J. Cividini, C. Appert-Rolland and H.J. Hilhorst
University Paris-Sud, CNRS; Laboratory of Theoretical Physics Batiment 210, F-91405
ORSAY Cedex, France.
Julien.Cividini@th.u-psud.fr, Cecile.Appert-Rolland@th.upsud.fr, Henk.Hilhorst@th.u-psud.fr
Abstract.
We introduce a new type of update for ceullar automata representing pedestrian traffic on a lattice. We investigate this update analytically and by Monte-Carlo simulations on a simple model. We are able to predict the fundamental diagram, and phase diagrams in several geometries.
Keywords: Pedestrian traffic, exclusion process, shuffle update
As part of the effort to model pedestrian motion, the past years have seen the development of cellular automata based models, among which the so-called ‘floor field’ model [1, 2]. These models represent pedestrians as particles that jump from site to site on a discrete lattice, with an exclusion principle that forbids two pedestrians to simultaneously occupy the same site. In general, the pedestrians’ positions are subject to a parallel update procedure, i.e., the particles attempt to jump at the same instants of time. Although this type of update ensures a quite regular motion under free flow conditions, it creates conflicts when two or more pedestrians try to move at the same time to the same target site. Introducing an arbitrary numerical decision procedure to resolve these conflicts can be considered as a drawback and alternative update procedures have been looked for. In reference [3], for example, large pedestrian simulations are performed by means of a random shuffle update. In this sequential update scheme an updating order for the pedestrians is drawn at random at the beginning of each time step.
The random shuffle update was first proposed and studied in [4, 5]. It was applied there to the Totally Asymmetric Simple Exclusion Process (TASEP), which can be adfa, p. 1, 2011.
© Springer-Verlag Berlin Heidelberg 2011

seen as a basic model for pedestrian traffic. A TASEP (for Totally Asymmetric
Simple Exclusion Process) is a cellular automaton in which particles move on a lattice by jumping from one box to the next one, always in the same direction, say from the left to the right. This is an exclusion process, so there can be only 0 or 1 particle on each site. The TASEP does not have to be totally deterministic, the movements of the pedestrians can occur at some probability different from one, for instance. Still, in this paper, we shall only consider the case where all attempted jumps are performed if they are possible.
We decided to introduce a new type of update scheme, which seems to have interesting properties : the period of update for each pedestrian is exactly 1,it is possible to give a physical interpretation of the update – at least in the FF phase and conflicts are avoided.
It is called 'frozen shuffle' update, and it consists in updating the particles in a randomly chosen order, which does not change at each timestep but is fixed once for all.
This order is implemented by giving each particle a “phase” τ between 0 and 1. This avoids priority issues between pedestrians : the first pedestrian to move will simply jump on its target before the other one. This update also lowers the statistical fluctuations, since all the pedestrians have barely the same velocity when they are not constrained. The dynamics is then deterministic once the order has been chosen, since the particles jump with probability 1 when they can. Besides, in the free flow phase, there exists a direct mapping between the discrete TASEP with frozen shuffle update and a model of moving rods in continuous space and time. Indeed, a particle moving at timestep t with a phase τ can be seen as a particle moving at continuous time t+ τ . We investigate this model analytically, and by Monte-Carlo simulation, for different boundary conditions and geometries.
Fig. 1.
2 configurations for pairs of particles. Particles 1 and 2 can move together because their phases are well ordered whereas 3 and 4 will spontaneously be separated by a hole. [8]

First, we study the TASEP with frozen shuffle update on the one-dimensional ring
[8], the simplest non-trivial geometry. Our goal is to plot the fundamental diagram, i.e. the mean particle current j as a function of the density ρ of the system. The stochastic part of the model lies on the fact that one value of the density corresponds to several updating orders of the particles. As a preliminary, we see that a pair of two neighbour particles will move as a block if the first one has the smaller phase, whereas the two particles will need a hole between them to move with velocity 1 if they are ordered the other way, as seen in figure 1. A sequence of an arbitrary number of adjacent particles with well-ordered phases will then move as a whole, and will never be split anymore. These stable blocks of adjacent particles will be called platoons.
For the whole system, we see that there are two cases depending on the density :
 If ρ < ½, all the configurations will end up in a free flow state, a state in which all the particles can move at each timestep.
 If ρ > ½, there are configurations in which there is at most one hole between two consecutive platoons, these configurations will be called jammed.
Fig. 2.
Current as a function of the density in periodic boundary conditions for different system sizes. The black lines are the results of the theory and the circles, squares and diamonds are numerical data. The transition takes place for a density of 2/3 for an infinite system. The curve becomes smooth for a finite system. From [8].

We notice that there are more and more jammed configurations as ρ goes to 1. In the case of an infinite system, as the spatial average realizes effectively an average over the disorder, we find a phase transition separating the free flow phase and the jammed phase. A simple combinatorics argument shows that the average platoon length is 2, so that an intuitive value for the critical density would be 2/3. We also were able to analytically predict finite-size corrections near the transition point [8].
We then carried out simulations and plotted the fundamental diagram. The results are shown in figure 2 with good agreement.
Secondly, we use open boundary conditions [10] for the TASEP lane and we fix the entrance and exit probabilities, ρ and j being functions of these rates. Since we now have to create particles, we must now insert them into the already-existing particles update chain.
We use the aforementioned equivalence of the model with a continuous model of pedestrians evolving in a continuous space-time to prescribe how particles are injected in the system. In particular, it does not only determine the timestep at which a new particle enters the system, but also the phase this particle will be given.
The entrance algorithm satisfies two inportant properties. Firstly, the particles entering the system do not create a jam unless an exterior perturbation blocks them. In our case, they all move with velocity 1 if there is no jam at the exit. Secondly, the probability that a particle enters at a given time is constant, provided the entrance site is free. This allows us to define an entrance rate α, the probability a particle enters the system at a given timestep where the entrance site is empty. However, note that a particle may also be injected if the first site is not empty at the beginning of the timestep. The exit rate β is defined as usual : the probability that a particle occupying the exit site goes out of the system at the next timestep.
Again we expect to see a phase transition between a jammed state, in which an infinite queue of particles propagates upstream, and a free flow. The jamming transition occurs when the entering current becomes greater than the maximum current the exit can sustain. After some computation, one obtains the simple condition α = β for the transition line. We have considered only the deterministic version of the model for which pedestrians move with probability 1 when it is possible. The usual maximalcurrent phase is therefore reduced to a point. An unusual property of our update is that the current in the jammed phase depends not only on the exit rate, but also on the entrance rate as the latter has an effect on the average platoon length, therefore on the mean current.
Again, we were able to predict the finite-size rounding for ρ(α) near the critical point. We could also compute the density profile using a domain-wall approach [6,7], which gives us the correlation length and some scaling law near the transition.

The results of the simulations are shown in figure 3 and 4. The agreement is very good for ρ(α), but some yet unexplained weak discrepancies between simulations and theory remain when it comes to the density profile.
Fig. 3.
Density as a function of the entrance rate for open boundary conditions, for
β
=0.4. The solid lines represent the theory and the dots are the results of simulations. See [9].

Fig. 4.
Density profile in the lane across the transition for β=0.4. k denotes the boxes of the lane. The circles are simulation results and the dotted lines are obtained using a domain wall approach. From [9].
Finally, we consider a set of two open TASEP lanes sharing one site we shall refer to as the crossing [11]. We can then vary the entrance and exit probabilities for each lane, and this should give us four possible states for the system, either lane being free or jammed. We shall refer to these regimes with the obvious abbreviations FF,FJ,JF and JJ. The simplest case is when the system is symmetrical between the two lanes : each lane has simply an effective exit rate of β/2. But we also studied an asymmetric system with exit probabilities set to one (in order to have only the jams created by the crossing and not by exit conditions). Eventually we vary each parameter independently.

Fig. 5.
Pairing effect in the crossing. The particles are represented by circles and the borders between platoons are emphazised as thick red lines. Suppose that we have two platoons P and
P' waiting at the crossing (a), and that the phase of 1 is smaller than the phase of 1'. 1 will hop forward, so that all the platoon P will follow it (b). Since 1 moves before 1' and 2, the crossing will be free when either 1' or 2 tries to jump, here 1' is faster than 2 (c). This will happen until all the particles in one of the platoons go out (d). Now remember that there is necessarily a hole between the end of a platoon having jumped at the previous timestep (2') and the beginning of the next one (3'). This ensures that the platoon P will go through the crossing before the new one (e). Eventually, the system will start again the process with two new platoons (f). [10]
In this geometry, the system shows an interesting property we called the pairing mechanism, explained in figure 5. This mechanism ensures that two incoming platoons will combine as they step in the crossing and go out at the same time. In simpler words, we have found a simple picture of the stationary JJ state, so that we can easily compute interesting properties of the system, the mean crossing occupation for example. It also gives us the outgoing current. We could obtain the current in the FJ/JF phase by similar means, and the method for the FF phase is the same as the onedimensional case. As a summary, we computed the current (and the density) in all phases, so this gives us the analytical phase diagram as shown in figure 6, which agrees with numerics.

Fig. 6.
Phase diagram of the crossing for β = 0.6. The lines are the theoretical boundaries and the diamonds, triangles and circles are the numerics [10].
To conclude, we have introduced in this paper a new update scheme for the
TASEP, namely the frozen shuffle update, which should be appropriate, in particular, for the modeling of pedestrians. We were able to fully determine the fundamental diagram on a ring, and all the macroscopic properties of the systems with open boundary conditions. A mapping with a continuous model of hard rods exists [8], which is exact for free flow configurations, and may be useful for the interpretation of the results in terms of pedestrian motion.

In further work, we plan to take into account the width of the corridors through multilane models, and to enlarge the crossing to a bigger square. Our aim would be to understand how macroscopic structures can spontaneously emerge in such systems, and how they are modified for various modifications of the dynamical rules. Another question would be the behavior of the system with one more stochastic component, i.e. if we allowed the particles to jump with probability p < 1.
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