Two dimensional outflows for cellular automata with shuffle updates
TTwo dimensional outflows for cellular automatawith shuffle updates
Chikashi Arita , Julien Cividini ∗ , C´ecile Appert-Rolland
21 Theoretical Physics, Saarland University, 66041 Saarbr¨ucken, Germany2 Laboratory of Theoretical Physics, CNRS (UMR 8627) andUniversity Paris-Sud, Building 210 91405 Orsay Cedex, France
Abstract
In this paper, we explore the two-dimensional behavior of cellular automata with shuffle updates.As a test case, we consider the evacuation of a square room by pedestrians modeled by a cellularautomaton model with a static floor field. Shuffle updates are characterized by a variable associatedto each particle and called phase , that can be interpreted as the phase in the step cycle in the frameof pedestrian flows. Here we also introduce a dynamics for these phases, in order to modify theproperties of the model. We investigate in particular the crossover between low- and high-densityregimes that occurs when the density of pedestrians increases, the dependency of the outflow in thestrength of the floor field, and the shape of the queue in front of the exit. Eventually we discuss therelevance of these results for pedestrians.
Modeling traffic systems has been a problem of growing interest in statistical physics during the last decade [1,2, 3, 4, 5]. Indeed, these problems usually involve a large number of interacting driven agents that exhibitcollective effects, and therefore belong to the realm of nonequilibrium statistical mechanics. Some examples ofthese collective effects are given by the spontaneous congestion of a highway [6, 7] or the pattern formationobserved in counter-propagating lanes of pedestrians [8, 9, 10] or at intersections [8, 9, 11, 12]. Such effects canoften be reproduced in minimal models inspired by statistical physics [13, 14, 15, 16, 17, 18, 19, 20]. Many ofthese models belong to the class of cellular automata. In these models, the moving agents (cars, pedestrians,molecular motors, etc) are usually modeled as particles hopping on a lattice according to predefined rules, andinteracting in particular through exclusion rules.Cellular automata are defined not only by the specification of the underlying lattice, and of the hopping rules,but also by the update order, i.e. the order in which the rules will be applied to the set of particles. It is knownthat update schemes can have a strong influence on the global dynamics of the system [21]. While randomsequential update is preferentially used in fundamental studies on out-of-equilibrium systems because of itscloseness to continuous time dynamics, other updates with lower fluctuations are used in traffic applications.The most widely used is the parallel update [3], for which all particles are updated at the same time. In contrastwith the random sequential update, the time step of parallel update can be given a physical meaning and beinterpreted as a reaction time.Some other update schemes have been proposed, such as the random shuffle update [22] and the frozen shuffleupdate [23], both inspired by pedestrian applications. We shall emphasize that all these shuffle updates canbe defined in terms of variables associated to the particles, and called phases . The phases determine the orderin which particles are updated. Both the random and frozen shuffle updates were investigated in detail in onedimension [24, 25, 26, 27, 23, 28, 29].In this paper, we want to explore the properties of these updates when applied to a two-dimensional model.As a test case, we chose to consider the evacuation of a square room. Pedestrians are modeled by a simpleso-called floor field model [16] that we will describe more precisely in section 2. ∗ Now in Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel In the random sequential update, one particle at a time step is chosen randomly and updated. a r X i v : . [ n li n . C G ] O c t D outflows for cellular automata with shuffle updates C Arita, J Cividini, C Appert-Rolland xy { { { { { (b)(a) Figure 1: (a) Geometry of the room for L = 11, l = 5 and (b) possible hopping directions in two particular configurations.The dots represent particles and the grey cells are walls. The cells of the room have coordinates ( x, y ) with | x | ≤ l and0 < y ≤ l + 1, and (0 ,
0) is the exit. The arrows represent the allowed hops for the particles in the centers. In the caseof hybrid shuffle update which is defined in subsection 2.4, the phase τ stays the same for the hops with blue arrows andis redrawn in the case of red arrows. When the density increases, a crossover occurs between low- and high-density regimes. We will characterizethis crossover, and the outflow in both regimes, including its dependency on the strength of the floor field. Wewill also discuss briefly the shape of the queue in front of the exit.Another aim of this paper is to illustrate by an example how it is possible to introduce a dynamics for thephases defining the update, and how this can modify the properties of the model. Eventually we shall discussthese results in view of pedestrian applications.This article is organized as follows. In section 2, we define the model in detail, and present the various shuffleupdate schemes used in the paper. In particular, we introduce some phase dynamics to define a hybrid shuffleupdate . Some theoretical and numerical results are presented in sections 3 and 4, where we study two different,low- and high-density regimes. In section 5, we turn to pedestrian applications and summarize some experimentalresults as well as modeling approaches from the literature. We discuss the most prominent features of evacuationflows that one would like to reproduce in models. In section 6 we give the conclusion of this article.
In this article we consider one of the simplest situations of pedestrians evacuating a room in the tradition ofcellular automata. The room is divided into cells. Pedestrians are simply represented by ‘particles’ hoppingstochastically from one cell to one of the neighbouring cells. Each cell can be occupied by at most one particle(‘simple exclusion’). Several evacuation models of this kind have been proposed [16, 17, 30, 31, 18], in which theparticles are usually updated in parallel, an update also traditionally used in car traffic problems.
We consider a square room of L × L cells, where L = 2 l + 1 is an odd number, see figure 1 for L = 11. Theroom has a unique exit located in the middle of a side, whose width is just the size of one cell. The cells ofthe room will be designated by their coordinates ( x, y ) with x = − l, − l + 1 , . . . , l and y = 1 , . . . , l + 1, with asupplementary cell (0 ,
0) representing the exit.In the following, particles will be allowed to hop from one cell to its empty neighbours, and a hop from ( x, y )to ( x (cid:48) , y (cid:48) ) will sometimes be denoted as ( x, y ) → ( x (cid:48) , y (cid:48) ). This latter notation implicitly assumes that x , y , x (cid:48) and y (cid:48) are such that the hop is indeed allowed, i.e. ( x, y ) and ( x (cid:48) , y (cid:48) ) are neighbours, ( x, y ) is occupied and ( x (cid:48) , y (cid:48) ) isempty. At initial time and during the whole time evolution the simple exclusion constraint is verified, i.e. there can beat most one particle on each cell.We direct the particles towards the exit by using a static floor field [16]: The probability that each particlehops to a certain target cell is determined by the distance between the target cell and the exit of the room.We define a parameter k ≥ norm by | r | ≡ (cid:112) x + y for r = ( x, y ). To each cell of the room we associate a weight w ( r ) ≡ e − k | r | . (1)A particle on cell ( x, y ) chooses its target cell among all of its von Neumann neighbourhood (cells ( x ± , y )and ( x, y ± r (cid:48) is chosen withprobability p ( r (cid:48) ) = Z − w ( r (cid:48) ) (2)with the normalization Z = w ( r ) + (cid:80) r (cid:48) w ( r (cid:48) ), where the summation runs over all the empty von Neumannneighbours of r . Figure 1 (b) summarizes the possible hops of a particle.Finally, cell (0 ,
0) is an exception to the rule (2): a particle standing on (0 ,
0) exits the system with probability1. In a larger simulation, its hopping probability would be determined by the downstream configuration.
When we use cellular automata, we need to specify an update scheme to fully define the model. In this paper,we shall exclusively consider shuffle updates.Shuffle updates are sequential updates, i.e. particles are updated one after the other. Here we chose toformulate these updates by associating to each particle i a phase τ i (0 ≤ τ i < τ i < τ i < · · · < τ i N , (3)where N is the number of particles present at a given time in the system. Note that with shuffle updates, ifsuccessive sites are occupied by particles with increasing phases, the set of particles can move as a whole onestep forward within one single time step.Equivalently to the previous definition, in a continuous time picture, one can consider that each particle i isupdated at time s + τ i , where s is the discrete time.Two variants of shuffle updates have been considered in the literature. In the frozen shuffle update, phasesare kept unchanged during the whole simulation. By contrast, in the random shuffle update, phases are drawnanew at the beginning of each time step.Shuffle updates have low fluctuations, in the sense that each particle is updated exactly once per time step.Still, for a given particle, the time between two updates can fluctuate between 0 and 2 for the random shuffleupdate, while it is always exactly one in the frozen shuffle update. Indeed, in the frozen shuffle update, a particle i is updated at times τ i , τ i + 1 , τ i + 2 , . . . .When particles hop on a one-dimensional lattice with a constant hopping probability p , the model is equivalentto the so-called totally asymmetric simple exclusion process [4]. In this 1D case, the fundamental diagram, i.e. the current J as a function of ρ , has already been calculated analytically for both updates in the deterministiccase ( p = 1). For the random shuffle update [24, 26], the current with deterministic hopping reads J ( ρ, p = 1) = (cid:40) ρ for ρ ≤ / , ρ (1 − ρ )2 ρ − (cid:104) exp (cid:16) ρ − ρ (cid:17) − (cid:105) for ρ > / . (4)In particular, this current has a maximum J = at ρ = . For the frozen shuffle update [23, 28], thefundamental diagram reads J ( ρ, p = 1) = (cid:26) ρ for ρ ≤ / , − ρ ) for ρ > / J = 2 /
3. This maximum corresponds to configurations where, after a short transient,particles form platoons . More precisely, an observer staying on a site and recording its occupation and the phases { τ i } of the particles that occupy it at each time step would see a sequence of particles with increasing phases,then a hole, then another sequence of particles with increasing phases, and so on. A sequence of particles withouta hole between them will be identified as a platoon. If we denote the average number of particles in a platoon by ν , then ν particles will exit every ν + 1 time steps, thus giving a maximal current νν +1 . For phases drawn from a See [32] for a comparison of the queue shapes when Euclidian or Manhattan distances are used.
3D outflows for cellular automata with shuffle updates C Arita, J Cividini, C Appert-Rollanduniform distribution of values between 0 and 1, ν can be shown to be equal to 2 [23], hence the maximal currentvalue.We see that J > J . Indeed, the repeated changes in the updating order of random shuffle update providesa supplementary source of blockings between particles, leading to less regular motion.For the frozen shuffle update, one must keep in mind that the flow may depend on the exact realization of theupdating order, i.e. on the specific choice for the phases. This is the case for closed systems, or for the evacuationproblem of this paper, or more generally for any system that involves a bounded number of particles whateverthe simulation duration is. The phases are like a frozen disorder , and one needs to take ensemble averages to getthe mean behavior [23].By contrast, in open systems with incoming flows [28], the time average would coincide with an ensembleaverage over the frozen disorder, i.e. over the phases, and a single simulation would be enough. Systems with random shuffle and frozen shuffle updates do not behave in the same way. For the first one,the renewing of the updating order introduces some randomness in the bulk particle dynamics while for thesecond, the disorder is injected at the boundaries and the bulk dynamics is deterministic (if the hopping rulesare deterministic).It is also possible to combine both aspects by introducing a dynamics of the phases themselves. The choiceof this dynamics is not unique. We will present one possible choice of hybrid shuffle updates in this section .This choice was guided by the request to have an update behaving similarly to the frozen shuffle update whenthe particle density is low, and closer to the random shuffle update inside congestions. We will discuss furtherour choice in section 5, but first we will define the hybrid shuffle update of this paper.We introduce the following rule. If a particle i attempts to hop from ( x, y ) to ( x + a, y ) (or ( x, y + a )), and ifthe neighbours of the arrival cell ( x + a, y ±
1) (or ( x ± , y + a )) are both occupied by other particles, then thephase of the hopping particle τ i is redrawn from a uniform distribution on 0 ≤ τ i < i.e. the hops are always allowed whenparticles chose a neighbouring empty cell, according to the probability (2). Only the order of update will bemodified, starting from the next time step.As particles on (0 ,
0) are supposed here to quickly step out, and thus not to be in the congestion anymore,we do not redraw the phase when a particle hops to (1 ,
0) even if (0 ,
0) and (2 ,
0) are occupied.Note that, in the one dimensional case, no renewal of phases occurs by definition. Therefore the hybrid shuffleupdate scheme is equivalent to the frozen one in one dimension, and we have J = J = 23 . (6)As we mentioned, the shuffle update schemes presented in section 2.3 were already studied in detail in onedimensional systems (at least for the deterministic TASEP). Therefore one of our particular interests in this articleis to investigate their behavior in a typical two dimensional system, i.e. an evacuation from a room though asmall exit. In the next two sections, we shall explore the behavior of the evacuation model, and distinguish twodynamical regimes. At initial time t = 0, N particles are dropped randomly on the lattice of figure 1 and their phases { τ i } are drawnrandomly from a uniform distribution. In this section and the next one, we show simulation results that we haveperformed with the room size L = 51 and various initial numbers N of particles, using either the random,frozen or hybrid shuffle update scheme. We will be interested in the total evacuation times T , i.e. the time whenthe last particle leaves the room, and the flux J of particles exiting the room, which is identical to the currentbetween (0 ,
1) and (0 , T versus the initial number N of particles. When the number ofparticles increases, there is a smooth transition between two qualitatively different regimes, referred to as the low-density and high-density regimes. Let us first study the low-density regime. In reference [33] we have introduced another hybrid shuffle update. The main difference is that here target sites are necessarilyempty, hence the phase dynamics cannot be based on the occupancy of the target site as in reference [33]. The choice made in thisarticle leads to higher flows in high density regime.
4D outflows for cellular automata with shuffle updates C Arita, J Cividini, C Appert-RollandFor low densities, the evacuation time is observed to be independent of the update scheme. Indeed, in thisfree flow regime, particles exit almost freely, and the velocity of non-interacting particles is the same for the threeupdates.The increase of the total evacuation time T with N is then simply due to the increase of the average of thedistance between the farthest particle and the door. A particle starting on cell ( x, y ) is at a Manhattan distance d = | x | + y from the exit. Neglecting all interactions with other particles, and assuming k = ∞ , this particle willneed a time t = d + 1 = | x | + y + 1 to evacuate the room. Recalling that the shape of the room is square with L cells ( L = 2 l + 1) and that the exit is located at the middle of one side, the number n ( d ) of cells located at aManhattan distance d from the exit is found to be n ( d ) = d − ≤ d ≤ l, l + 1 for l + 1 ≤ d ≤ l + 1 , l + 4 − d for 2 l + 2 ≤ d ≤ l + 1 . (7)Note that the n ( d ) takes a maximum value L when l + 1 ≤ d ≤ l + 1. Then we shall show that the average totalevacuation time T is given by T = 1 + (cid:18) L N (cid:19) − l +1 (cid:88) d =1 d (cid:20)(cid:18)(cid:80) dd (cid:48) =1 n ( d (cid:48) ) N (cid:19) − (cid:18)(cid:80) d − d (cid:48) =1 n ( d (cid:48) ) N (cid:19)(cid:21) , (8)where (cid:18) ab (cid:19) = a ! b !( b − a )! is a binomial coefficient. In equation (8) the difference of binomial coefficients is thenumber of initial configurations where the farthest particle is at distance exactly d from the exit. One obtainsthe probability that the farthest distance is d , by dividing it by the total number of initial configurations (cid:18) L N (cid:19) ,and then the summation over d = 1 , . . . , l + 1 with multiplication by d gives the average farthest distance. Theexpression (8) of T versus N is plotted for L = 51 in figure 2 (a). We observe that the agreement is good when N is small, i.e. as long as particles (almost) do not interact. However, the true evacuation time becomes largerthan predicted by (8) as N increases, as a result of the collective effects occurring in the high-density regime.We now discuss in more detail the crossover towards this large N regime. Because of the geometry of theroom, the number of particles arriving at the exit is not constant in time. A jam will form at the exit if theincoming flow J in ( t ) becomes larger than the maximal current that the exit bottleneck can sustain. This limitvalue is called the capacity of the bottleneck, and is equal to the outflow in the high-density regime, denoted by J . The value of J will be determined in subsection 4.1. For now it is enough to know that J takes somenumerical value that depends on the update scheme, but not on L or t .Let us first compute the incoming flow J in of particles. For the time being, we assume that particles movewithout interacting with each other. In the case k = ∞ , the time t needed by a non-interacting particle to arriveat the exit site (0 ,
0) is equal to the Manhattan distance d = | x | + y between its initial position ( x, y ) and theexit. Then, for an initial uniform density N/L of particles, the inflow coming to the exit at time t is J in ( t ) = n ( t ) × N/L , (9)where n ( t ) is given by (7). Between an initial increase and a final decrease, the function J in ( t ) reaches a plateauvalue J in max = N/L .When J in max > J , congestion occurs, i.e. one enters the high-density regime. The crossover thus occurs,when the number of particles is N c = J L. (10)In the next section we shall focus on the high-density regime and determine the update-dependent value of J .This will allow us in particular to locate the crossover between the two regimes.For finite k , a similar transition is observed, see figure 2 (c,d). The form (8) is no longer valid in the low-density regime for finite k . Instead of this we compared the genuine evacuation times of N particles (coloredmarkers) with the single-particle problem ( i.e. initially only one particle is put randomly in the room). The solidlines were obtained by averaging the maximum of evacuation times T i of N independent simulation runs of thesingle-particle problem, i.e. formally E [max { T , · · · , T N } ] . (11) In this section we study the evacuation from the room in the high-density regime.5D outflows for cellular automata with shuffle updates C Arita, J Cividini, C Appert-Rolland
T N (a)
RH F T N (b) T N (c)
RH F T N (d)
10 1010 Figure 2:
Total evacuation time T versus the number of particles N with random (red × ), frozen (blue (cid:13) ) and hybrid(green (cid:3) ) shuffle updates in linear (left) and logarithmic (right) scales. The room size is L = 51 , and the parameter k ischosen as k = ∞ (a,b) and k = 3 (c,d). The plot markers were obtained by averaging over 10 simulation runs. The blacksolid lines in (a,b) and (c,d) correspond to (8) and (11), respectively. For comparison, we drew dashed lines J = const . × t in (b) and (d). In (a) and (c), the arrows indicate the crossover points (10) between the low- and high-density phases, seethe last paragraph of Subsection 4.1. k = ∞ ) From figure 2 it can be seen that in the high-density regime the evacuation time grows linearly with the number ofparticles. Moreover, it can be checked in figure 3 that the particle current flowing out of the system (the outflow J X ) is constant in time except in the initial and final transients, so that the evacuation time is approximatelygiven by T = N/J X , where J X depends on the update schemes, i.e. X =R (random shuffle), F (random shuffle)and H (hybrid shuffle). It remains to compute the value of this current for these three updates, which is donefor k = ∞ in the following paragraphs.We start with the case of random shuffle update. The outflow will be determined by the dynamics aroundthe exit. We therefore write an approximate master equation for the P ab ( t ), where the letters a and b stand forthe occupations of cells (0 ,
1) and (0 , e.g. we write a = 1 or a = 0 depending on whether (0 , , , − ,
1) are always occupied. Under this first order approximation and with the simple choice k = ∞ , cells(0 ,
1) and (0 ,
0) are never simultaneously empty at integer times and we can write a closed equation for the threeremaining probabilities as (cid:126)P ( t + 1) = M (cid:126)P ( t ) , where (cid:126)P ( t ) = (cid:32) P ( t ) P ( t ) P ( t ) (cid:33) , M = (cid:32) / / / / / (cid:33) . (12)Let us explain for instance how the first column of the matrix can be obtained. It corresponds to the transitionprobabilities from state to states , and . Starting from state , the particle in (0 ,
1) will surely hop to (0 , , , ,
2) and ( − , ,
1) draws the highest phase (probability 1 / whereas in the opposite case (probability 3 /
4) at least one particle from (1 , ,
2) or( − ,
1) will be able to hop towards (0 ,
1) and the next state will be .By calculating the stationary solution i.e. (cid:126)P = M (cid:126)P , we find a stationary current J = P + P = 4371 = 0 . . . . . (13)This prediction is in good agreement with the simulations, see figure 3(a).6D outflows for cellular automata with shuffle updates C Arita, J Cividini, C Appert-Rolland k =1 k =2 k =10 J =43/71 J t (a) k = , theory) k =1 k =2 k =10 J t (b) k =1 k =2 k =10 J =2/3 J t (c)
Figure 3: Outflows J X as functions of time for random shuffle (a), frozen shuffle (b) and hybrid shuffle (c) updatesfor three values of k = 1, 2 and 10 and for a room of size L = 51 with quarter-filling, N = ( L − / L always has an odd number of sites). We are thus clearly in the high-density regime. The theoreticalpredictions for k = ∞ are J = 43 /
71 for random shuffle update and J = 1 for frozen shuffle. For comparison,we also drew the line J = 2 / { τ i } of the particles that occupyit at each time step would see a sequence of particles with increasing phases, then a hole, then another sequenceof particles with increasing phases, and so on. A sequence of particles without a hole between them will beidentified as a platoon. Platoons are self-organized in a stronger sense than the 1D case. The ordering of theparticles is not conserved in 2D. And thus they become very large. Numerically we observe that the average sizeof these structures keeps growing with the number of particles in the room. This fact leads to the current J = 1 (14)which is expected to become the exact value in the limit N → ∞ .In the high-density regime of the two-dimensional evacuation, the hybrid shuffle update is very similar to therandom shuffle update because there is a queue near the exit. When a particle i hops, most of the neighbouringcells are occupied, so that its phase τ i will often be redrawn. The outflow of the hybrid case is therefore similar tothat of the random case, as shown in figure 3. The remaining difference comes from the fact that, in the hybridcase, phases are often but not systematically redrawn. Still, as we shall see below, it is enough to strongly limitthe 2D outflow as we wanted. While we were able to predict the flow precisely for the random case in (12), itis not so easy to do it for the hybrid shuffle update, because of the remaining ‘correlations’ of the phases. Bysimulation results, the two dimensional current for the hybrid case is measured as J ≈ . . (15)The maximal one dimensional flow J for the hybrid shuffle update scheme is equal to 2 /
3, as stated in (6).The interactions between particles result from the dynamics of the phases { τ i } .Now we go back to the problem of the crossover between low- and high-density regimes. By substitutingthe values of the outflows in the high-density regime (13), (14) and (15) into (10), we estimate the transitionsbetween the low- and high-density regimes to occur at N c ≈ . ,
51 and 32 . L = 51. These values agree with the numerical data, see figure 2 (a). k In the preceding subsection we studied the case k = ∞ . Here we consider the effect of varying k on the totalevacuation time T .In figure 4 (a) we plotted the total evacuation time as a function of k for the three updates. The evacuationtime is monotonous and decreases with increasing k . In this congested two-dimensional situation the hybridshuffle case is closer to the random shuffle one, as expected.Now we apply an approximation, which is similar to what we did for k = ∞ : all the sites ( x, y ) with | x | + | y | = 2are assumed to be occupied at any integer time. For the calculation of hopping probabilities, we also use anapproximation that we take into account only the two choices, i.e. to stay at the same site and to move theneighbouring site closer to the exit. Then the probability of particles’ hops (0 , → (0 , − , → (0 , + kT / N (a) kT ( k ) {1(b) T ( ) Figure 4: (a) Total evacuation time per particle as a function of k and (b) its relative variation with k for the random (red × ), frozen (blue (cid:13) ) and hybrid (green (cid:3) ) shuffle update schemes with L = 51 and N = ( L − /
4. For better visibility,we used logarithmic scales for k and T ( k ) /T ( ∞ ) − T /N = 1 /J with equation (13) in the random shuffle case (dashed line). For comparison,we also drew the dotted line T /N = 1 .
5, which is obtained if we assume that the outflow is identical to the 1D flow in thehybrid shuffle case. To obtain the data points we averaged over 10 simulation runs. Due to the finiteness of simulationruns, fluctuations for large k are observed in the log scale plots. (0 , → (0 ,
1) and (1 , → (0 ,
1) (denoted by p , p , p and p , respectively) are given as p = 11 + e − k , p = p = e − k e − k + e −√ k , p = e − k e − k + e − k . (16)Now we extend the master equation (12) to general k . Since (0 ,
0) and (0 ,
1) can be simultaneously empty, weadd P ( t ) to the probability vector: (cid:126)P ( t + 1) = M (cid:126)P ( t ) , (cid:126)P ( t ) = P ( t ) P ( t ) P ( t ) P ( t ) . (17)For the random-shuffle case, the elements of the transition probability matrix are given as M = s s s + s +3 s p s +2 s +3 s p − s − s − p − p − s − s − s p − s − s − s p , (18)where s = q + q + q , s = q q + q q + q q , s = q q q , q i = 1 − p i . Solving the balance equation (cid:126)P = M (cid:126)P ,we finally find the outflow as J R ( k ) = P + P = p (1 − s )+ p (1 − s )(9+ s − s − s )120(1 − s )+2 p (42+3 s +2 s − s +2 s s +3 s s +12 s )+ p (9+ s − s − s ) . (19)The total evacuation time is then estimated by T R ( k ) = N/J R ( k ).For the frozen and hybrid cases, the correlation among phases prevents us from obtaining M explicitly. Infigure 4, however, we luckily observe that the plots T H ( k ) /T H ( ∞ ) − T R ( k ) /T R ( ∞ ) − k is estimated as T H ( k ) = T H ( ∞ ) J R ( ∞ ) /J R ( k ) by using (19). In the frozen case, when a particle of a platoon does not hop,another one may squeeze in and modify the structure of the platoons, hence a stronger effect of finite k values.Finally we discuss on the crossover points between the high- and low-density regimes. For finite k , the inflowof the low-density regime at time t , J in ( t ), is not given by (9). We rather determine it from single-particlesimulations, using the approximation J in ( t ) ≈ N × P ( T i = t ) , (20)where P ( T i = t ) is the probability of the evacuation time T i in each simulation run of the single-particle problem.Then the crossover points are estimated from max t J in ( t ) = J out i.e. N c ≈ J out (cid:14) max t P ( T i = t ) (21)with the outflows J out measured in the three updates with finite k . For k = 3, the crossover points are numericallyestimated to be N c ≈ . , . .
2, which provide good estimations, see figure 2 (c).8D outflows for cellular automata with shuffle updates C Arita, J Cividini, C Appert-Rolland
Some of the most impressive collective effects exhibited by pedestrian crowds are found in evacuations, seereference [34] for review. Countless human disasters have happened during mass events, where the motion of thecrowd lead to stampedes [35, 36].Modeling is useful to predict the crowd behavior, but also as a tool to understand the link between individualbehavior and collective effects. In particular, interactions between individuals play a key role in evacuationprocesses.Several types of interactions take place during evacuation processes. At very high density, the flow at abottleneck can be completely stopped due to the formation of arches, in which the pedestrians are compactedand cannot move because of physical contact with their neighbours.At low densities, pedestrians interact without physical contact through so-called social interactions, in orderto avoid collisions while minimizing the inconvenience of deviating from their optimal trajectory. A full charac-terization of these interactions is difficult, as they are non local, non isotropic, they include some anticipation,and they are non-additive when more than two pedestrians are involved.Another way to characterize the interactions is to observe their consequences on the collective behavior andto infer from their most relevant characteristics of interactions.
One interesting feature is the role of an obstacle suitably located near the exit of a room. Some experiments havebeen performed, which compare the outflows from the room, with and without obstacle. When pedestrians are ina panic-like condition, i.e. with strong physical contacts, the presence of a well located obstacle can surprisinglyincrease the outflow by about 30% [35]. But for obvious security reasons, this effect has not been systematicallyconfirmed .In more “normal” conditions, the difference between outflows with and without obstacle are far less pro-nounced [35, 31, 32] (the outflow increase is estimated around 4% in [31], 6% in [32]) and it should still beconfirmed whether this weak increase is also observed in real life conditions. However, even if the outflows aremore or less equal, this is still surprising, as we could expect that an obstacle would decrease the outflow (andindeed this is the case for ill-located or ill-sized obstacles [32]). Another effect that was conjectured from numerical observations is the so-called faster-is-slower effect, namely thefact that a stronger will of pedestrians to go out may lead to longer evacuation times. An experiment of aircraftevacuation showed a trend to have larger evacuation times in competitive trials compared to non-competitiveones, but the result was strongly geometry dependent and opposite effects could also be observed [39]. Morerecently, it was shown [40, 41] that indeed a faster-is-slower effect could be obtained with humans, leading to anincrease of evacuation times of typically 15 to 20% under competitive egress, and related to physical contacts andthe formation of clogs. A faster-is-slower effect can also be observed in some other systems that exhibit clogging,namely granular matter or sheep [41]. By contrast, as ants do not clog but rather keep their density always moreor less constant even under stress [42], no significant faster-is-slower effect was observed for ants [43], except inthe trivial limit where the stress source has an impact on the physiology of the ants, and thus on their steppingbehavior [42]. In this paper, we rather focus on normal evacuation conditions of pedestrians, i.e. with limitedphysical contacts, for which no faster-is-slower effect was ever evidenced. It is an open question to characterizethe interactions between pedestrians in these more ordinary situations.
An experimental observation that may seem less spectacular but has strong implications for modeling, is that,if one compares the outflow of a single line and of two-dimensional flows, one finds that 2D outflows are similaror even sometimes smaller than 1D outflows . Indeed, in [31], two experiments are performed and the reported A similar effect was observed for ants [37] and for granular matter [38], but the systems are too different to extrapolate topedestrians. This remark applies for disordered random 2D flows. If pedestrians are pre-ordered into several incoming lines, higher flows canbe achieved [31, 32], but this situation is quite special and we shall not consider it here. Still, it is an open question to know howto account for this phenomenon in simulations. Due to the discrete lattice, straight motions in directions not parallel to the latticehave to be decomposed into hops of different orientation. Besides, the exclusion rule is too crude to account for the subtle zipper
9D outflows for cellular automata with shuffle updates C Arita, J Cividini, C Appert-Rollanddecrease of the 2D outflow compared to the 1D outflow is respectively 4% and 14%.The fact that 2D outflows are not larger than 1D outflows is quite surprising. In fact, in two-dimensionalflows, more pedestrians arrive at the exit. This overfeeding effect should lead to higher flows . Actually this iswhat we observe when we compare the 1D and 2D outflows obtained in sections 2.3 and 4 : Both for the frozenand random shuffle updates, the 2D outflow is larger than the one-dimensional one. This makes it more clearnow why we have introduced the hybrid shuffle update defined in section 2.4. One sees from figure 3 (c) thatfor this update, the 2D and 1D outflows (both measured for the deterministic case k = ∞ to compare equivalentmodels) are similar.The fact that 2D outflows are not larger than 1D outflows indicates that there are some interactions betweenpedestrians beyond the hard-core ones, that compensate the overfeeding. As these interactions decrease the flow,they are often assimilated with friction. One must be careful that this notion of friction can refer to differenttypes of interactions (with physical contacts or without in ‘normal’ flows), with different consequences on thecollective behavior as we explained above. The choice of the friction implemented in the simulations implicitlyassumes whether physical contacts are considered or not and it is not clear yet whether a single model can accountfor all these different regimes. Most cellular automata models for pedestrians use parallel update. This update has the particularity thatconflicts occur: when two pedestrians have the same target site, the exclusion rule requires to select which onewill actually hop. The need to include a conflict resolution procedure is in general considered as a drawback insimulations. However, in the case of pedestrian flows, it was suggested that these conflicts could have a physicalrelevance [17, 30, 44]. Indeed, they occur preferentially in these converging areas where friction should occur.An easy way to tune the friction level is to allow with a certain probability none of the pedestrians involved tomove. This clearly reduces the overall flow when the number of conflicts increases.
One interesting feature of the frozen shuffle update is the possibility to interpret the phases τ i as the phases in thestepping cycle of pedestrians. This allows in particular, for 1D deterministic flows, to map the cellular automatondynamics in the free flow phase onto a continuous space and time dynamics [28], reproducing the regular motionof free pedestrians. This also gives a physical interpretation to the priority given to one pedestrian when twopedestrians meet. The time step of the update, which plays the role of a reaction time, is then directly connectedto the stepping period – actually it is known that pedestrians cannot turn/change velocity at any points of thestepping cycle but only at specific moments [45]. In the frame of this interpretation of the phases, it becomesnatural to modify the phases in the jammed phase, as done by hybrid shuffle updates. Indeed there is no reasonwhy a pedestrian should keep its stepping phase while being blocked by its predecessor.It should be noted that cellular automata are rather mesoscopic models that do not aim to reproduce theprecise microscopic dynamics but simply to give the correct average flows and densities at specific key locations.Still, it would be interesting to see if features coming from the stepping behavior of pedestrians could be retainedand give interesting properties to the models, closer to what happens at the scale of individuals.The hybrid shuffle update presented in this paper illustrates that indeed some interactions leading to flowreduction in the congested phase can be implemented through a partial renewal of the phases. The choice wehave made is not unique (see [33] for another one). One difficulty is to trigger friction preferentially in convergentflows – and not in all high density situations . Still, we have not explored all the possibilities opened by hybridupdates. While in this paper the phase dynamics consists simply in a partial redrawing of the phases from auniform distribution, one could also consider some evolution rules for the phases where the new phase valuewould depend for example on its previous value, on the phases of its neighbours, or possibly on the phase of thelast visitor of the target cell. Flow-density relations have been measured in several experiments. However a consensus is difficult to reach,first, because the fundamental diagram depends on the geometry of the system, and second, because the precise effects that are favored with pre-ordered lines at the bottleneck [31]. Indeed, as we said in the previous footnote, this increase is what happens with pre-ordered lines [31, 32]. For the parallel update, conflict occurrence automatically coincides with convergence of the flow.
10D outflows for cellular automata with shuffle updates C Arita, J Cividini, C Appert-Rolland t =0 t =300 t =600 t =900(a) t =0 t =200 t =400 t =600(b) t =0 t =300 t =600 t =900(c) Figure 5: Typical queue shapes for random shuffle (a), frozen shuffle (b) and hybrid shuffle (c) updates withparameters L = 51 , N = ( L − / k = 5.method used to measure velocity and density may have a strong influence on the results [18, 46]. However, nowthat many data have been collected, one can nevertheless make a few observations.A first set of experiments deals with one-dimensional flows, where pedestrians walk in a line without passingeach other [47, 48, 49]. If one would crudely approximate the fundamental diagram in this one-dimensional caseby a triangular shape, the density for which the maximal flow should be clearly smaller than half the maximaldensity [50]. This is not the case for the one-dimensional TASEP, for which the density of maximum flow is ρ = 1 / ρ = 2 / We end this discussion by a few observations on the queue shapes. Though having a realistic queue shape infront of the bottleneck is not necessarily required as long as outflows are correct, it would be more satisfactoryto approach the shape of real queues. In figure 5, we show the positions of particles at different moments oftime, as obtained with the model of this paper for k = 5. The queues look like half-circles for the three updatesconsidered. This is what is usually observed in simulations. In experiments, shapes like droplets have ratherbeen observed (see figure 4 in [54]). This difference of queue shape between simulations and experiments wasalready mentioned in [34]. We have shown in reference [33] a simulation snapshot exhibiting a queue with dropletshape, as a result of some modifications of the rules, especially adding diagonal hopping. Indeed, it was shownin [55] how the queue shape can vary strongly (including very unrealistic shapes) depending on the allowed hopsand their associated rates. Another way to decrease the density on each side of the exit is by penalizing sharpturns [31]. 11D outflows for cellular automata with shuffle updates C Arita, J Cividini, C Appert-RollandRandom Frozen Hybrid J / / / J /
71 1 0 . k = ∞ ). All the values of J are exact in the limit of an infinite system. Thevalue of J for frozen shuffle update is also expected to be exact in the limit of an infinite system. For randomshuffle update, J = was led by an approximation, and the 2D value for hybrid shuffle update was measurednumerically. In this article, we have investigated the properties of shuffle updates in two-dimensions through the study of asimple evacuation problem. A particular attention has been given to evacuation times and outflows, which wehave studied both numerically and analytically.In this study, we have considered a floor field model with only a static field. In general floor field models, a dynamic floor field is also considered, which allows to transform some long range interactions between pedestriansinto purely local ones [16]. Indeed our aim was not to design a full pedestrian model but to explore howinteractions between pedestrians could be modified through the update scheme itself. More generally, it isan important issue to understand how to model properly inter-pedestrian interactions, as they determine inparticular the outflow of evacuation problems.We have used a unified picture to define the various shuffle updates, by associating a phase τ i to each particle.Then updates differ only by the dynamics of these phases. For the random shuffle update, all the phases areredrawn in every time step. For the frozen one, phases are initially ( t = 0) given to particles, and we keep themtill the end of evacuation. For the hybrid one, we partially redraw some of the phases. The last one interpolatesbetween the frozen at low density and the random in congested configurations. Using the hybrid shuffle updatethe phases { τ i } keep their interpretation as trackers of the walking cycle, and the dynamics of the phases in thehigh density phase simply corresponds to a loss of memory of the stepping cycle phase when pedestrians areforced to slow down or stop.It is clear that many updating procedures that interpolate between frozen shuffle and random shuffle exist,and that the choice we have presented in this article is only a particular instance of them, that we have chosen toillustrate how model properties can be modified by including some phase dynamics. We found that this test casehas similar 1D and 2D outflows. Table 1 shows the flows for the shuffle updates, summarizing subsections 2.3,2.4 and 4.1.Other features could be included, in particular via the phase dynamics. For example, in reference [56]measurements of the time gap between two consecutive pedestrians are presented. In our model, phases allow tohave an interpretation of the dynamics in continuous time, and hence it is possible to define time gaps. Attemptscould be made to monitor the distribution of this observable, e.g. by drawing the phase τ i of a blocked particlein correlation with the phase of the blocking particle.The idea of deriving an updating order from the stepping dynamics was even pushed further in the so-calledoptimal steps model [57, 58, 59], for which not only the phase but also the duration of the stepping cycle and thelength of the steps vary from one individual to another one. Though in the case of a constant step length theoptimal steps model can also be applied to cellular automata models, it is more generally defined in a continuousspace. It would be interesting to study the influence of this update on outflows too.Improvement of pedestrian models would require a better understanding of flow-density relations. Beyondthe fact that a better consensus should be reached on which fundamental diagram the models should verify, onemust be aware that fundamental diagrams are supposed to be obtained in stationary regimes. In practice, inexperiments, one realizes short-term and localized averages. In the extreme case where instantaneous individualsmeasurements are realized, one observes quite different flow-velocity relations. Indeed, the fluctuations aroundthe mean behavior can be quite strong [49]. It is an open question to determine whether real pedestrian flows –including the important case of evacuations – can be fully described by implementing stationary laws, or whetherthey could be dominated by transients, which can give quite different flow values, not necessarily well reproducedby models calibrated in the stationary regime. Another issue to improve cellular automata models would be tostudy more systematically the isotropy of their properties [60].12D outflows for cellular automata with shuffle updates C Arita, J Cividini, C Appert-Rolland Acknowledgements
We thank Ludger Santen for useful discussions and Daniel R. Parisi for private communication. We are gratefulto an anonymous referee for critical reading of the manuscript and many valuable comments.