p = 0 .
3. Underthese conditions, only 4 particles per open period can leave the lattice in the uncontrolledcase (the left panel). However, in the controlled case (the right panel), 5 particles leavethe lattice during the second open period, because the control mitigates jamming nearthe exit.Next, we vary the values of α and investigate the improvements of Q ( p ) to themodel. We define a ( p ) as the ratio of the change of Q ( p ) from Q (1); that is, a ( p ) = Q ( p ) − Q (1) Q (1) . (8)From the definition of a ( p ), the cases of a ( p ) > a ( p ) <
0) indicate that the control iseffective (detrimental).Referring to the last section, the uncontrolled flow is Q (1) = α/ (1 + α ) in theLD, and Q (1) = β ∗ / Q (0 < p <
1) from Q (1) for α ∈ { . , . , } . We select these values of α to investigatethe LD case, the case near the boundary and the HD case without the control. The elocity control for improving flow through a bottleneck Figure 7:
Space-time diagrams of the model. The left and right panels show the results ofthe uncontrolled ( p = 1) and controlled ( p = 0 .
3) cases, respectively. Black (white)squares indicate sites of n i ( t ) = 1 ( n i ( t ) = 0). We extract only 20 sites from theright boundary per two cycles (40 time steps) in the steady state with α = 1, β ∗ = 0 .
6, and T = 20. conditions ( α, β ∗ ) ∈ { (0 . , . , (0 . , . , (1 , . } correspond to the LD, HD, and HD,respectively, because the cases α < β ∗ / (3 − β ∗ ) and α > β ∗ / (3 − β ∗ ) represent the LDand HD phases when p = 1. For ( α, β ∗ ) ∈ { (0 . , . , (0 . , . , (1 , . } , Q (1) is equalto 1/6, 1/5, and 1/5, respectively. Figure 8:
Simulated values of a ( p ) as a function of p for various α ∈ { . . } . The other parameters are set as L = 200, β ∗ = 0 . T = 20, and s = 0. The simulated values of a ( p ) are plotted in Fig. 8. We first observe that in the case α = 0 . a ( p ) monotonically increases as p approaches 1, indicating that the controlis detrimental rather than effective. On the other hand, in the cases α =0.4 and 1,the control is effective at appropriate values of p , and there exists an optimal p , p opt ,that maximizes a ( p ). When p is too small, however, a ( p ) decreases because the gapsbetween particles become excessively long and lead to a decrease in the flow. We have elocity control for improving flow through a bottleneck a never exceeds 0 with Woelki’s method (controlling input rateaccording to the lattice density) [25] in our model (see Appendix C for more details) .Since the scope of this study is to investigate the effect of our control, we hereafterset α = 1, with which the most remarkable flow improvement is observed.Note that under the control, Q ( p ) can take β ∗ / β ∗ / a ( p ) istheoretically 0.5. Therefore, we consider that a ( p ) ≈ . T = 20 achieved by ourcontrol is not a subtle increase but an effective improvement.Figure 9 shows the configurations in front of the exit at t = nT , when β isswitched from 0 to 1. The upper (lower) panel illustrates the case Q ( p ) = β ∗ / Q ( p ) = β ∗ / α = 1 and p = 0, Q ( p ) becomes β ∗ / a ( p ) equals 0.5, because every particle maintains a one-site gap. However, Q ( p )plummets as p increases marginally from 0 because some particles move forward andare blocked by their leading particles. This results in the occurrence of the SlS rule,which is propagated through the lattice. Eventually, we can regard the state with p = 0as a sort of metastable state. Such a special case is out of our scope, so that we searchthe range from p = 0 . p = 1 in Fig. 8 and 10. exit (cid:1) β * exit (cid:1) β * T T
Figure 9:
Configurations of particles in the HD phase. The upper panel illustrates theuncontrolled case ( Q ( p ) = β ∗ / . Q ( p ) = β ∗ / . β ∗ T (= τ ) is a multiple of 6. Q ( p ) for various T Next, we investigate the effect of T on a ( p ) with s = 0 for constant β ∗ = 0 .
6. Figure 10plots the simulated values of a ( p ) as a function of p for various T ∈ { , , } .Two phenomena are observed in Fig. 10. First, the maximum a ( p ) increases as T decreases because particles less readily accumulate in front of the exit with a shorterclosing period. Second, p opt decreases as T increases. As a closing period grows, largergaps between particles produced with smaller values of p are required in order to removejams. Eventually, the relatively long jams with a relatively long T become difficult toabsorb; that is, a ( p ) approaches 0. elocity control for improving flow through a bottleneck Figure 10:
Simulated values of a ( p ) as a function of p for various T ∈{
10 (red), 20 (green),40 (blue) } . The other parameters are set as L = 200, α = 1, β ∗ = 0 .
6, and s = 0. C(1)C(2)C(3)exit P
Figure 11:
Three different particle configurations, labeled C(1), C(2), and C(3) at t = nT + τ ,and the configuration of the starting point of a jam, labeled P. We also discuss the relation between T and p opt in numerical simulations andtheoretical analyses. In the HD phase, we assume that particles relatively near theexit are always separated by a two-site gap, because almost all of the particles areinfluenced by the SlS rule until they approach the exit. Under this assumption, theconfiguration of two particles near the exit at t = nT + τ , when β is switched from 1 to0, can take three different states (labeled C(1), C(2), and C(3) in Fig. 11) with equalprobability. Neglecting the occasions when the following particle catches up with theleading particle between t = nT + τ and t = ( n + 1) T in C(2) and C(3), the states C(1),C(2), and C(3) reach the state P in times 2 /p , 3 /p , and 4 /p on average, respectively.Note that if the state P is reached during a closing period, Q approaches β ∗ / . t = ( n + 1) T , when β is switched from 0 to 1 and theleading particle leaves the lattice. In this case, the state P can be avoided with the elocity control for improving flow through a bottleneck T − τ = (1 − β ∗ ) T = 2 p ×
13 + 3 p ×
13 + 4 p × . (9)Substituting p = p opt into Eq. (9), we obtain the simplified expression: p opt = min (cid:26) − β ∗ ) T , (cid:27) . (10) Figure 12:
Simulated (red dots) and theoretical (green curve) values of p opt as a functionof T . The red bars show the range of p which achieves more than 95% of theimprovement with p opt . The other parameters are set as L = 200, α = 1, β ∗ = 0 . s = 0. The theoretical results are given by Eq. (10). exitexit Figure 13:
The upper panel describes the configuration near the exit at t = nT + τ in our assumption, whereas the lower one describes the ideal configuration at t = ( n + 1) T . The particles further from the exit have to move more sites as inthis figure. Figure 12 plots the simulated and theoretical p opt as a function of T with red bars.The red bars show the range of p which achieves more than 95% of the improvementwith p opt . Note that the detailed calculation scheme is described in Appendix D. Ingeneral, the analytical line shows the same trend as the simulation results, althoughthe simulation results are less (higher) than the analytical line when T <
25 (
T > elocity control for improving flow through a bottleneck
T <
25. Basically, in our approximation, we assume that p is a deterministic valuerather than a probability. However, because p is a probability in practice, the number ofhops performed by one particle in one closing period have some variance. This varianceinfluences the flow as follows; if the actual number of sites which each particle hops inone closing period becomes larger than the average number (= p opt ( T − τ )), the SlSeffect occurs near the exit (see Fig. 11) and deteriorates the flow. Therefore, particleshave to hop with a smaller p to decrease the possibility of the occurrence of the SlSeffect. As a result, the simulated values of p opt becomes less than the analytical line.Meanwhile, when T >
25, the configuration of particles far from the exit, whichis neglected in our approximation, also influences on the flow. Note that this will bediscussed further in Subsec. 4.5. We calculate p opt in our approximation consideringonly two particles near the exit. However, other particles behind the second particlehave to hop with a larger p in order to achieve an appropriate gap between particles,otherwise their gap exceeds more than one site and deteriorates the flow (see Fig. 13).This is because we assume that particles are separated by a two-site gap at t = nT + τ .Eventually, the optimal p for other particles behind the second particle becomes larger,pushing up the simulation results of p opt over the theoretical line. Q for various s In actual situations, the values of s depend on particles’ property, i.e., the law of inertiaand the reaction delay. Therefore, we investigate the change of a varying s in thissubsection. Figure 14:
Simulated values of a as a function of s for various p ∈ { . . . } . The other parameters are set as L = 200, α = 1, β ∗ = 0 . T = 20,and s = 0. We plot the simulated values of a obtained as a function of s for p ∈ { . . . } with T = 20. The results are plotted in Fig. 14. Note that we elocity control for improving flow through a bottleneck Q and a , not Q ( p ) and a ( p ), in Subsec. 4.3, 4.4, and 4.5 because both Q and a depend also on other parameters.The control is effective when s is relatively small, that is, when the SlS effect isrelatively large, and becomes detrimental as s increases. When s is relatively large( s > . a recovers toward 0 (finally reaching a = 0 at s = 1). This phenomenon isexplained as follows. In the region where s > .
7, the SlS effect is drastically weakened.Therefore, the flow is scarcely affected by the configuration of particles, and the controlexerts little influence on the flow. Q for various r In this subsection, we introduce a new important parameter, r . The parameter r definesthe proportion of particles which obey the control. Each particle entering the latticeobeys the control with probability r , and disobeys it with probability 1 − r . All particlesin the lattice obey the control when r = 1; conversely, no particles obey the control when r = 0. The obedience or disobedience of each particle is determined at the left boundaryand fixed while the particle resides on the lattice. (a) (b)Figure 15: (a) Simulated values of a as a function of s and r . The other parameters are setas L = 200, α = 1, β ∗ = 0 . T = 20, and p = 0 .
3. (b) Simulated values of a asa function of r for s ∈ {
0, 0 .
2, 0 . } . The other parameters are set as L = 200, α = 1, β ∗ = 0 . T = 20, and p = 0 . Here, we investigate the coupled effects of s and r on a . Figure 15a presents thesimulated values of a as a function of s and r with T = 20 and p = 0 .
3. Between s = 0 and s ≈ .
2, we find that a ≥ r . On the otherhand, between s ≈ . s ≈ . a becomes negative when r becomes large. Notethat when s ≈ a also diminishes to nearly 0, becausethe SlS effect becomes very weak and the control exerts little influence on the flow (seeSubsec. 4.3).To clarify the r -dependence of a , we plot a versus r for various values of s ∈ {
0, 0 . . } in Fig. 15b. In the case s = 0 (indicating the strongest SlS effect), a monotonically elocity control for improving flow through a bottleneck r . As a result, the flow is maximized when all particles obey the control,i.e., r = 1. Meanwhile, in the case s = 0 .
2, the maximum a is achieved at r = 0 .
32. Thisresult indicates that we should control an appropriate portion of particles as s increasesfrom 0. Finally, in the case s = 0 . a monotonically decreases with r . Therefore, whenthe SlS effect is relatively small, the control can conversely deteriorate the flow. Finally, we introduce another important parameter l in this subsection. The parameter l defines the length of the section from the exit, where the control works. The controlis valid throughout the lattice when l = L , and completely invalid when l = 0. Figure16 describes the schematic of l . (cid:1) v=1 α β =0l sites (controlled section) (cid:1) v=1 v=pL-l sites (uncontrolled section) Figure 16:
Schematic definition of the controlled and the uncontrolled section during theclosing period with r = 1. In this subsection, we investigate the effect of l on a . Note that we consider themost fundamental case, studied in Subsec. 4.1 and 4.2, where s = 0 and r = 1.Figure 17a plots the simulated values of a as a function of l for various T ∈{ , , } . Here, referring to Fig. 12, we set p to its p opt at each value of T , obtainedby the simulations (see Fig. 10). Specifically, for T = 10, 20, and 40 in Fig. 17a, we set p = 0.47, 0.32, and 0.21, respectively. In Fig. 17a, we observe that a is maximized at someoptimal value of l ( l opt ) depending on T . Specifically, we find that l opt ( T = 10) = 7, l opt ( T = 20) = 12, and l opt ( T = 40) = 24. Therefore, we conjecture that l opt ≈ × β ∗ T, (11)where 1 denotes the hopping probability during an open period.Figure 17b compares the simulated l opt with red bars and the predictions of Eq.(11) as a function of T . The red bars show the range of l which achieves more than 95%of the improvement with l opt . Note that the detailed calculation scheme is described inAppendix D.The simulation results are in excellent agreement with Eq. (11). To explain thisphenomenon, we focus on the fact that particles located at most 1 × β ∗ T sites fromthe exit at t = nT , when β is switched from 0 to 1, might leave the lattice during thenext opening period. For example, the particle occupying the ( L − × β ∗ T )th site at t = nT can exit the lattice at t = nT + τ unless it is blocked by the leading particle elocity control for improving flow through a bottleneck (a) (b)Figure 17: (a) Simulated values of a as a function of l for ( T, p ) ∈ { (10, 0.47) (red), (20,0.32) (green), (40, 0.21) (blue) } . The other parameters are set as L = 200, α = 1, β ∗ = 0 . s = 0, and r = 1. (b) Simulated (red dots) and theoretical (green line)values of l opt as a function of T . The red bars show the range of l which achievesmore than 95% of the improvement with l opt . The other parameters are set as L = 200, α = 1, β ∗ = 0 . s = 0, and r = 1. The theoretical results are obtainedby Eq. (11). during that open period. The number of particles leaving the lattice at t = nT dependson the configuration of particles within 1 × β ∗ T sites from the exit during that period.Conversely, particles more than 1 × β ∗ T sites from the exit can never reach the exitwithin that period. Therefore, at the sites further than 1 × β ∗ T sites from the exit, thecontrol is meaningless and can be rather detrimental.
5. Conclusion
The present paper analyzes an effective control method, which improves the flow overa lattice with a bottleneck. The flow is enhanced by appropriately changing particles’hopping probability on the lattice, depending on the state of the bottleneck. Specifically,when β = 0 ( β = 1), the hopping probability is set to v = p ∈ (0 ,
1) ( v = 1). Notethat our model differs from the related works [21–25], in which the hopping probabilityis fixed and other properties, such as the input probability (rate), are altered.Herein, we report a number of important results. In Subsec. 4.1 and 4.2, weobserved that the control improves the flow if p is appropriately chosen. We also foundthat an optimal value of p ( p opt ) which maximally improves the flow in the HD phase(without the control) depends on T . Specifically, the value of p opt and the maximalvalue of a ( p ) decreases if τ or β ∗ is constant. Conversely, in the LD phase (withoutthe control), the control impedes the flow. In Subsec. 4.3 and 4.4, we investigated theinfluence of s and r on the flow. The flow was improved only when the SlS effect wasrelatively large. Especially, in the case where the SlS effect is maximum ( s = 0), theflow improvement became the largest when all particles obeyed the control ( r = 1). Asthe SlS effect weakened (i.e., s increased), the optimal r diminished and finally reached elocity control for improving flow through a bottleneck
180 (indicating loss of the control). Finally, in Subsec. 4.5, we found an optimal controllength l opt that maximizes the flow. Similar to p opt , l opt is also T -dependent.According to these results, the effect of our control is maximized if we select anappropriate hopping probability and controlled section when the lattice is congestedand the SlS effect is relatively large. We stress again that the essence of the resultslies in that the flow improves even if deceleration by p is stochastic, although the mostideal control is to make all particles keep an appropriate gap deterministically duringdeceleration.In vehicular traffic contexts, it is reported that traffic flow through a bottleneckcan be improved by appropriately decelerating vehicles. One of the proposed methodsis variable speed limit (VSL) [36] control, where the limited speed of vehicles is changedaccording to the state of a bottleneck. It is presumable to interpret our control as a sortof VSL control. Applying the results in this study for an actual traffic flow through atraffic light, we can show that our VSL control improves the flow by approximately 4%in the following conditions: T ≈
20 sec., τ ≈
12 sec., v ≈
60 km/h (the maximal speed)or 12 km/h (the limited speed during a red-light period), and l ≈
180 m. We note thatthe length of one site and the length of one time step are set as 7.5 m [52] and 0.5 s,respectively.We admit that more refined models are required to study a real traffic accurately.Such approaches, for example, introducing inhomogeneity of particles, remain as futureworks. However, we would like to mention that our simple model allows us to discussthe effect of VSL control as a first step.
Acknowledgements
The authors greatly thank Takahiro Ezaki for helpful discussions and beneficialcomments. This work was supported by JSPS KAKENHI Grant No. 25287026 and15K17583.
Appendix A. Dependence on τ In this appendix, we vary the values of τ (= β ∗ T ) ∈ { , , } and investigate theimprovements of Q ( p ) to the model, fixing L = 200, α = 1, and T = 20. We note that m ( m ∈ N ) particles can leave the lattice with s = 0 for τ ∈ { m − , m − , m } inone open period, i.e., Q (1) = 0 . τ ∈ { , , } with s = 0.The simulated values of improvement ratio a ( p ) are plotted in Fig. A1 for (a) s = 0and (b) s = 0 . τ ∈ { , } if we appropriatelyselect p , whereas the control never improves the flow for τ = 10. In the case of τ = 3 m − τ = 3 m − m (= 3) particles to leave thelattice with s = 0. elocity control for improving flow through a bottleneck (a) (b)Figure A1: Simulated values of improvement ratio a as a function of p for various τ ∈{ , , } with (a) s = 0 and (b) s = 0 .
15. The other parameters are set as L = 200, α = 1, and T = 20. On the other hand, in Fig. A1b, although the improvements are clearly lessenedfor τ = 11 ,
12 compared to the left panel, the flow improves by 1 % at most for τ = 10.From the above discussion, we conclude that the effect of the control is (i) largebut greatly depends on τ when s = 0 and (ii) small but the dependence on the selectionof τ gradually disappears as s becomes large. Appendix B. Dependence on SlS rules
Throughout the paper, we assume a constant s . In this appendix, we vary s dependenton β . Specifically, s = s c during close periods and s = s o during open periods, where s o = s c . We describe two figures corresponding to Fig. 8, setting (a) ( s c , s o ) = (0 . , s c , s o ) = (0 , . (a) (b)Figure B1: Simulated values of a ( p ) as a function of p for various α ∈ { . . } . The other parameters are set as L = 200, β ∗ = 0 . T = 20, and (a)( s c , s o ) = (0 . ,
0) and (b) ( s c , s o ) = (0 , . elocity control for improving flow through a bottleneck s during open periods. Therefore, weconclude that the SlS coefficient during open periods greatly affects the flow in ourmodel.Note that our SlS rule is basically based on Benjamin-Johnson-Hui (BJH)model [45]. There are other typical SlS rules, which are implemented in Takayasuand Takayasu (T ) model [44] and Appert-Rolland and Santen (AS) model [46]. In theformer, a blocked particle cannot hop until its leading particle stay two-sites ahead ofit, whereas in the latter the SlS effect remains until it finally hops. Both models havestronger SlS effects than BJH model. Therefore, we are convinced that our control canhave a positive effect on the flow change if our SlS rule is modified based on either T or AS models. Appendix C. Implementation of Woelki’s method
In Woelki’s approach, input rate is decreased (increased) if the average density of thesystem exceeds (falls below) a threshold density ρ t . Noting the difference from Woelki’sapproach, we focus on the control of a hopping probability (particle’s velocity), whileWoelki focused on the control of an input rate. Moreover, the SlS effect is not consideredin his work. Figure C1:
Improvement ratio a as a function of the threshold density ρ t with Woelki’scontrol rule. The other parameters are set as L = 200, α = 1, β ∗ = 0 .
6, and T = 20. Changing only the global density of the lattice is ineffective in theTASEP with the SlS rule. As a result of simulations of our model applied with Woelki’s rule, the flow isnot improved if we control input probability depending on the global density as withWoelki’s approach. Figure C1 shows the results of simulations by Woelki’s rule, wherethe horizontal axis represents threshold density ρ t and the vertical axis representsimprovement ratio of the flow a . We observe that there is no flow improvement ( a > ρ t . elocity control for improving flow through a bottleneck Appendix D. Calculation of p opt ( l opt ) Here, we explain how we pick up p opt ( l opt ) and depict red bars in Fig. 12 and Fig.17b. First, we calculate the improvement ratio a in increments of ∆ p = 0 .
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