Comparison of escalator strategies in models using a modified totally asymmetric simple exclusion process
aa r X i v : . [ n li n . C G ] A p r Comparison of escalator strategies in models usinga modified totally asymmetric simple exclusion process
Hiroki Yamamoto , ∗ Daichi Yanagisawa , , and Katsuhiro Nishinari , School of Medicine, Hirosaki University 5 Zaifu-cho Hirosaki city, Aomori Prefecture, 036-8562, Japan Research Center for Advanced Science and Technology, The University of Tokyo,4-6-1 Komaba, Meguro-ku, Tokyo 153-8904, Japan Department of Aeronautics and Astronautics, School of Engineering, The University of Tokyo,7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan (Dated: April 9, 2019)We develop a modified version of the totally asymmetric simple exclusion process (TASEP) anduse it to reproduce flow on an escalator with two distinct lanes of pedestrian traffic. The model isused to compare strategies with two standing lanes and a standing lane with a walking lane, usingtheoretical analysis and numerical simulations. The results show that two standing lanes are betterfor smoother overall transportation, while a mixture of standing and walking is advantageous onlyin limited cases that have a small number of pedestrians. In contrast, with many pedestrians, theindividual travel time of the first several entering particles is always shorter with distinct standingand walking lanes than it is with two standing lanes.
I. INTRODUCTION
An escalator is an essential system for pedestrian trans-portation in many public facilities, such as train stations,shopping malls, and airports, that enable people to effi-ciently move. The capacity of an escalator has been in-vestigated throughly [1, 2]. Such studies mainly focus onthe maximum capacity of the system.When considering the operation of an escalator af-ter it is installed, individual pedestrian behaviors mustalso be understood. In many countries, for example, eti-quette dictates that people should stand on one side andwalk on the other side of an escalator [3–7]. Somewhatcounter-intuitively, however, some practitioners have re-cently started to encourage people to stand on both sidesfor smoother transportation and safety [8–10].Research about escalators in the fields of transporta-tion engineering and nonequilibrium statistical mechan-ics has only recently begun [11–17] . Individual behaviorhas also been considered. Refs. [11–13] investigate theflow of traffic mainly from numerical simulations, whileRefs. [14, 15] study pedestrian choices between escalatorsand stairways. Especially, in Refs. [16, 17], they inves-tigate the escalator etiquette (standing on one side andwalking on the other side) and conclude that walking isnot beneficial in some cases. However, their results arebasically obtained using only numerical simulations withsome specific parameters. We consider it to be essentialto anew investigate which escalator strategy is suitablefor various situations using numerical simulations withextensive parameters and theoretical analyses.In the present study, we analyze a novel escalatormodel that reproduces individual pedestrians’ behaviorson both macro- (total transportation time or flow) andmicro-scales (individual transportation time). Theoret-ical analysis and numerical simulations are used. To ∗ [email protected]; present an escalator, we construct a two-lane modelwith a modified totally asymmetric simple exclusion pro-cess (TASEP), which is a stochastic process on a one-dimensional lattice in which particles are allowed to hopin one direction (left to right in the present study). In thefield of nonequilibrium statistical mechanics, researchershave applied TASEP extensively to various themes suchas molecular-motor traffic [18–21], vehicular traffic [22–25], and exclusive queuing processes [26–28], and the pro-cess is especially useful since it can be solved exactly [29–31]. Our model differs from the original TASEP withopen boundaries in two respects.First, the updating rules for particles are different.Specifically, in addition to the original hopping probabil-ity, particles can deterministically hop one site forwardeven when the right-neighboring site is occupied, whichis an important feature of an escalator.Second, our model consists of two lanes that can havedifferent hopping probabilities. The multi-lane TASEPhas itself been investigated vigorously [32–35]; however,most of them assume the same hopping probability inall lanes. Fundamental behaviors of this model like thephase transitions are investigated in Refs. [32, 33]. Mean-while, Refs. [34, 35] analyze actual traffic flows using themulti-lane TASEP. We emphasize that most studies ofthe multi-lane model allow particles to switch lanes, whileour model prohibits this behavior.In the present study, we investigate three escalatorstrategies; (i) Strategy SS: two standing lanes, (ii) Strat-egy SW: one standing lane with one walking lane, and(iii) Strategy WW: two walking lanes. We note thatStrategy SS and SW are our focus, because they are morecommon.Using both theoretical analysis and numerical simula-tions with this model, we find that Strategy SS is gen-erally more advantageous in terms of reducing the total(macro-scale) transportation time, especially with a rela-tively large number of particles, which model pedestrians.Conversely, in terms of reducing the individual (micro-scale) transportation time, Strategy SW offers better re-sults for the first-entering particles, which tend to preferwalking, on the escalator.The rest of this paper is organized as follows. SectionII defines the modified two-lane TASEP. Then Sec. IIIdiscusses the behavior of the basic one-lane model withour modified update rules. In Sec. IV, we examine thetotal transportation time given with our modified two-lane model. Then, we proceed to discussion of individualtransportation times in Sec. V. Finally, Sec. VI givesconcluding remarks. II. MODEL DESCRIPTIONA. Original TASEP with open-boundary conditions
The original TASEP with open-boundary conditionsis defined as a one-dimensional lattice (lane) of L sites,which are labeled from left to right as i = 0 , , ......, L − i th site attime t is occupied by a particle, its state is representedas s i ( t ) = 1; otherwise, its state is s i ( t ) = 0.Discrete time and parallel updating are adopted in thepresent study. During parallel updating, the states of allthe particles on the lattice are determined simultaneouslyin the next time step. Particles enter the lattice from theleft boundary with input probability α , and they leavethe lattice from the right boundary with output proba-bility β . In the bulk, particles whose right-neighboringsites are empty can hop to the rightward site with proba-bility p ; otherwise they remain at their present site. Themodified TASEP differs from the original process in thefollowing two respects. (cid:1) p p (cid:1) (cid:2) L sites i = 0 i = 1 i = L- FIG. 1. (Color Online) Schematic illustration of the originalTASEP with open-boundary conditions.
B. Modification 1: Updating rules
Our modified updating rules have two-step structure.First, particles hop deterministically one site forward re-gardless of the right-neighboring site, which reproducesan important feature of an escalator.In addition, particles may hop one more site forwardwith hopping probability p , which is fixed throughout each lattice in our model, if the right-neighboring site isempty. This possibility represents walking on an escala-tor.So, particles can hop one or two sites for each timestep. We note that in our model particles have two op-portunities for hopping and can hop one or two sites ineach time step, unlike the Nagel-Schreckenberg model forvehicular traffic [36], in which particles hop equal to ormore than zero site only once in each time step. TableI summarizes the modified updating rules in comparisonwith the original updating rules. TABLE I. (Color Online) Updating rules for the redparticles in the bulk of our model for comparison with theoriginal TASEP. The notation ‘Prob.’ represents theprobability of each configuration at time ( t + 1). We notethat blue particles are not depicted at time ( t + 1). Our modified TASEP Original TASEPTime t Time t+ Prob. Prob. p Case
Time t+ p p pp ppp At the right boundary, a particle can leave the systemfrom the ( L − L − L − β . In the present model, a particle occupying the( L − t leaves the system with probability p and hops to the ( L − − p at time ( t + 1)if s L − ( t ) = 0; otherwise it hops to the ( L − L − t must leave the system. Table II extracts and sum-marizes the updating rules around the right boundary. C. Modification 2: Two lanes with two types ofparticles
Second, our modified model consists of two lanes. Thestate of the i th site of the lane 1 and 2 at time t arerepresented as s i ( t ) and s i ( t ), respectively. Each lanecan act as a standing lane ( p = 0) or a walking lane(0 < p ≤ TABLE II. (Color Online) Updating rules for the redparticles around the right boundary. L- L- L- L- L- L- p p p p Time t Case Time t+ Time t+ L- L- L- L- L- L- L- L- lanes, corresponding to Strategy WW (see Fig. 2 (c)).Strategy WW is generally not adopted in real situations;however, it is investigated here for the sake of compari-son.With Strategy SS and WW, particles can enter eitherof the two lanes if the first site is empty. Specifically, aparticle enters lane 1 (lane 2) if ( s i ( t ) , s i ( t )) = (0 ,
1) (if( s i ( t ) , s i ( t )) = (1 , s i ( t ) , s i ( t )) = (0 ,
0) indiscriminately, i.e.,with probability 1/2.We emphasize that particles can always enter the sys-tem with Strategy SS and WW. This is because all theparticles in the system will hop one or two sites forwardevery time step, so at least one of the two left bound-aries will always be vacant; that is, ( s ( t ) , s ( t )) = (0 , , , r and standing-preference particles with probability1 − r . Standing (walking) particles can enter the stand-ing (walking) lane if the leftmost site of the correspondinglane is empty; otherwise they cannot. Unlike Strategy SSand WW, particles frequently cannot enter the system inthis case because of the preference. We note that in thesimulations below the preference of a particle is deter-mined just before it enters the system; this preference isreset if the particle cannot enter and is redetermined atthe next chance of entering. III. ONE-LANE MODELWITH THE MODIFIED UPDATING RULES
In this section, we briefly discuss the steady-state flowof the basic one-lane model with the modified updatingrules.The basic one-lane model with our modified updatingrules, as illustrated in Fig. 3. In this system, particlesmust hop one or two sites forward. Therefore, the steady- (cid:1) (cid:1) L sites (cid:1) (a) Strategy SS 111 111 (cid:1) r (cid:1) (cid:1) -r L sites (cid:1) (b) Strategy SW 1- p p p p
11 111 (cid:1) (cid:1) (cid:1) L sites (cid:1) (c) Strategy WW 11- p p pp p p p p p p FIG. 2. (Color Online) Schematic illustration of themodified TASEP with (a) Strategy SS: two standing lanes,(b) Strategy SW: one standing lane (upper) and one walkinglane (lower), and (c) Strategy WW: two walking lanes. Wenote that red (green) particles prefer standing (walking). (cid:1) p (cid:1) L sites (cid:1) (cid:1) p p p FIG. 3. (Color Online) Schematic illustration of the one-lanemodel with our modified updating rules. state flow is clearly equal to or more than that of theoriginal TASEP with hopping probability p = 1.The original TASEP with p = 1 for various ( α, β ) ex-hibits only two phases; the low-density (LD) phase, inwhich the system is governed by the left boundary, andthe high-density (HD) phase, in which the system is gov-erned by the right boundary, and the maximal current(MC) phase, in which the system is governed by thebulk, never occurs. Therefore, the one-lane model is al-ways governed by the left boundary, since a queue cannever form near the right boundary. This fact, counter-intuitively, indicates that the flow is determined regard-less of p , so the flow is always constant. This can beexplained as follows. (cid:1)(cid:2)(cid:3)(cid:4)(cid:4) (cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:9)(cid:4)(cid:3)(cid:10)(cid:5)(cid:11) global density (right) (cid:1) (cid:2) (cid:3) (cid:4)(cid:4) (cid:5) (cid:6) (cid:7) (cid:3) g l oba l den s i t y p FIG. 4. (Color Online) Simulation values of dwell time (red,left axis) with the sample standard deviation and globaldensity (blue, right axis) as functions of p . The dwell time iscalculated by averaging the times of 1000 particles startingfrom the (10 + 1)th particle, and the global densitycalculated by averaging over 10 time steps after evolvingthe system for 10 time steps. First, walking clearly reduces the dwell time of eachparticle in the system, which is defined as the time gapbetween the time when the particle enters the systemand the time when it leaves. The dwell time decreasesas p increases. In addition, the global density of thelane, which is defined as the average number of occupiedcells over the space [0 , L −
1] in one time step, decreasesas p increases, due to the longer gap between particles.Figure 4 shows the average dwell time and the globaldensity as functions of p , and reproduce the behaviordiscussed above. Since (i) the average velocity of particlesis proportional to the dwell time, and (ii) the flow isrepresented as a multiplication of the average velocityand the global density, the flow remains constant.The steady-state flow Q of the one-lane model withour updating rules satisfies Q = α (1 − Q ) , (1)resulting in Q = α α . (2) Eq. (2) is equal to that of the original TASEP in the LDphase with parallel updating, indicating that the one-lanesystem always exhibits the LD phase.Figure 5 compares the simulation (dots) and theoret-ical (curve) values of Q as functions of α for various p ∈ { , . , } . In the simulations, we set L = 200 andthe flow is obtained by averaging over 10 time steps af-ter evolving the system for 10 time steps (and similarlyhereafter).In Fig. 5, the simulation values show very good agree-ment with the theoretical curve, and at the same time,the simulations also confirm that Q is independent on p . (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:2)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:13)(cid:4)(cid:5)(cid:7)(cid:13)(cid:14)(cid:15)(cid:12)(cid:8)(cid:11)(cid:12)(cid:7)(cid:2)(cid:16)(cid:6)(cid:5)(cid:10)(cid:16)(cid:4)(cid:11)(cid:17)(cid:12) (cid:1) p = p = p = (cid:1) Q (cid:1) FIG. 5. (Color Online) Simulation (dots) and theoretical(curve) values of Q as functions of α for various p ∈ { , . , } . IV. TOTAL (MACRO-SCALE)TRANSPORTATION TIMEWITH THE TWO-LANE MODEL
In this section, we investigate the total (macro-scale)transportation time T of N particles with the two-lanemodel. Specifically, T is defined as the time gap be-tween the start of the simulation and the time at whichthe final leaving particle leaves the system. The totaltransportation times with Strategy SS, SW, and WWare represented as T SS , T SW , and T WW , respectively. A. Steady-state flow Q SS , Q SW , and Q WW Before examining T , we briefly discuss the steady-stateflow in the two-lane model.Since particles can always enter the system with Strat-egy SS and WW (see Subsec. II C), the following relationholds Q SS = Q WW = α, (3)where Q SS and Q WW are defined as the steady-state flowof the two-lane model with Strategy SS and WW, respec-tively. We emphasize that counter-intuitively, Q WW = Q SS because of the independence of the flow from p (seethe previous section), so walking can have no effect onincreasing the steady-state flow.Second, with Strategy SW, particles prefer to entera standing lane or a walking lane with some probabil-ity. Given that a particle prefers standing (walking) withprobability 1 − r ( r ), the input probability of the stand-ing (walking) lane reduces to (1 − r ) α ( rα ). Therefore,the steady-state flows of the standing lane Q S and thewalking lane Q W in the two-lane model are given as Q S = (1 − r ) α − r ) α , (4)and Q W = rα rα , (5)which are derived from Eq. (2) by replacing α with (1 − r ) α and rα , respectively.Consequently, the steady-state flow of the two-lanemodel with Strategy SW, Q SW , can be calculated as Q SW = Q S + Q W = (1 − r ) α − r ) α + rα rα , (6)which takes its maximum value when r = 0 .
5. The de-tailed properties of the function of Eq. (6) are discussedin Appendix A.Figure 6 compares the simulation (dots) and theoreti-cal (curves) values of (a) Q SS , Q SW for p ∈ { . , } , and Q WW for p ∈ { . , } as functions of α , and (b) Q SW forvarious α ∈ { . , . , } and p ∈ { . , } as functions of r .The simulation values again show very good agreementwith the theoretical curves.From Eqs. (3) and (6), we immediately obtain Q SW < rα + (1 − r ) α = α = Q SS (= Q WW ) , (7)indicating that Strategy SS (WW) is always advanta-geous in terms of steady-state flow.The absolute advantage of Strategy SS (WW) overStrategy SW is explained with the behavior at the en-trances of the lanes. Table III summarizes the modelbehavior at the left boundary, extracting the first andsecond site with α = 1.With Strategy SS (WW), since consecutive pairs ofparticles mutually enter either lane, two particles can en-ter the system in two time steps, as shown in the upperpanel of Tab. III.On the other hand, with Strategy SW they cannot en-ter the system with two time steps but with three timesteps, if both of two consecutive particles’ preference arethe same, as shown in the lower panel of Tab. III. Thetwo particles can enter the system in two time steps iftwo consecutive particles separately prefer walking andstanding. Therefore, Q SW is maximized if r = 0 .
5, which (cid:1) Q (cid:1)(cid:1) , Q (cid:1) (cid:2) , Q (cid:2)(cid:2) (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:2)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:13)(cid:4)(cid:5)(cid:7)(cid:13) Q (cid:1)(cid:1) Q (cid:1)(cid:2) ( p = 0.5) (cid:1) Q (cid:1)(cid:2) ( p = 1) (a) (cid:1) Q (cid:2)(cid:2) ( p = 0.5) Q (cid:2)(cid:2) ( p = 1) (cid:14)(cid:15)(cid:12)(cid:8)(cid:11)(cid:12)(cid:7)(cid:2)(cid:16)(cid:6)(cid:5)(cid:10)(cid:16)(cid:4)(cid:11)(cid:17)(cid:12)(cid:13) Q (cid:1)(cid:1) , Q (cid:2)(cid:2) Q (cid:1)(cid:2) r Q (cid:1) (cid:2) (cid:1) = (cid:1) = (cid:1) = (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:2)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:13)(cid:4)(cid:5)(cid:7)(cid:13) (cid:14)(cid:15)(cid:12)(cid:8)(cid:11)(cid:12)(cid:7)(cid:2)(cid:16)(cid:6)(cid:5)(cid:10)(cid:16)(cid:4)(cid:11)(cid:17)(cid:12)(cid:13) (b) p = 0.5 p = 1 FIG. 6. (Color online) Simulation (dots) and theoretical(red/orange curves) values of (a) Q SS (blue circles), Q SW for p ∈ { . , } , and Q WW for p ∈ { . , } as functions of α ,and (b) Q SW for various α ∈ { . , . , } and p ∈ { . , } as functions of r .TABLE III. (Color Online) Comparison of the left boundarybetween Strategy SS and SW, with constant α = 1. Two redparticles can enter either of two standing lanes withStrategy SS. Two green particles, which prefer walking inthis figure, attempt to enter the walking lane with StrategySW. We note that the upper (lower) lane is a standing(walking) lane for Strategy SW. SWSS
Time t Strategy Time t+ Time t+ (cid:1)(cid:1) (cid:1)(cid:1) (cid:1)(cid:1) (cid:1)(cid:1) (cid:1)(cid:1) (cid:1)(cid:1) clearly maximizes the probability that the preferences ofthe two consecutive particles differ.Considering that the system is governed by the leftboundary, these facts finally lead to Q SS > Q SW , ex-plaining the absolute advantage of Strategy SS (WW)over Strategy SW. B. Total (macro-scale) transportation time T
1. Approximate theoretical analyses of T In this subsection, we theoretically calculate the ap-proximations of T SS (total transportation time withStrategy SS), T SW (total transportation time with Strat-egy SW), and T WW (total transportation time with Strat-egy WW) and consider the relations among them.Considering that the steady-state flow Q expresses theaverage number of particles that pass a certain point (theleft boundary) each time step, the average required timesteps T s ( N ) for the N th particle to enter the systemfrom the time at which the first-entering particle entersin steady-state flow can be calculated as T s ( N ) ≈ N − Q . (8)On the other hand, the required time steps T f ( p ) forthe final-leaving particle of both lanes, which is not al-ways identical to the final-entering particle, to reach theright boundary can be represented as T f = T fS = L ,T f ( p ) = T fW ( p ) ' L p for a walking lane , (9)where T fS ( T fW ) is a (an) definitive (approximate) value,and when calculating T fW the possibility that a walkingparticle is blocked by the particle ahead of it is ignored,resulting in a slight underestimation of T fW (see also Sec.IV B 2).Assuming that (i) 1 /α time steps are needed on av-erage for the first particle to enter the system, and that(ii) particles enter both lanes during the steady-state flowafter the first particle enters the system, T SS ( N, α ) and T WW ( N, α, p )— their arguments can be abbreviated un-less otherwise specified (similarly for other variables)—can be written as T SS ( N, α ) ≈ α + T sSS ( N ) + T fS = Nα + L T WW ( N, α, p ) ≈ α + T sWW ( N ) + T fW = Nα + L p , (11)from Eqs. (3), (8), and (9). We note that the first-in-first-out condition—i.e., the entering sequence is identicalto the leaving sequence—must be satisfied when StrategySS is modeled, whereas the relation is mostly satisfiedwhen Strategy WW is modeled.From Eqs. (10) and (11), T SS and T WW differ only dueto the difference in the second term, resulting in T SS ≈ T WW if N is sufficiently large.Under the same assumption, we then consider T SW .Unlike T SS and T WW , the first-in-first-out condition gen-erally does not hold since the hopping probabilities of the two lanes are different, leading to difficulty in calculating T SW .For approximate calculations of T SW , we need to con-sider the final-entering standing-preference particle andthe final-entering walking-preference one. Let us define N as a threshold number. Specifically, if a standing-preference particle is (not) included in the last N enter-ing particles, the final leaving particle is on average iden-tical to the final-entering standing-preference (walking-preference) particle. We note that if r = 1, N is always N = 0.Figure 7 gives a schematic illustration of N . Thisfigure focuses on the last k entering particles, wherethe k th ( k = 1 , ......, N + 1) entering particle from thefinal-entering particle is identical to the final-enteringstanding-preference particle. We note that k = N + 1represents that all N particles prefer walking, which isdefined for the sake of convenience. (cid:1) Last k entering particles (cid:2) f f (cid:1) Final leaving particle (cid:2) (cid:1)(cid:1) (cid:1)(cid:1)(cid:1)(cid:1) f ff f L -2 L -1 (cid:1) k particles (cid:1) k (cid:2) N (cid:1) N (cid:1) (cid:1) k (cid:1) N + walking-preferparticles L -2 L -1 FIG. 7. (Color online) Schematic illustration for explaining N . The red (green) particles are standing (walking)particles, and the red (green) particle labeled ‘f’ is thefinal-entering standing-preference (walking-preference)particle. If 1 ≤ k < N , the final-leaving particle is, onaverage, identical to the final-entering standing-preferenceparticle; otherwise the final-leaving particle is identical tothe final-entering walking-preference particle. Similarly to the assumptions as those applied to T SS and T WW , if we assume that (iii) particles tend to en-ter the system every 1 /Q SW time steps, and that (iv) astanding-preference (walking-preference) particle on av-erage stays in the system for L/ L/ (1 + p )) time steps, N satisfies ( N − × Q SW ≈ L − L p . (12)Consequently, N reduces to N ≈ (cid:18) L − L p (cid:19) Q SW + 1= pL p (cid:26) rα rα + (1 − r ) α − r ) α (cid:27) + 1 . (13)We note that although N is defined to be consecutive,we use its integer part when it is used as the upper limitof summation, which must be an integer. T SW ≈ N +1 X k =1 P ( k ) t SW ( k )= 1 α + N X k =1 { P ( k ) × ( T sSW ( N − k + 1) + T fS ) } + − N X k =1 P ( k ) ! × ( T sSW ( N ) + T fW )= 1 α + N − Q SW − P N k =1 (1 − r N − k ) r k Q SW + (1 − r N ) L r N L p = 1 α + N − rα rα + (1 − r ) α − r ) α − P N k =1 (1 − r N − k ) r krα rα + (1 − r ) α − r ) α + (1 − r N ) L r N L p (16) T SW > α + N X k =1 { P ( k ) × ( T sSW ( N − N ) + T fS ) } + − N X k =1 P ( k ) ! × ( T sSW ( N ) + T fW )= 1 α + N X k =1 P ( k ) × (cid:18) N − Q SW − N Q SW + L (cid:19) + − N X k =1 P ( k ) ! × (cid:18) N − Q SW + L p (cid:19) ≥ α + N X k =1 P ( k ) × (cid:18) N − Q SW − N Q SW + L (cid:19) + − N X k =1 P ( k ) ! × (cid:18) N − Q SW + L p (cid:19) = 1 α + N − Q SW + L p ≥ α + N − Q WW + L p = T WW (17) T SW − T SS = 1 α + N − Q SW − P N k =1 (1 − r N − k ) r k Q SW + (1 − r N ) L r N L p − (cid:18) NQ SS + L (cid:19) = ( N − (cid:18) Q SW − Q SS (cid:19) − P N k =1 (1 − r N − k ) r k Q SW | {z } first part − pr N p L | {z } second part (18) Regarding T SW as a random variable, i.e., T SW = t SW ( k ) ( k = 1 , , ......, N + 1), t SW ( k ) can be representedas t SW ( k ) ≈ α + T sSW ( N − k + 1) + T fS for 1 ≤ k < N , α + T sSW ( N ) + T fW for N ≤ k ≤ N + 1 , (14)where N = min( N, N ).Because each particle prefers standing (walking) withprobability 1 − r ( r ), the probability P ( k ) that the k th-entering particle from the final one is approximately iden-tical to the final-entering standing-preference particle canbe calculated as P ( k ) = (cid:26) (1 − r ) r k − for 1 ≤ k ≤ N,r k − for k = N + 1 , (15)satisfying P N +1 k =1 P ( k ) = 1.Using Eqs. (14) and (15), the expected value of T SW can be calculated as Eq. (16). The detailed derivation of T SW is given in Appendix B. Next, we compare T SS , T SW , and T WW , using Eqs.(10), (11), and (16). First, T SS clearly exceeds T WW . Inaddition, because of Eq. (17), T SW also exceeds T WW .The relation between T SS and T SW is somewhat com-plicated. Using Eqs. (10) and (16), T SW − T SS can becalculated as in Eq. (18).First, the second part of Eq. (18) is always equal toor less than 0 (the equal sign holds only when r = 0),and can be regarded as a function of L . This is dueto the shorter dwell time of walking-preference particlescompared to standing-preference particles (see also Fig.4). We hereafter refer to this effect as the ‘positive effectof walking,’ and it disappears gradually as r decreases or N increases, since r N ≈ N >
1, the first term inthe first part is always positive, due to Q SS > Q SW (seeEq. (7)), and can be regarded as a function of N .However, the existence of the second term in the firstpart of Eq. (18) invites the result that the first part ofEq. (18) becomes negative. The numerator of this termsatisfies the following relation: N X k =1 (1 − r N − k ) r k < N (1 − r N ) < N . (19) T (cid:1)(cid:1) , (cid:3) (cid:4) (cid:5) / T (cid:1)(cid:1) , (cid:6) (cid:7) (cid:8) (cid:9) N (cid:1) = (cid:1) = (cid:1) = (a) T (cid:1) (cid:10) , (cid:3) (cid:4) (cid:5) / T (cid:1) (cid:10) , (cid:6) (cid:7) (cid:8) (cid:9) N ( (cid:1) , r, p ) = (0.2, 0.8, 0.5)( (cid:1) , r, p ) = (0.5, 0.2, 0.5)( (cid:1) , r, p ) = (1, 1, 0.2) (b) T (cid:10)(cid:10) , (cid:3) (cid:4) (cid:5) / T (cid:10)(cid:10) , (cid:6) (cid:7) (cid:8) (cid:9) N ( (cid:1) , p ) = (0.2, 0.5)( (cid:1) , p ) = (0.5, 0.5)( (cid:1) , p ) = (1, 0.2) T (cid:1)(cid:1) , (cid:3) (cid:4) (cid:5) / T (cid:10)(cid:10) , (cid:3) (cid:4) (cid:5) N ( (cid:1) , p ) = (0.2, 0.5)( (cid:1) , p ) = (0.5, 0.5)( (cid:1) , p ) = (1, 0.2) (d)(c) FIG. 8. (Color Online) Calculated values of the ratios (a) T SS , sim /T SS , theo for α ∈ { . , . , } , (b) T SW , sim /T SW , theo for various( α, r, p ) ∈ { (0 . , . , , (0 . , . , . , (1 , , . } , (c) T WW , sim /T WW , sim for various( α, p ) ∈ { (0 . , , (0 . , . , (1 , . } , (d) T SS , sim /T WW , theo for various( α, p ) ∈ { (0 . , , (0 . , . , (1 , . } . All the plots start from N = 10.lim N →∞ T SW T SS = lim N →∞ α + N − Q SW − P N k =1 (1 − r N − k ) r k Q SW + (1 − r N ) L + r N L pNQ SS + L = lim N →∞ N − Q SW NQ SS = Q SS Q SW > From the definition of N , this numerator increasesmonotonically and converges to a constant value; specifi-cally, P N k =1 (1 − r N − k ) r k when N ≥ N , as N increases.From Eq. (13), this constant value can be regarded as amonotonically increasing function of L .Therefore, for sufficiently large N , the first part ofEq. (18) exceeds 0. We hereafter refer this effect asthe ‘negative effect of preference,’ because a differencearises due to Q SS > Q SW , which is introduced by thewalking/standing preference (see Eq. (7)).Consequently, T SW − T SS for N > N can be rewrittenas T SW − T SS = f ( N ) − g ( L ) , (20)where f ( N ) and g ( L ) are, respectively, monotonically in-creasing functions of N and L . This fact implies that forsufficiently large N the positive effect of walking becomesnegligible and T SW − T SS > T SW > T SS ), while thisrelation can be reversed for small N . From Eqs. (10) and (16), T SW /T SS converges to a certain value as N in-creases, as in Eq. (21).The discussion in this subsection indicates that (i)Strategy WW is always advantageous over Strategy SSand SW, and that (ii) Strategy SS is advantageous overStrategy SW if N is sufficiently large; otherwise StrategySW performs better than Strategy SS.
2. Validation of approximations of T This subsection discusses the validity of the theoreti-cal approximations of T SS , T SW , and T WW by comparingthem with the results of the numerical simulations.Figure 8 plots the calculated values of theratios (a) T SS , sim /T SS , theo for various α ∈{ . , . , } , (b) T SW , sim /T SW , theo for various( α, r, p ) ∈ { (0 . , . , , (0 . , . , . , (1 , , . } ,(c) T WW , sim /T WW , sim for various ( α, p ) ∈ { (0 . , , (0 . , . , (1 , . } , and (d) T SS , sim /T WW , theo for various ( α, p ) ∈ { (0 . , , (0 . , . , (1 , . } . Weaverage the simulation values of T over 1000 trials(and do similarly below for the simulations of T unlessotherwise specified).Figure 8 (a) shows that the simulation values of T SS agree with the theoretical ones very well, even for small N . Conversely, Figs. 8 (b) and (c) show that the simu-lation values of T SW and T WW agree with the theoreticalones very well for large N ; however, the simulation di-verges from the analysis for small N and especially if p is small. This behavior can be explained as follows.From Eq. (9), T fS is deterministic, whereas T fW is ex-pected. In addition, although the final-leaving particle isassumed to be able to hop forward freely in the theoreti-cal calculations, it may be blocked by the particle aheadof it in a walking lane, so T fW > L/ (1+ p ). These featuresmean that T SW and T WW can diverge in the theoreticalapproximations, especially for small N and small p , asthese conditions enhance the influence of T fW on T SW and T WW .In Fig. 8 (d), T SS is confirmed to approach T WW as N increases (strictly speaking, T WW > T SS ). From here on,we do not consider T WW , since T WW < T SS and T WW
5, and p = 0 . T SW /T SS as functions N T T (cid:1)(cid:2) ( r = 0.2) T (cid:1)(cid:1) T (cid:1)(cid:2) ( r = 0.5) T (cid:1)(cid:2) ( r = 0.8) T (cid:1)(cid:2) ( r = 1) (a) N T T (cid:1)(cid:2) ( r = 0.8) T (cid:1)(cid:1) T (cid:1)(cid:2) ( r = 1) N = N (cid:3)(cid:4) N = N (cid:3)(cid:4) (b) FIG. 9. (Color Online) (a) Simulation values of T SS (red)and T SW for various r ∈ { . , . , . , } as functionsof N , fixing α = 0 . p = 0 .
5. All the plots start from N = 10. (b) Zoomed Fig. 9 (a), in the area enclosed with ablack dotted circle in Fig. 9 (a). We note that the curves for r = 0 . r = 0 . N = N cr , which will be discussed in Subsec.IV G, with black arrows. of α for various N ∈ { , , } . The simulationresults show very good agreement with the theoreticalcurves. The inequality T SW /T SS > T SW /T SS < T SW /T SS > T SS
5, respectively. Allthe plots start from α = 0 . Then, taking note of the following two relations:lim α → Q SW Q SS = lim α → (cid:26) r rα + 1 − r − r ) α (cid:27) = 1 (23)andlim α → N = lim α → N = lim α → (cid:20) pL p (cid:26) rα rα + (1 − r ) α − r ) α (cid:27) + 1 (cid:21) = 1 , (24)we have Eq. (25). In Fig. 10, all the plots (curves) areobserved to approach 1 as α approaches 0. Eq. (23)implies that for small α , the negative effect of preferencedisappears.Here, we define a new value α = α , where α (0 <α ≤
1) satisfies N ( α = α ) = 2 ⇔ pL p (cid:26) rα rα + (1 − r ) α − r ) α (cid:27) = 1 . (26)For sufficiently large N , T SW − T SS always exceeds 0when α < α ≤
1, which is discussed in detail in Ap-pendix C. On the other hand, when 0 < α ≤ α , from Eq. (18), T SW − T SS = ( N − (cid:18) Q SW − Q SS (cid:19) − rp p L N − Q SW × g ( α ) > , (27)where g ( α ) is defined as g ( α ) = 1 − Q SW Q SS − rpL (1 + p )( N − . (28)The behavior of g ( α ) is discussed in detail in AppendixD. We note that N = N = 1 for the upper limit ofsummation when α ≤ α because we need to use theinteger part for that case. Eq. (27) indicates that forsufficiently large N ; especially, N > pL p max (cid:18) r − r , (cid:19) + 1 , (29) T SW /T SS always exceeds 1, and converges to 1 with α →
0, because the negative effect of preference is al-ways greater than the positive effect of walking when0 < α ≤ N , as α decreases, T SW /T SS can become below 1, because the positive effectof walking is greater than the negative effect of walking,and finally T SW /T SS converges to 1 from Eq. (25). E. Effect of r In this subsection, we investigate the effect of r on T SW for various N . We set α = 0 . p = 0 . r = 0 ( r = 1).Figure 11 compares the simulation (circles) and theo-retical (curves) values of the ratio T SW /T SS as functionsof r for various N ∈ { , , } . We emphasize that T SS is constant for all values of r . The simulation re-sults again show very good agreement with the theoreti-cal curves even though they diverge in very limited casesof small N and large r ( N = 10 and r > . N and large r , since the influence of the negative effect ofpreference is small and r N approaches 1. In addition,1although the final leaving particle is assumed to be ableto hop forward freely in the theoretical calculations, itcan be blocked by the particle ahead of it in a walkinglane, leading to a slight underestimation of T SW (see Sub-sec. IV B 2). Consequently, small N and large r enhancethe influence of L/ (1 + p ) and increase the likelihood ofunderestimating it when calculating the theoretical val-ues of T SW , so the theoretical curve remains below thesimulation values. r T (cid:1) (cid:2) / T (cid:1)(cid:1) N = 1000 N = 200 N = 10 (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:2)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:13)(cid:4)(cid:5)(cid:7)(cid:13)(cid:14)(cid:15)(cid:12)(cid:8)(cid:11)(cid:12)(cid:7)(cid:2)(cid:16)(cid:6)(cid:5)(cid:10)(cid:16)(cid:4)(cid:11)(cid:17)(cid:12)(cid:13) r = r (cid:1)(cid:2) FIG. 11. (Color Online) Simulation (circles) and theoretical(curves) values of the ratio T SW /T SS as functions of r forvarious N ∈ { , , } . The otherparameters are fixed as α = 0 . p = 0 .
5, respectively.We visually mark the vicinities of r = r cr , which will bediscussed in Subsec. IV G, with black arrows. In Fig. 11, we see that T SW /T SS > T SS < T SW ) forlarge N ( N = 200 and 1000 in this figure) regardless of r , whereas T SW /T SS < T SS > T SW ) for small N andlarge r ( N = 10 and r > . T SW /T SS takes its mini-mum value near r = 0 . N ( N = 200and N = 1000 in this figure), because the negative effectof preference is least active with r being slightly morethan 0.5, due to the positive effect of walking.On the other hand, T SW /T SS can take a minimumvalue with r = 1 for sufficiently small N ( N = 10 inthis figure), due to the enhanced positive effect of walk-ing for small N and large r . Due to the positive effectof walking, T SW /T SS ( r = 0) > T SW /T SS ( r = 1) even forlarge N . F. Effect of p In this subsection, we investigate the effect of p (in awalking lane) on T SW for various N . We set α = 0 .
5, and r = 0 .
5. We emphasize that T SS is constant regardless of p , as it is when changing r .Figure 12 plots the simulation (circles) and theoretical(curves) values of T SW /T SS as functions of p for various N ∈ { , , } . The simulation values again showgood agreement with the theoretical curves. p T (cid:1) (cid:2) / T (cid:1)(cid:1) N = 1000 N = 200 N = 10 (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:2)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:13)(cid:4)(cid:5)(cid:7)(cid:13)(cid:14)(cid:15)(cid:12)(cid:8)(cid:11)(cid:12)(cid:7)(cid:2)(cid:16)(cid:6)(cid:5)(cid:10)(cid:16)(cid:4)(cid:11)(cid:17)(cid:12)(cid:13) FIG. 12. (Color Online) Simulation (circles) and theoretical(curves) values of T SW /T SS as a function of p for various N ∈ { , , } . The otherparameters are fixed as α = 0 . r = 0 .
5, respectively.
In Fig. 12, we see that T SW /T SS > T SS < T SW ) and T is hardly affected by p , because the steady-state flowdoes not depend on p and the positive effect of walkingis very slight when r = 0 . N ≥ T SW /T SS = 1 ( T SS = T SW ) with p =0, at which both lanes act as standing lanes even whenStrategy SW is modeled. In this case, particles can entereither of the two lanes indiscriminately with Strategy SS,while particles still prefer either of the two lanes whenStrategy SW is modeled. G. Reversal point N cr and r cr In this subsection, we theoretically determine N cr ( α, r, p ) and r cr ( N ), at which T SS = T SW (see alsoFigs. 9—11). These theoretical values are then com-pared with the simulation values.From Eqs. (10) and (16), N cr for 0 < r < T SS ( N = N cr ) = T SW ( N = N cr ) ⇔ N cr α + L
1= 1 α + N cr − rα rα + (1 − r ) α − r ) α + (1 − r N ) L r N L p − r − r { − N r N − + ( N − r N } rα rα + (1 − r ) α − r ) α , (30)where N = min( N, N ) = min( N cr , N ). N cr is generally difficult to obtain for 0 < r <
1; how-ever, for the case r = 1, the general form of N cr can be2 (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:2)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:13)(cid:4)(cid:5)(cid:7)(cid:13)(cid:14)(cid:15)(cid:12)(cid:8)(cid:11)(cid:12)(cid:7)(cid:2)(cid:16)(cid:6)(cid:5)(cid:10)(cid:16)(cid:4)(cid:11)(cid:17)(cid:12) N r (cid:1) (cid:2) (d) (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:2)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:13)(cid:4)(cid:5)(cid:7)(cid:13)(cid:14)(cid:15)(cid:12)(cid:8)(cid:11)(cid:12)(cid:7)(cid:2)(cid:16)(cid:6)(cid:5)(cid:10)(cid:16)(cid:4)(cid:11)(cid:17)(cid:12) r (cid:1) (cid:1) (cid:2) (b) (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:2)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:13)(cid:4)(cid:5)(cid:7)(cid:13)(cid:14)(cid:15)(cid:12)(cid:8)(cid:11)(cid:12)(cid:7)(cid:2)(cid:16)(cid:6)(cid:5)(cid:10)(cid:16)(cid:4)(cid:11)(cid:17)(cid:12) (cid:1) (cid:1) (cid:1) (cid:2) (a) (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:2)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:13)(cid:4)(cid:5)(cid:7)(cid:13)(cid:14)(cid:15)(cid:12)(cid:8)(cid:11)(cid:12)(cid:7)(cid:2)(cid:16)(cid:6)(cid:5)(cid:10)(cid:16)(cid:4)(cid:11)(cid:17)(cid:12) p (cid:1) (cid:1) (cid:2) (c) FIG. 13. (Color Online) Simulation (circles) and theoretical (curve) values of (a) N cr as a function of α with( r, p ) = (0 . , . N cr as a function of r with ( α, p ) = (0 . , . N cr as a function of p with ( α, r ) = (0 . , r cr as a function of N with ( α, p ) = (0 . , . N cr satisfy T SS ( N = N cr ) > T SW ( N = N cr )and T SS ( N = N cr + 1) < T SW ( N = N cr + 1), and that those of r cr satisfy T SS ( r = r cr ) < T SW ( r = r cr ) and T SS ( r = r cr + 0 . > T SW ( r = r cr + 0 . obtained easily as T SS ( N = N cr ) = T SW ( N = N cr ) ⇔ N cr α + L α + N cr − α α + L p ⇔ N cr = pL p + 1 . (31)Similarly, r cr (0 < r cr <
1) satisfies the following rela-tion: T SS ( r = r cr ) = T SW ( r = r cr ) ⇔ Nα + L
1= 1 α + N − r cr α r cr α + (1 − r cr ) α − r cr ) α + (1 − r cr N ) L r N cr L p − r cr − r cr { − N r N − + ( N − r N cr } r cr α r cr α + (1 − r cr ) α − r cr ) α . (32)We note that r cr = 1 if no value in 0 < r cr < r for a fixed N . Figure 13 compares the simulation (circles) and theo-retical (curves) values of (a) N cr as a function of α , (b) N cr as a function of r , (c) N cr as a function of p , and (d) r cr as a function of N . When calculating N cr and r cr , thesimulation values of T were obtained as averages takenover 10 trials. In all figures, the simulations agree wellwith the theoretical curves.From the definition of N cr ( r cr ), T SS > T SW (Strat-egy SW is advantageous), if N < N cr ( r > r cr ); other-wise T SS < T SW (Strategy SS is advantageous), for fixed( α, r, p ) (for fixed ( N, α, p )). Therefore, these results in-dicate that Strategy SS is generally advantageous espe-cially for large N ; however, in limited cases Strategy SWcan perform better than Strategy SS, mainly for small α ,large r , large p , and small N . V. INDIVIDUAL (MICRO-SCALE)TRANSPORTATION TIMEWITH THE TWO-LANE MODEL
Now, we introduce the important novel quantity τ ( n, α, r, p ). The quantity τ is defined as the time gapbetween the start of the simulation and the time when3the n th-leaving particle leaves the system, which we referto as the individual transportation time. The quantity n is assumed to be sufficiently small compared to the totalnumber of particles when considering τ ( n, α, r, p ). Theindividual transportation times with Strategy SS and SWare represented as τ SS and τ SW , respectively, as when wewere discussing T .The quantity τ SS ( n = N ) clearly coincides with T SS ( N ) as the first-in-first-out condition is always sat-isfied with Strategy SS; however, τ SW ( n = N ) is gen-erally below T SW ( N ) since the first-in-first-out condi-tion might not hold with Strategy SW. We note that τ SW ( n = N ) = T SW ( N ) only for r = 0 and r = 1, forwhich the first-in-first-out condition must be satisfied, so r is set to 0 < r < A. Approximate theoretical analyses of n cr In this subsection, we theoretically determine approx-imate values for n cr , for which τ SS ( n = n cr ) = τ SW ( n = n cr ).Unlike we did when approximating T , we regard thesteady-state flow Q as the average number of particlesthat pass the right boundary in each step.In addition, we assume that (i) 1 /α time stepsare needed on average for the first-entering standing-preference (walking-preference) particle to enter the sys-tem [37], and that (ii) particles leave the system in thesteady state after the first-leaving particle leaves the sys-tem.Furthermore, the number of particles is assumed to besufficiently large. Especially, the number of all walking-preference particles needs to be larger than the averagenumber of walking-preference particles that leave the sys-tem until the time at which the first-leaving standing-preference particle leaves the system.We next define the threshold n = N ; specifically,the N th-leaving walking-preference particle leaves thesystem approximately at the same time when the first-leaving standing-preference particle leaves. Therefore, N satisfies( N − × Q W ≈ α + L − α − L p , (33)which reduces to N ≈ rα rα pL p + 1 . (34)Based on the above assumptions, the system behaviorcan be in three states; (i) both lanes are not yet in steadystate, (ii) only the walking lane is in steady state, and(iii) both lanes are in steady state. Table IV summarizesthe states (flows) of the system and relates them to timesteps in terms of the model parameters.Consequently, if we note the first N particles leave thesystem with Q W and the next ( n − N ) particles leave TABLE IV. States (flows) of the system at ranges of timesteps. ‘Stand,’ ‘Walk,’ and ‘Entire’ represent the standinglane, walking lane, and entire system, respectively. Inaddition, the notations ‘R.P.’ and ‘S.S.’ stand relaxationprocess and steady state, respectively.No. Time Stand Walk Entire1 0 ≤ t < α + L p R.P. (0) R.P. (0) R.P. (0)2 1 α + L p ≤ t < α + L Q W ) R.P. ( Q W )3 1 α + L ≤ t S.S. ( Q S ) S.S. ( Q W ) S.S. ( Q SW ) with Q SW , τ SW reduces to τ SW ( n, α, r, p ) ≈ α + L p + n − rα rα for n ≤ N , α + L n − N rα rα + (1 − r ) α − r ) α for n > N , (35)from Eqs. (5) and (6).On the other hand, since τ SS ( n = N ) = T SS ( N ), τ SS ( n, α, p ) can be written as τ SS ( n, α, p ) = T SS ( N = n ) ≈ L nα , (36)from Eq. (10).From the definition of n cr , and Eqs. (35) and (36), n cr satisfies the following equation: τ SS ( n = n cr ) = τ SW ( n = n cr ) ⇔ L n cr α = 1 α + L n cr − N rα rα + (1 − r ) α − r ) α . (37)The quantity τ SW is clearly smaller than τ SS for n < N ,so n cr > N . Therefore, n cr ( α, r, p ) finally reduces to n cr ( α, r, p ) = pLrα { − r ) α } [ { (1 − r ) + r } α + r (1 − r ) α ](1 + p ) + 1 , (38)using Eqs. (34) and (37). B. Comparison with simulation results
In this subsection, we compare the theoretical approx-imations, obtained in the previous subsection, with sim-ulation results.First, Fig. 14 shows the values of the ratio τ SW , sim /τ SW , theo as a function of n for ( α, r, p ) ∈{ (0 . , . , . , (0 . , . , . } . The simulation values4show very good agreement with the theoretical ones for n >
10. We note that the comparison regarding τ SS isabbreviated since τ SS ( n = N ) = T SS ( N ). τ (cid:1) (cid:2) , (cid:4) (cid:5) (cid:6) / τ (cid:1) (cid:2) , (cid:7) (cid:8) (cid:9) (cid:10) n ( (cid:1) , r, p ) = (0.2, 0.8, 0.5)( (cid:1) , r, p ) = (0.5, 0.2, 0.5) FIG. 14. (Color Online) Calculated values of the ratio τ SW , sim /τ SW , theo as functions of n for ( α, r, p ) ∈{ (0 . , . , , (0 . , . , . } , withfixed N = 1000. Both plots begin at n = 2. Next, Fig. 15 compares the simulation values of τ SS and τ SW as functions of n , with fixed N = 1000. InFigs. 15 (a) and (b), we see a point τ SS ( n ) = τ SW ( n ), inwhich n is defined as n = n cr . This phenomenon can beexplained qualitatively as follows.At first, some of the first entering waking-preferenceparticles leave the system when Strategy SW is simulatedmore quickly than when Strategy SS is simulated. In thiscase, for small n the positive effect of walking has thedominant effect on τ SW , so τ SS > τ SW . After some timehas elapsed, the negative effect of preference becomesdominant, resulting in τ SS < τ SW .Finally, Fig. 16 compares the simulation (dots) andtheoretical (curve) values for n cr as a function of r . InFig. 16, the simulation values show relatively good agree-ment with the theoretical curves. The upper (lower)bound of the curve of n cr indicates the number of parti-cles that can leave the system more quickly with StrategySS (Strategy SW) than with Strategy SW (Strategy SS),even if N is large.In addition, n cr takes the maximal value around r =0 . τ SW in Eq.(35) for n > N , the numerator is maximized when r = 1(see also Eq. (34)), while the denominator is maximizedwhen r = 0 . n, α, p ), τ SW takes its maximum value in the range 0 . < r <
1, whichimplies that n cr also takes its maximal value in the range0 . < r < n (cid:1) (a) τ (cid:1)(cid:2) τ (cid:1)(cid:1) n (cid:1) n = n (cid:1)(cid:2) (b) n = r N (cid:3) τ (cid:1)(cid:2) τ (cid:1)(cid:1) FIG. 15. (Color Online) (a) Simulation values of τ SS (red)and τ SW (blue) as functions of n , with fixed α = 0 . r = 0 . p = 0 .
5. (b) Zoomed-in inset Fig. 15 (a), focusing onthe range enclosed by a black dotted circle in Fig. 15 (a).The vicinities of n = rN and n = n cr are marked with blackarrows, respectively. r n (cid:1) (cid:2) (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:2)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:13)(cid:4)(cid:5)(cid:7)(cid:13)(cid:14)(cid:15)(cid:12)(cid:8)(cid:11)(cid:12)(cid:7)(cid:2)(cid:16)(cid:6)(cid:5)(cid:10)(cid:16)(cid:4)(cid:11)(cid:17)(cid:12) FIG. 16. (Color Online) Simulation (dots) and theoretical(curve) values of n cr as a function of r . The otherparameters are fixed as α = 0 . p = 0 .
5. We note thatthe simulation values of n cr satisfy τ SS ( n = n cr ) > τ SW ( n = n cr ) and τ SS ( n = n cr + 1) < τ SW ( n = n cr + 1). VI. CONCLUSION
We have analyzed three strategies for movement on anescalator: (i) two standing lanes (Strategy SS), (ii) onestanding lane plus one walking lane (Strategy SW), and(iii) two walking lanes (Strategy WW). These strategieswere modeled with a modified two-lane TASEP. The spe-cific contributions of this study are as follows.In Sec. III, we found that the one-lane model with ourmodified updating rules exhibits only the LD phase forall values of p , and that the steady-state flow for any p isidentical to that of the original open-boundary TASEPwith p = 1. This indicates that walking on an escalatordoes not affect steady-state flow, which is consistent withthe results in [17].In Sec. IV, we considered the total transportationtime T using theoretical analysis and numerical simu-lations. In steady state, Q SS is equal to Q WW and isalways greater than Q SW . For example, Q SW decreasesat 33%—50% compared to Q SS in the most-congestedsituations ( α = 1). This finding is consistent with the re-sults of a pilot performed in the London underground in2015 [8]; when ‘standing only’ was specified for escalatorsat Holborn station, about 30% more customers could usean escalator during the busiest times when both laneswere used only for standing.Since Q SW = Q WW , counter-intuitively, T SS ≈ T WW for sufficiently large N , and T SS is generally smaller than T SW , indicating that Strategy SS is advantageous forlarge N . On the other hand, in limited cases for small N , Strategy SW can be more useful. Those differencesarise because the negative effect of preference generallyexceeds the positive effect of walking with Strategy SWfor large N , while this relation is reversed for small N .We have determined the reversal point N = N cr and r = r cr , which satisfies T SS ( N = N cr ) = T SW ( N = N cr )and T SS ( r = r cr ) = T SW ( r = r cr ). These values confirmthat Strategy SW is advantageous mainly for small α ,large r , large p , and small N .In Sec. V, we considered the individual transportationtime τ using theoretical analysis and numerical simula-tions. Even if N is large, in which Strategy SS is ad-vantageous in terms of reducing T , the first-leaving n cr particles, mainly including walking-preference particles,can leave the system more quickly under Strategy SWthan under Strategy SS. This behavior arises because thepositive effect of walking is more active than the negativeeffect of preference in this case.Applying the results of the present paper to real situa-tions, we find that encouraging pedestrians to only standon an escalator, which has recently been suggested bypractitioners, is indeed beneficial for improving pedes-trian flow, especially when a large number of pedestrians(large N ) are using a facility. Conversely, providing awalking lane can improve pedestrian flow in limited casesin which the entrance is relatively uncongested (small α ), the fraction of walking-preference pedestrians is high(large r ), and the walking velocity is large (large p ) with a small number of pedestrians (small N ). In addition,the first-entering walking-preference pedestrians tend tobenefit from Strategy SW, even if the number of pedes-trians is large (large N ).More realistic models and comparisons with field datawill be needed to more clearly relate our findings to thereal world. For example, combining multiple modelsof escalators and floor fields, like a cellular-automatonpedestrian model [38], remains as a future work, alongwith fitting using actual data. Even so, the simple modeldiscussed above offers useful insights as a first step. ACKNOWLEDGMENTS
We greatly appreciate Airi Goto for her fruitful com-ments. This work was partially supported by JST-MiraiProgram Grant Number JPMJMI17D4, Japan, JSPSKAKENHI Grant Number JP15K17583, and MEXTas gPost-K Computer Exploratory Challengesh (Ex-ploratory Challenge 2: Construction of Models for In-teraction Among Multiple Socioeconomic Phenomena,Model Development and its Applications for EnablingRobust and Optimized Social Transportation Systems)(Project ID: hp180188).
Appendix A: Properties of Q SW In this appendix, we briefly give the details of the prop-erties of Q SW , i.e., Eq. (6), as a function of α and r .Defining f ( α, r ) as Q SW = f ( α, r ) = (1 − r ) α − r ) α + rα rα , (A1) f ( α, r ) can be rewritten as f ( α, r ) = 2 −
11 + rα −
11 + (1 − r ) α . (A2)Therefore, for fixed r (0 ≤ r ≤ f ( α, r ) = f ( α ) isobviously a monotonically increasing function of α (0 ≤ α ≤ α is constant hereafter.Replacing rα with x for simplicity, f ( α, r ) = f ( x ) canbe written as f ( x ) = α − x α − x + x x , (A3)where 0 ≤ x ≤ α . Taking the first derivation of f ( x ), df ( x ) /dx is calculated as df ( x ) dx = ( α − x ) { α + 4 x ( α − x ) } (1 + x ) (1 + α − x ) . (A4)The results of the first derivative test are summarizedin Tab. V. f ( x ) takes its maximum value 2 α/ (2+ α ) with x = α/ r = 0 . f ( α, r ) takes its maximum value ( f ( α, r ) =2 /
3) when α = 1 and r = 0 .
5. Figure 17 shows thecalculated values of f ( α, r ) in the ( α, r ) plane. FromFig. 17, f ( α, r ) is sensitive to r for large α .6 T SW ≈ N +1 X k =1 P ( k ) t SW ( k )= 1 α + N X k =1 { P ( k ) × ( T sSW ( N − k + 1) + T fS ) } + − N X k =1 P ( k ) ! × ( T sSW ( N ) + T fW )= 1 α + N X k =1 (cid:26) (1 − r ) r k − (cid:18) N − kQ SW + L (cid:19)(cid:27) + − N X k =1 (1 − r ) r k − ! × (cid:18) N − Q SW + L p (cid:19) = 1 α + N − Q SW − N X k =1 ( k − − r ) r k − Q SW + (1 − r N ) L r N L p (B1)= α + N − rα rα + (1 − r ) α − r ) α − r − r { − N r N − + ( N − r N } rα rα + (1 − r ) α − r ) α + (1 − r N ) L r N L p for r = 11 α + N − α α + L p for r = 1= 1 α + N − rα rα + (1 − r ) α − r ) α − P N k =1 (1 − r N − k ) r krα rα + (1 − r ) α − r ) α + (1 − r N ) L r N L p TABLE V. Result of the first derivative test of f ( x ) x α αf ( x ) ր f (cid:16) α (cid:17) ց df ( x ) dx + + 0 − − (cid:1)(cid:2)(cid:3) (cid:1)(cid:2)(cid:4) (cid:1)(cid:2)(cid:5) (cid:1)(cid:2)(cid:6) (cid:1)(cid:2)(cid:7) (cid:1)(cid:2)(cid:8) (cid:1)(cid:2)(cid:9) (cid:1)(cid:2)(cid:10) (cid:1)(cid:2)(cid:11) (cid:3)(cid:1)(cid:1)(cid:2)(cid:3)(cid:1)(cid:2)(cid:4)(cid:1)(cid:2)(cid:5)(cid:1)(cid:2)(cid:6)(cid:1)(cid:2)(cid:7)(cid:1)(cid:2)(cid:8)(cid:1)(cid:2)(cid:9)(cid:1)(cid:2)(cid:10)(cid:1)(cid:2)(cid:11)(cid:3)(cid:1) (cid:1)(cid:2)(cid:8)(cid:1)(cid:2)(cid:5)(cid:1)(cid:2)(cid:4)(cid:1)(cid:1)(cid:2)(cid:9)(cid:1)(cid:2)(cid:3) (cid:1) r (cid:1)(cid:2)(cid:6)(cid:1)(cid:2)(cid:7) f ( (cid:1) , r ) FIG. 17. (Color Online) Theoretical values of f ( α, r ) forvarious ( α, r ). Appendix B: Detailed calculation of T SW In this appendix, we discuss the detailed calculationsfor obtaining the general form of T SW .The specific calculations are summarized as Eq. (B1).We note that the third term of the fourth line in Eq. (B1) is the summation of an arithmetico-geometric sequence,and therefore, we can obtain a closed-form expression. Appendix C: Discussion of the behavior of T SW − T SS when α < α ≤ for sufficiently large N In this appendix, we briefly discuss the behavior of T SW − T SS when α < α ≤ N .For sufficiently large N ; especially N ≫ N , N = N ,due to N = min( N , N ). Therefore, using Eq. (18), T SW − T SS can be represented as follows: T SW − T SS = ( N − (cid:18) Q SW − Q SS (cid:19) − P N k =1 (1 − r N − k ) r k Q SW − pr N p L A − r − r ) α + r rα − B − − r ) α − rα − C, (C1)where A , B , and C can be regard as constant values;specifically, A = { − r (1 − r ) } ( N − > , (C2) B = N X k =1 (1 − r N − k ) r k > , (C3)and C = pr N p L > . (C4)7Noting that the the denominators of the first (second)terms of Eq. (C1) clearly increase (decrease) monotoni-cally with respect to α , from Eq. (C1)—(C4) we see that T SW − T SS is a monotonically increasing function of α .Considering that T SW − T SS > α = α andsufficiently large N , T SW − T SS always exceeds 0 for α < α ≤ Appendix D: Discussion of the behavior of g ( α ) when < α ≤ α for sufficiently large N Here, we discuss the details of the behavior of g ( α )when 0 < α ≤ α for sufficiently large N . The function g ( α ) can be rewritten as follows: g ( α ) = 1 − − r − r ) α − r rα − rpL (1 + p )( N − (cid:26) − r − r ) + rα rα (cid:27) = 1 − rpL (1 + p )( N − − − r − rpL (1+ p )( N − − r ) α − r − rpL (1+ p )( N − rα = 1 − rpL (1 + p )( N − − D − r ) α − E rα , (D1)where D and E are represented as D = 1 − r − rpL (1 + p )( N −
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E >
0; specifically,
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